## Biography

**Runyao Duan** is currently a Professor and the Director of the Centre for Quantum Software and Information (QSI), Faculty of Engineering and Information Technology (FEIT), University of Technology Sydney (UTS). Previously he was the Director of Quantum Computation Laboratory of Centre for Quantum Computation and Intelligent Systems (QCIS), the forerunner of QSI. He received the BS and PhD from the Department of Computer Science and Technology, Tsinghua University, Beijing, China in the years of 2002 and 2006, respectively. On graduation he joined the same department as an Assistant Professor. From October 2007 to April 2008, he was a visiting Research Scientist in the University of Michigan. In December 2008, he moved to UTS as a Senior Lecturer (continuing position), and was promoted to Associate Professor in August 2010. Since July 2012, he has become an ARC (Australian Research Council) Future Fellow and Professor.

Prof Duan has been working in the fields of quantum information theory since 2002, and has made several fundamental and methodological contributions in the areas of quantum operation discrimination, quantum state discrimination, zero-error communication via noisy quantum channels, and quantum entanglement transformation. Up to now, he has published about 80 papers in top-tier international referred journals including *Physical Review Letters, **IEEE Transactions on Information Theory*, and *ACM Transactions on Programming Languages and Systems, etc*. His research works were presented at international competitive conferences including POPL, QIP, AQIS, and ISIT. He served and chaired the Steering Committee of QIP conferences (2013-2016), and was the head of QIP2015 Local Organizing Committee. He also served the Program Committee of AQIS2012, AQIS2014-6, and QIP2017.

#### Can supervise: YES

#### Research Interests

**1.** Quantum information theory

**2.** Quantum state/operation discrimination

**3.** Quantum zero-error information theory

**4.** Measurement-based quantum computation

#### Publications

Fang, K, Wang, X, Tomamichel, M & Duan, R 2019, 'Non-Asymptotic Entanglement Distillation', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 65, no. 10, pp. 6454-6465.View/Download from: Publisher's site

Liu, S, He, K & Duan, R 2019, 'Implementing termination analysis on quantum programming', *Science China Information Sciences*, vol. 62, no. 12.View/Download from: Publisher's site

#### View description

© 2019, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature. Termination analysis is an essential part in programming. Especially quantum programming concerning measurement, entanglement and even superposition are the foundations of bizarre behaviours in quantum programs. In this paper, we analyse and extend the theoretical theorems on termination analysis proposed by Ying et al. into computational theorems and algorithms. The new algorithm without the Jordan decomposition process has a significant acceleration with polynomial complexity both on terminating and almost-surely terminating programs. Moreover, the least upper bound of termination programs steps is studied and utilized to output the substituted matrix representation of quantum programs. We also implement four groups of experiments to illustrate the advantages of the new algorithm in case of processing a simplified quantum walk example comparing with the original counterpart.

Liu, S, Li, Y & Duan, R 2019, 'Distinguishing unitary gates on the IBM quantum processor', *Science China Information Sciences*, vol. 62, no. 7.View/Download from: Publisher's site

#### View description

© 2019, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature. An unknown unitary gate, which is secretly chosen from several known ones, can always be distinguished perfectly. In this paper, we implement such a task on IBM’s quantum processor. More precisely, we experimentally demonstrate the discrimination of two qubit unitary gates, the identity gate and the 23π-phase shift gate, using two discrimination schemes — the parallel scheme and the sequential scheme. We program these two schemes on the ibmqx4, a 5-qubit superconducting quantum processor via IBM cloud, with the help of the QSI modules. We report that both discrimination schemes achieve success probabilities at least 85%.

Wang, X, Fang, K & Duan, R 2019, 'Semidefinite Programming Converse Bounds for Quantum Communication', *IEEE Transactions on Information Theory*, vol. 65, no. 4, pp. 2583-2592.View/Download from: Publisher's site

#### View description

© 2018 IEEE. We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive one-shot semidefinite programming (SDP) converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes, Nat. Commun. 7, 2016]. As applications, we study quantum communication over depolarizing channels and amplitude damping channels with finite resources. Second, we find an SDP-strong converse bound for the quantum capacity of an arbitrary quantum channel, which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases. Furthermore, we prove that the SDP-strong converse bound improves the partial transposition bound introduced by Holevo and Werner. Third, we prove that this SDP strong converse bound is equal to the so-called max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP strong converse bound is weaker than the Rains information, but it is efficiently computable for general quantum channels.

Gour, G, Jennings, D, Buscemi, F, Duan, R & Marvian, I 2018, 'Quantum majorization and a complete set of entropic conditions for quantum thermodynamics.', *Nature communications*, vol. 9, no. 1.View/Download from: Publisher's site

#### View description

What does it mean for one quantum process to be more disordered than another? Interestingly, this apparently abstract question arises naturally in a wide range of areas such as information theory, thermodynamics, quantum reference frames, and the resource theory of asymmetry. Here we use a quantum-mechanical generalization of majorization to develop a framework for answering this question, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. We also investigate some of the applications of this framework, and remarkably find that, in the context of quantum thermodynamics it provides the first complete set of necessary and sufficient conditions for arbitrary quantum state transformations under thermodynamic processes, which rigorously accounts for quantum-mechanical properties, such as coherence. Our framework of generalized thermal processes extends thermal operations, and is based on natural physical principles, namely, energy conservation, the existence of equilibrium states, and the requirement that quantum coherence be accounted for thermodynamically.

Wang, X & Duan, R 2018, 'Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity', *IEEE Transactions on Information Theory*, vol. 64, no. 3, pp. 1454-1460.View/Download from: Publisher's site

#### View description

© 1963-2012 IEEE. Quantum Lovász number is a quantum generalization of the Lovász number in graph theory. It is the best known efficiently computable upper bound of the entanglement-assisted zero-error classical capacity of a quantum channel. However, it remains an intriguing open problem whether quantum entanglement can always enhance the zero-error capacity to achieve the quantum Lovász number. In this paper, by constructing a particular class of qutrit-to-qutrit channels, we show that there exists a strict gap between the entanglement-assisted zero-error capacity and the quantum Lovász number. Interestingly, for this class of quantum channels, the quantum generalization of fractional packing number is strictly larger than the zero-error capacity assisted with feedback or no-signaling correlations, which differs from the case of classical channels.

Wang, X, Xie, W & Duan, R 2018, 'Semidefinite programming strong converse bounds for classical capacity', *IEEE Transactions on Information Theory*, vol. 64, no. 1, pp. 640-653.View/Download from: Publisher's site

#### View description

IEEE We investigate the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the oneshot & #x03F5;-error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds of general quantum channels for entanglement-assisted codes and unassisted codes. Furthermore, we derive two SDP strong converse bounds for the classical capacity of general quantum channels: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive an improved upper bound on the classical capacity of the amplitude damping channel. We also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel

Li, Y, Qiao, Y, Wang, X & Duan, R 2018, 'Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces', *Communications in Mathematical Physics*, vol. 358, no. 2, pp. 791-814.View/Download from: Publisher's site

#### View description

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings, utilizing powerful results from the structure of matrix spaces. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation when multiple copies are provided. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate. Computing this rate turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipa...

Acín, A, Duan, R, Roberson, DE, Sainz, AB & Winter, A 2017, 'A new property of the Lovász number and duality relations between graph parameters', *Discrete Applied Mathematics*, vol. 216, no. 3, pp. 489-501.View/Download from: Publisher's site

#### View description

© 2016 Elsevier B.V.We show that for any graph G, by considering "activation" through the strong product with another graph H, the relation α(G)≤θ(symbol)(G) between the independence number and the Lovász number of G can be made arbitrarily tight: Precisely, the inequality α(G(squared times)H)≤θ(symbol)(G(squared times)H)=θ(symbol)(G)θ(symbol)(H) becomes asymptotically an equality for a suitable sequence of ancillary graphs H.This motivates us to look for other products of graph parameters of G and H on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that α(G(squared times)H)≤α*(G)α(H), with the fractional packing number α*(G), and for every G there exists H that makes the above an equality; conversely, for every graph H there is a G that attains equality.These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which α and α* are dual to each other, and the Lovász number θ(symbol) is self-dual. We also show duality of Schrijver's and Szegedy's variants θ(symbol)- and θ(symbol)+ of the Lovász number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.

Lai, CY & Duan, R 2017, 'On the one-shot zero-error classical capacity of classical-quantum channels assisted by quantum non-signalling correlations', *Quantum Information and Computation*, vol. 17, no. 5-6, pp. 380-398.View/Download from: Publisher's site

#### View description

© Rinton Press. Duan and Winter studied the one-shot zero-error classical capacity of a quantum channel assisted by quantum non-signalling correlations, and formulated this problem as a semidefinite program depending only on the Kraus operator space of the channel. For the class of classical-quantum channels, they showed that the asymptotic zero-error classical capacity assisted by quantum non-signalling correlations, minimized over all classicalquantum channels with a confusability graph G, is exactly log v(G), where v(G) is the celebrated Lovász theta function. In this paper, we show that the one-shot capacity for a classical-quantum channel, induced from a circulant graph G defined by equal-sized cyclotomic cosets, is log⌊v(G)⌋, which further implies that its asymptotic capacity is log v(G). This type of graphs include the cycle graphs of odd length, the Paley graphs of prime vertices, and the cubit residue graphs of prime vertices. Examples of other graphs are also discussed. This gives Lovász v function another operational meaning in zero-error classical-quantum communication.

Li, Y, Wang, X & Duan, R 2017, 'Indistinguishability of bipartite states by positive-partial-transpose operations in the many-copy scenario', *Physical Review A*, vol. 95, no. 5, pp. 052346-1-052346-6.View/Download from: Publisher's site

Wang, X & Duan, R 2017, 'Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose.', *Physical Review Letters*, vol. 119, no. 18, pp. 1-5.View/Download from: Publisher's site

#### View description

We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the 3⊗3 antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a by-product, we find that for this class of states, both the Rains's bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. So, in general, there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further show a computable sufficient condition for the irreversibility of entanglement distillation by LOCC (or PPT) operations.

Wang, X & Duan, R 2017, 'Nonadditivity of Rains' bound for distillable entanglement', *Physical Review A*, vol. 95, no. 6, pp. 062322-1-062322-5.View/Download from: Publisher's site

#### View description

Rains' bound is arguably the best known upper bound of the distillable entanglement by operations completely preserving positivity of partial transpose (PPT) and was conjectured to be additive and coincide with the asymptotic relative entropy of entanglement. We disprove both conjectures by explicitly constructing a special class of mixed two-qubit states. We then introduce an additive semidefinite programming lower bound (EM) for the asymptotic Rains' bound, and it immediately becomes a computable lower bound for entanglement cost of bipartite states. Furthermore, EM is also proved to be the best known upper bound of the PPT-assisted deterministic distillable entanglement and gives the asymptotic rates for all pure states and some class of genuinely mixed states.

Xie, W, Fang, K, Wang, X & Duan, R 2017, 'Approximate broadcasting of quantum correlations', *PHYSICAL REVIEW A*, vol. 96, no. 2.View/Download from: Publisher's site

Yu, N, Duan, R & Xu, Q 2017, 'Bounds on the Distance Between a Unital Quantum Channel and the Convex Hull of Unitary Channels', *IEEE Transactions on Information Theory*, vol. 63, no. 2, pp. 1299-1310.View/Download from: Publisher's site

#### View description

© 2016 IEEE. Motivated by the recent resolution of asymptotic quantum birkhoff conjecture (AQBC), we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels. We provide two lower bounds on this distance by employing techniques from quantum information and operator algebras, respectively. We then show how to apply these results to construct some explicit counterexamples to AQBC. We also point out an interesting connection between the Grothendieck's inequality and AQBC.

Duan, R & Winter, A 2016, 'No-Signalling Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovasz Number', *IEEE Transactions on Information Theory*, vol. 62, no. 2, pp. 891-914.View/Download from: Publisher's site

#### View description

We study the one-shot zero-error classical capacity of a quantum channel

assisted by quantum no-signalling correlations, and the reverse problem of

exact simulation of a prescribed channel by a noiseless classical one. Quantum

no-signalling correlations are viewed as two-input and two-output completely

positive and trace preserving maps with linear constraints enforcing that the

device cannot signal. Both problems lead to simple semidefinite programmes

(SDPs) that depend only on the Kraus operator space of the channel. In

particular, we show that the zero-error classical simulation cost is precisely

the conditional min-entropy of the Choi-Jamiolkowski matrix of the given

channel. The zero-error classical capacity is given by a similar-looking but

different SDP; the asymptotic zero-error classical capacity is the

regularization of this SDP, and in general we do not know of any simple form.

Interestingly however, for the class of classical-quantum channels, we show

that the asymptotic capacity is given by a much simpler SDP, which coincides

with a semidefinite generalization of the fractional packing number suggested

earlier by Aram Harrow. This finally results in an operational interpretation

of the celebrated Lovasz $\vartheta$ function of a graph as the zero-error

classical capacity of the graph assisted by quantum no-signalling correlations,

the first information theoretic interpretation of the Lovasz number.

Duan, R, Severini, S & Winter, A 2016, 'On zero-error communication via quantum channels in the presence of noiseless feedback', *IEEE Transactions on Information Theory*, vol. 62, no. 9, pp. 5260-5277.View/Download from: Publisher's site

#### View description

We initiate the study of zero-error communication via quantum channels when

the receiver and sender have at their disposal a noiseless feedback channel of

unlimited quantum capacity, generalizing Shannon's zero-error communication

theory with instantaneous feedback.

We first show that this capacity is a function only of the linear span of

Choi-Kraus operators of the channel, which generalizes the bipartite

equivocation graph of a classical channel, and which we dub "non-commutative

bipartite graph". Then we go on to show that the feedback-assisted capacity is

non-zero (with constant activating noiseless communication) if and only if the

non-commutative bipartite graph is non-trivial, and give a number of equivalent

characterizations. This result involves a far-reaching extension of the

"conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys.

8:475-478].

We then present an upper bound on the feedback-assisted zero-error capacity,

motivated by a conjecture originally made by Shannon and proved later by

Ahlswede. We demonstrate this bound to have many good properties, including

being additive and given by a minimax formula. We also prove that this quantity

is the entanglement-assisted capacity against an adversarially chosen channel

from the set of all channels with the same Choi-Kraus span, which can also be

interpreted as the feedback-assisted unambiguous capacity. The proof relies on

a generalization of the "Postselection Lemma" [Christandl/Koenig/Renner, PRL

102:020504] that allows to reflect additional constraints, and which we believe

to be of independent interest.

We illustrate our ideas with a number of examples, including

classical-quantum channels and Weyl diagonal channels, and close with an

extensive discussion of open questions.

Wang, X & Duan, R 2016, 'Improved semidefinite programming upper bound on distillable entanglement', *PHYSICAL REVIEW A*, vol. 94, no. 5.View/Download from: Publisher's site

Ban, Y & Chen, X 2015, 'Counter-diabatic driving for fast spin control in a two-electron double quantum dot', *Scientific Reports*, vol. 4, no. 1.View/Download from: Publisher's site

Chitambar, EA, Duan, R & Hsieh, M 2014, 'When Do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination?', *IEEE Transactions On Information Theory*, vol. 60, no. 3, pp. 1549-1561.View/Download from: Publisher's site

#### View description

In this paper, we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the perfect distinguishability problem of two-qubit ensembles-both for separable operations and LOCC-by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then, for the well-known task of minimum error discrimination, it is shown that almost all two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that any two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as non-locality without entanglement. We next consider an example of N parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant trine ensemble, and we establish a tight connection between the N-copy ensemble and Shor's lifted single-copy ensemble. For any finite N, we prove that optimal identification of the states cannot be achieved by LOCC; however, as N?8, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum error probability is considered.

Yu, N, Duan, R & Ying, M 2014, 'Distinguishability of Quantum States by Positive Operator-Valued Measures with Positive Partial Transpose', *IEEE Transactions on Information Theory*, vol. 60, no. 4, pp. 2069-2079.View/Download from: Publisher's site

#### View description

We study the distinguishability of bipartite quantum states by Positive

Operator-Valued Measures with positive partial transpose (PPT POVMs). The

contributions of this paper include: (1). We give a negative answer to an open

problem of [M. Horodecki $et. al$, Phys. Rev. Lett. 90, 047902(2003)] showing a

limitation of their method for detecting nondistinguishability. (2). We show

that a maximally entangled state and its orthogonal complement, no matter how

many copies are supplied, can not be distinguished by PPT POVMs, even

unambiguously. This result is much stronger than the previous known ones

\cite{DUAN06,BAN11}. (3). We study the entanglement cost of distinguishing

quantum states. It is proved that $\sqrt{2/3}\ket{00}+\sqrt{1/3}\ket{11}$ is

sufficient and necessary for distinguishing three Bell states by PPT POVMs. An

upper bound of entanglement cost of distinguishing a $d\otimes d$ pure state

and its orthogonal complement is obtained for separable operations. Based on

this bound, we are able to construct two orthogonal quantum states which cannot

be distinguished unambiguously by separable POVMs, but finite copies would make

them perfectly distinguishable by LOCC. We further observe that a two-qubit

maximally entangled state is always enough for distinguishing a $d\otimes d$

pure state and its orthogonal complement by PPT POVMs, no matter the value of

$d$. In sharp contrast, an entangled state with Schmidt number at least $d$ is

always needed for distinguishing such two states by separable POVMs. As an

application, we show that the entanglement cost of distinguishing a $d\otimes

d$ maximally entangled state and its orthogonal complement must be a maximally

entangled state for $d=2$,which implies that teleportation is optimal; and in

general, it could be chosen as $\mathcal{O}(\frac{\log d}{d})$.

Yu, N, Guo, C & Duan, R 2014, 'Obtaining a W state from a Greenberger-Horne-Zeilinger state via stochastic local operations and classical communication with a rate approaching unity', *Physical Review Letters*, vol. 112, no. 16.View/Download from: Publisher's site

#### View description

We introduce a notion of the entanglement transformation rate to characterize the asymptotic comparability of two multipartite pure entangled states under stochastic local operations and classical communication (SLOCC). For two well known SLOCC inequivalent three-qubit states |GHZ=(1/2)(|000+|111) and |W=(1/3)(|100+|010+|001), we show that the entanglement transformation rate from |GHZ to |W is exactly 1. That means that we can obtain one copy of the W state from one copy of the Greenberg-Horne-Zeilinger (GHZ) state by SLOCC, asymptotically. We then apply similar techniques to obtain a lower bound on the entanglement transformation rates from an N-partite GHZ state to a class of Dicke states, and prove the tightness of this bound for some special cases which naturally generalize the |W state. A new lower bound on the tensor rank of the matrix permanent is also obtained by evaluating the tensor rank of Dicke states. © 2014 American Physical Society.

Ying, M, Yu, N, Feng, Y & Duan, R 2013, 'Verification of quantum programs', *Science Of Computer Programming*, vol. 78, no. 9, pp. 1679-1700.View/Download from: Publisher's site

#### View description

This paper develops verification methodology for quantum programs, and the contribution of the paper is two-fold. Sharir, Pnueli and Hart [M. Sharir, A. Pnueli, S. Hart, Verification of probabilistic programs, SIAM Journal of Computing 13 (1984) 292314] presented a general method for proving properties of probabilistic programs, in which a probabilistic program is modeled by a Markov chain and an assertion on the output distribution is extended to an invariant assertion on all intermediate distributions. Their method is essentially a probabilistic generalization of the classical Floyd inductive assertion method. In this paper, we consider quantum programs modeled by quantum Markov chains which are defined by super-operators. It is shown that the SharirPnueliHart method can be elegantly generalized to quantum programs by exploiting the SchrödingerHeisenberg duality between quantum states and observables. In particular, a completeness theorem for the SharirPnueliHart verification method of quantum programs is established.

Yu, N, Duan, R & Ying, M 2013, 'Five two-qubit gates are necessary for implementing the Toffoli gate', *PHYSICAL REVIEW A*, vol. 88, no. 1.View/Download from: Publisher's site

Duan, R, Severini, S & Winter, A 2013, 'Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number', *IEEE Transactions On Information Theory*, vol. 59, no. 2, pp. 1164-1174.View/Download from: Publisher's site

#### View description

We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lova´sz' famous ? function on general operator systems, as the norm-completion (or stabilization) of a naive generalization of ?. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lova´sz' original ? in the classical case, give several applications, and propose to study the operator systems associated with channels as noncommutative graphs, using the language of Hilbert modules.

Feng, Y, Duan, R & Ying, M 2012, 'Bisimulation For Quantum Processes', *ACM Transactions pn Programming Language and Systems (TOPLAS)*, vol. 34, no. 4, pp. 1-43.View/Download from: Publisher's site

#### View description

Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the oth

Ying, M, Feng, Y, Duan, R, Li, Y & Yu, N 2012, 'Quantum Programming: From Theories To Implementations', *Chinese Science Bulletin*, vol. 57, no. 16, pp. 1903-1909.View/Download from: Publisher's site

#### View description

This paper surveys the new field of programming methodology and techniques for future quantum computers, including design of sequential and concurrent quantum programming languages, their semantics and implementations. Several verification methods for qu

Yu, N, Duan, R & Ying, M 2012, 'Four locally indistinguishable ququad-ququad orthogonal maximally entangled states.', *Physical Review Letters*, vol. 109, no. 2, pp. 020506-020506.View/Download from: Publisher's site

#### View description

We explicitly exhibit a set of four ququad-ququad orthogonal maximally entangled states that cannot be perfectly distinguished by means of local operations and classical communication. Before our work, it was unknown whether there is a set of d locally indistinguishable d⊗d orthogonal maximally entangled states for some positive integer d. We further show that a 2⊗2 maximally entangled state can be used to locally distinguish this set of states without being consumed, thus demonstrate a novel phenomenon of entanglement discrimination catalysis. Based on this set of states, we construct a new set K consisting of four locally indistinguishable states such that K(⊗m) (with 4(m) members) is locally distinguishable for some m greater than one. As an immediate application, we construct a noisy quantum channel with one sender and two receivers whose local zero-error classical capacity can achieve the full dimension of the input space but only with a multi-shot protocol.

Lu, C, Chen, JF & Duan, R 2012, 'Some Bounds On The Minimum Number Of Queries Required For Quantum Channel Perfect Discrimination', *Quantum Information and Computation*, vol. 12, no. 1-2, pp. 138-148.

#### View description

We prove a lower bound on the q-maximal fidelities between two quantum channels epsilon(0) and epsilon(1) and an upper bound on the q-maximal fidelities between a quantum channel epsilon and an identity tau. Then we apply these two bounds to provide a si

Chen, J, Chen, X, Duan, R, Ji, Z & Zeng, B 2011, 'No-go Theorem For One-way Quantum Computing On Naturally Occurring Two-level Systems', *Physical Review A*, vol. 83, no. 5, pp. 0-0.View/Download from: Publisher's site

#### View description

The ground states of some many-body quantum systems can serve as resource states for the one-way quantum computing model, achieving the full power of quantum computation. Such resource states are found, for example, in spin- 5 2 and spin- 3 2 systems. It is, of course, desirable to have a natural resource state in a spin- 1 2 , that is, qubit system. Here, we give a negative answer to this question for frustration-free systems with two-body interactions. In fact, it is shown to be impossible for any genuinely entangled qubit state to be a nondegenerate ground state of any two-body frustration-free Hamiltonian. What is more, we also prove that every spin- 1 2 frustration-free Hamiltonian with two-body interaction always has a ground state that is a product of single- or two-qubit states. In other words, there cannot be any interesting entanglement features in the ground state of such a qubit Hamiltonian.

Yu, N, Duan, R & Ying, M 2011, 'Any 2 Circle Times N Subspace Is Locally Distinguishable', *Physical Review A*, vol. 84, no. 1, pp. 1-3.View/Download from: Publisher's site

#### View description

A subspace of a multipartite Hilbert space is said to be locally indistinguishable if any orthonormal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any m . n bipartite system with m > 2 and n > 2 has a locally indistinguishable subspace. However, it has been an open problem since 2005 whether there is a locally indistinguishable bipartite subspace with a qubit subsystem.We settle this problem in negative by showing that any 2 . n bipartite subspace contains a basis that is locally distinguishable. As an interesting application, we show that any quantum channel with two Kraus operators has optimal environment-assisted classical capacity.

Chen, L, Chitambar, E, Duan, R, Ji, Z & Winter, A 2010, 'Tensor rank and stochastic entanglement catalysis for multipartite pure states.', *Physical review letters*, vol. 105, no. 20, p. 200501.View/Download from: Publisher's site

#### View description

The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state |W3>=1/√3(|100> + |010> + |001>) and its N-partite generalization |W(N)>. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of |W3> have a rank of either 15 or 16, (ii) two copies of |W(N)> have a rank of 3N - 2, and (iii) n copies of |W(N)> have a rank of O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple-copy bunches or when assisted by some catalyzing state. This effect is impossible for bipartite pure states.

Chen, L, Chitambar, EA, Duan, R, Ji, Z & Winter, A 2010, 'Tensor Rank And Stochastic Entanglement Catalysis For Multipartite Pure States', *Physical Review Letters*, vol. 105, no. 20, pp. 1-4.View/Download from: Publisher's site

#### View description

The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investig

Chen, X, Duan, R, Ji, Z & Zeng, B 2010, 'Quantum State Reduction For Universal Measurement Based Computation', *Physical Review Letters*, vol. 105, no. 2, pp. 1-4.View/Download from: Publisher's site

#### View description

Measurement based quantum computation, which requires only single particle measurements on a universal resource state to achieve the full power of quantum computing, has been recognized as one of the most promising models for the physical realization of

Duan, R, Xin, Y & Ying, M 2010, 'Locally Indistinguishable Subspaces Spanned By Three-Qubit Unextendible Product Bases', *Physical Review A*, vol. 81, no. 3, pp. 1-10.View/Download from: Publisher's site

#### View description

We study the local distinguishability of general multiqubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that

Li, Y, Duan, R & Ying, M 2010, 'Local Unambiguous Discrimination With Remaining Entanglement', *Physical Review A*, vol. 82, no. 3, pp. 1-6.View/Download from: Publisher's site

#### View description

A bipartite state, which is secretly chosen from a finite set of known entangled pure states, cannot immediately be useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we i

Yu, N, Duan, R & Ying, M 2010, 'Optimal Simulation Of A Perfect Entangler', *Physical Review A*, vol. 81, no. 3, pp. 1-4.View/Download from: Publisher's site

#### View description

A2 circle times 2 unitary operation is called a perfect entangler if it can generate a maximally entangled state from some unentangled input. We study the following question: How many runs of a given two-qubit entangling unitary operation are required to

Chitambar, EA, Duan, R & Shi, Y 2010, 'Multipartite-To-Bipartite Entanglement Transformations And Polynomial Identity Testing', *Physical Review A*, vol. 81, no. 5, pp. 1-4.View/Download from: Publisher's site

#### View description

We consider the problem of deciding if some multiparty entangled pure state can be converted, with a nonzero success probability, into a given bipartite pure state shared between two specified parties through local quantum operations and classical commun

Duan, R & Shi, Y 2010, 'When Is There A Multipartite Maximum Entangled State?', *Quantum Information and Computation*, vol. 10, no. 11-12, pp. 925-935.

#### View description

For a multipartite quantum system of the dimension d(1) circle times d(2) circle times ... circle times d(n), where d(1) >= d(2) >= ... >= d(n) >= 2, is there an entangled state maximum in the sense that all other states in the system can be obtained from the state through local quantum operations and classical communications (LOCC)? When d(1) >= n i=2d(i), such state exists. We show that this condition is also necessary. Our proof, somewhat surprisingly, uses results from algebraic complexity theory.

Yu, N, Chitambar, EA, Guo, C & Duan, R 2010, 'Tensor Rank Of The Tripartite State Vertical Bar W >(Circle Times N)', *Physical Review A*, vol. 81, no. 1, pp. 1-3.

#### View description

Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its nonadditivity as an entanglement measure has recently been observed. In this Brief Report, we estimate the tensor rank of multiple copies of the

Ying, M, Feng, Y, Duan, R & Ji, Z 2009, 'An Algebra Of Quantum Processes', *Acm Transactions On Computational Logic*, vol. 10, no. 3, pp. 1-36.View/Download from: Publisher's site

#### View description

We introduce an algebra qCCS of pure quantum processes in which communications by moving quantum states physically are allowed and computations are modeled by super-operators, but no classical data is explicitly involved. An operational semantics of qCCS is presented in terms of (nonprobabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and approximate reduction-bisimilarity are preserved by various constructors of quantum processes. This provides us with a formal tool for observing robustness of quantum processes against inaccuracy in the implementation of its elementary gates.

Duan, R, Feng, Y & Ying, M 2009, 'Perfect Distinguishability of Quantum Operations', *PHYSICAL REVIEW LETTERS*, vol. 103, no. 21.View/Download from: Publisher's site

Duan, R, Feng, Y, Xin, Y & Ying, M 2009, 'Distinguishability of Quantum States by Separable Operations', *IEEE Transactions On Information Theory*, vol. 55, no. 3, pp. 1320-1330.View/Download from: Publisher's site

#### View description

In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of (D - 1) pure states, where D is the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 circle times 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 circle times 3 nonlocal separable operations was known [Bennett et al, Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.

Feng, Y, Duan, R & Ying, M 2009, 'Locally undetermined states, generalized Schmidt decomposition, and application in distributed computing', *Quantum Information and Computation*, vol. 9, no. 11, pp. 997-1012.

#### View description

Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, local determinability of pure states is closely related to a generalized notion of Schmidt decomposition. Furthermore, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computation. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable and they are not allowed to use local ancillary quantum systems, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state

Chitambar, E & Duan, R 2009, 'Nonlocal entanglement transformations achievable by separable operations.', *Physical review letters*, vol. 103, no. 11, p. 110502.View/Download from: Publisher's site

#### View description

The weird phenomenon of "quantum nonlocality without entanglement" means that local quantum operations assisted by classical communication constitute a proper subset of the class of separable quantum operations. Despite considerable recent advances, little is known to what extent the class of separable operations differs from local quantum operations and classical communication. In this Letter we show that separable operations are generally stronger than local quantum operations and classical communication when distilling a mixed state into a pure entangled state and thus confirm the existence of entanglement monotones that can increase under separable operations. Our finding can also be interpreted as confirming the ability of separable operations to enhance the entanglement of mixed states relative to certain measures, a sensible but important fact that has never been rigorously proven before.

Chitambar, EA & Duan, R 2009, 'Nonlocal Entanglement Transformations Achievable by Separable Operations', *Physical Review Letters*, vol. 103, no. 11, pp. 1-4.View/Download from: Publisher's site

#### View description

The weird phenomenon of "quantum nonlocality without entanglement" means that local quantum operations assisted by classical communication constitute a proper subset of the class of separable quantum operations. Despite considerable recent advances, little is known to what extent the class of separable operations differs from local quantum operations and classical communication. In this Letter we show that separable operations are generally stronger than local quantum operations and classical communication when distilling a mixed state into a pure entangled state and thus confirm the existence of entanglement monotones that can increase under separable operations. Our finding can also be interpreted as confirming the ability of separable operations to enhance the entanglement of mixed states relative to certain measures, a sensible but important fact that has never been rigorously proven before.

Chen, JF, Duan, R, Ji, Z, Ying, M & Yu, JX 2008, 'Existence Of Universal Entangler', *Journal of Mathematical Physics*, vol. 49, no. 1, pp. 1-7.View/Download from: Publisher's site

#### View description

A gate is called an entangler if it transforms some (pure) product states to entangled states. A universal entangler is a gate which transforms all product states to entangled states. In practice, a universal entangler is a very powerful device for gener

Chitambar, EA, Duan, R & Shi, Y 2008, 'Tripartite Entanglement Transformations And Tensor Rank', *Physical Review Letters*, vol. 101, no. 14, pp. 1-4.View/Download from: Publisher's site

#### View description

A basic question regarding quantum entangled states is whether one can be probabilistically converted to another through local operations and classical communication exclusively. While the answer for bipartite systems is known, we show that for tripartit

Ji, Z, Wang, G, Duan, R, Feng, Y & Ying, M 2008, 'Parameter Estimation of Quantum Channels', *IEEE Transactions On Information Theory*, vol. 54, no. 11, pp. 5172-5185.View/Download from: Publisher's site

#### View description

The efficiency of parameter estimation of quantum channels is studied in this paper. We introduce the concept of programmable parameters to the theory of estimation. It is found that programmable parameters obey the standard quantum limit strictly; hence, no speedup is possible in its estimation. We also construct a class of nonunitary quantum channels whose parameter can be estimated in a way that the standard quantum limit is broken. The study of estimation of general quantum channels also enables an investigation of the effect of noises on quantum estimation.

Duan, R, Feng, Y & Ying, M 2008, 'Local Distinguishability Of Multipartite Unitary Operations', *Physical Review Letters*, vol. 100, no. 2, pp. 1-4.View/Download from: Publisher's site

#### View description

We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguished by local operations and classical communication when a finite number of runs is allowed. Intuitively, this result indicate

Chitambar, E, Duan, R & Shi, Y 2008, 'Tripartite entanglement transformations and tensor rank', *Physical Review Letters*, vol. 101, no. 14.View/Download from: Publisher's site

#### View description

A basic question regarding quantum entangled states is whether one can be probabilistically converted to another through local operations and classical communication exclusively. While the answer for bipartite systems is known, we show that for tripartite systems, this question encodes some of the most challenging open problems in mathematics and computer science. In particular, we show that there is no easy general criterion to determine the feasibility, and in fact, the problem is NP hard. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the maximum rate to convert the Greenberger-Horne-Zeilinger state to a triangular distribution of three EPR states. Our results are based on connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory. © 2008 The American Physical Society.

Duan, R & Shi, Y 2008, 'Entanglement Between Two Uses Of A Noisy Multipartite Quantum Channel Enables Perfect Transmission Of Classical Information', *Physical Review Letters*, vol. 101, no. 2, pp. 1-4.View/Download from: Publisher's site

#### View description

Suppose that m senders want to transmit classical information to n receivers with zero probability of error using a noisy multipartite communication channel. The senders are allowed to exchange classical, but not quantum, messages among themselves, and t

Duan, R, Feng, Y & Ying, M 2008, 'Local distinguishability of multipartite unitary operations.', *Physical review letters*, vol. 100, no. 2, p. 020503.View/Download from: Publisher's site

#### View description

We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguished by local operations and classical communication when a finite number of runs is allowed. Intuitively, this result indicates that the lost identity of a nonlocal unitary operation can be recovered locally. No entanglement between distant parties is required.

Wu, X & Duan, R 2008, 'Exact Quantum Search By Parallel Unitary Discrimination Schemes', *Physical Review A*, vol. 78, no. 1, pp. 1-8.View/Download from: Publisher's site

#### View description

We study the unsorted database search problem with items N from the viewpoint of unitary discrimination. Instead of considering the famous O(root N) Grover bounded-error algorithm for the original problem, we seek the results for the exact algorithms, i.

Xin, Y & Duan, R 2008, 'Local Distinguishability Of Orthogonal 2 Circle Times 3 Pure States', *Physical Review A*, vol. 77, no. 1, pp. 1-10.

#### View description

We present a complete characterization for the local distinguishability of orthogonal 2 circle times 3 pure states except for some special cases of three states. Interestingly, we find there is a large class of four or three states that are indistinguish

Duan, R, Feng, Y, Ji, Z & Ying, M 2007, 'Distinguishing Arbitrary Multipartite Basis Unambiguously Using Local Operations And Classical Communication', *Physical Review Letters*, vol. 98, no. 23, pp. 1-4.

#### View description

We show that an arbitrary basis of a multipartite quantum state space consisting of K distant parties such that the kth party has local dimension d(k) always contains at least N=Sigma(K)(k=1)(d(k)-1)+1 members that are unambiguously distinguishable using

Duan, R, Feng, Y, Ji, Z & Ying, M 2007, 'Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication (vol 98, art no 230602, 2007)', *PHYSICAL REVIEW LETTERS*, vol. 99, no. 1.View/Download from: Publisher's site

Feng, Y, Duan, R, Ji, Z & Ying, M 2007, 'Probabilistic Bisimulations For Quantum Processes', *Information And Computation*, vol. 205, no. 11, pp. 1608-1639.View/Download from: Publisher's site

#### View description

Modeling and reasoning about concurrent quantum systems is very important for both distributed quantum computing and quantum protocol verification. As a consequence, a general framework formally describing communication and concurrency in complex quantum

Feng, Y, Duan, R, Ji, Z & Ying, M 2007, 'Proof Rules For The Correctness Of Quantum Programs', *Theoretical Computer Science*, vol. 386, no. 1-2, pp. 151-166.View/Download from: Publisher's site

#### View description

We apply the notion of quantum predicate proposed by D'Hondt and Panangaden to analyze a simple language fragment which may describe the quantum part of a future quantum computer in Knill's architecture. The notion of weakest liberal precondition semanti

Duan, R, Feng, Y & Ying, M 2007, 'Entanglement Is Not Necessary For Perfect Discrimination Between Unitary Operations', *Physical Review Letters*, vol. 98, no. 10, pp. 1-4.

#### View description

We show that a unitary operation (quantum circuit) secretly chosen from a finite set of unitary operations can be determined with certainty by sequentially applying only a finite amount of runs of the unknown circuit. No entanglement or joint quantum ope

Duan, R, Feng, Y & Ying, M 2007, 'Entanglement is not necessary for perfect discrimination between unitary operations (vol 98, pg 100503, 2007)', *PHYSICAL REVIEW LETTERS*, vol. 98, no. 12.View/Download from: Publisher's site

Duan, R, Feng, Y & Ying, M 2007, 'Entanglement is not necessary for perfect discrimination between unitary operations.', *Physical review letters*, vol. 98, no. 10, p. 100503.View/Download from: Publisher's site

#### View description

We show that a unitary operation (quantum circuit) secretly chosen from a finite set of unitary operations can be determined with certainty by sequentially applying only a finite amount of runs of the unknown circuit. No entanglement or joint quantum operations are required in our scheme. We further show that our scheme is optimal in the sense that the number of the runs is minimal when discriminating only two unitary operations.

Ying, M, Chen, JF, Feng, Y & Duan, R 2007, 'Commutativity Of Quantum Weakest Preconditions', *Information Processing Letters*, vol. 104, no. 4, pp. 152-158.View/Download from: Publisher's site

#### View description

The notion of quantum weakest precondition was introduced by D'Hondt and P. Panangaden [E. D'Hondt, P. Panangaden, Quantum weakest preconditions, Mathematical Structures in Computer Science 16 (2006) 429-451], and they presented a representation of weake

Xin, Y & Duan, R 2007, 'Conditions For Entanglement Transformation Between A Class Of Multipartite Pure States With Generalized Schmidt Decompositions', *Physical Review A*, vol. 76, no. 4, pp. 1-3.

#### View description

We generalize Nielsen's majorization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett. 83, 436 (1999)] to a special class of multipartite pure states with generalized Schmidt decompositions.

Duan, R, Ji, Z, Feng, Y & Ying, M 2006, 'Some Issues In Quantum Information Theory', *Journal Of Computer Science And Technology*, vol. 21, no. 5, pp. 776-789.View/Download from: Publisher's site

#### View description

Quantum information theory is a new interdisciplinary research field related to quantum mechanics, computer science, information theory, and applied mathematics. It provides completely new paradigms to do information processing tasks by employing the pri

Feng, Y, Duan, R & Ji, Z 2006, 'Optimal Dense Coding With Arbitrary Pure Entangled States', *Physical Review A*, vol. 74, no. 1, pp. 1-5.

#### View description

We examine dense coding with an arbitrary pure entangled state sharing between the sender and the receiver. Upper bounds on the average success probability in approximate dense coding and on the probability of conclusive results in unambiguous dense codi

Ji, Z, Feng, Y, Duan, R & Ying, M 2006, 'Boundary Effect Of Deterministic Dense Coding', *Physical Review A*, vol. 73, no. 3, pp. 1-3.

#### View description

We present a rigorous proof of an interesting boundary effect of deterministic dense coding first observed by S. Mozes, J. Oppenheim, and B. Reznik [Phys. Rev. A 71, 012311 (2005)]. Namely, it is shown that d(2)-1 cannot be the maximal alphabet size of a

Ji, Z, Feng, Y, Duan, R & Ying, M 2006, 'Identification And Distance Measures Of Measurement Apparatus', *Physical Review Letters*, vol. 96, no. 20, pp. 1-4.

#### View description

We propose simple schemes that can perfectly identify projective measurement apparatuses secretly chosen from a finite set. Entanglement is used in these schemes both to make possible the perfect identification and to improve the efficiency significantly

Duan, R, Feng, Y & Ying, M 2006, 'Partial Recovery Of Quantum Entanglement', *IEEE Transactions On Information Theory*, vol. 52, no. 7, pp. 3080-3104.View/Download from: Publisher's site

#### View description

Suppose Alice and Bob try to transform an entangled state shared between them into another one by local operations and classical communications. Then in general a certain amount of entanglement contained in the initial state will decrease in the process

Feng, Y, Duan, R & Ying, M 2006, 'Relation Between Catalyst-Assisted Transformation And Multiple-Copy Transformation For Bipartite Pure States', *Physical Review A*, vol. 74, no. 4, pp. 1-7.

#### View description

We show that in some cases, catalyst-assisted entanglement transformation cannot be implemented by multiple-copy transformation for pure states. This fact, together with the result we obtained in R. Y. Duan, Y. Feng, X. Li, and M. S. Ying, Phys. Rev. A 7

Duan, R, Feng, Y, Ji, Z & Ying, M 2005, 'Efficiency Of Deterministic Entanglement Transformation', *Physical Review A*, vol. 71, no. 2, pp. 1-7.

#### View description

We prove that sufficiently many copies of a bipartite entangled pure state can always be transformed into some copies of another one with certainty by local quantum operations and classical communication. The efficiency of such a transformation is charac

Feng, Y, Duan, R & Ji, Z 2005, 'Condition And Capability Of Quantum State Separation', *Physical Review A*, vol. 72, no. 1, pp. 1-6.

#### View description

The linearity of quantum operations puts many fundamental constraints on the information processing tasks we can achieve on a quantum system whose state is not exactly known, just as we observe in quantum cloning and quantum discrimination. In this paper

Duan, R, Feng, Y & Ying, M 2005, 'Entanglement-Assisted Transformation Is Asymptotically Equivalent To Multiple-Copy Transformation', *Physical Review A*, vol. 72, no. 2, pp. 1-5.

#### View description

We show that two ways of manipulating quantum entanglement-namely, entanglement-assisted local transformation [D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566 (1999)] and multiple-copy transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen

Duan, R, Feng, Y, Li, X & Ying, M 2005, 'Multiple-Copy Entanglement Transformation And Entanglement Catalysis', *Physical Review A*, vol. 71, no. 4, pp. 1-11.

#### View description

We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys. Rev. A 65, 052315 (2002)] can be implemented by a suitable entanglement-assisted local transformation [D. Jonathan and M. B. Plenio, Phys. R

Duan, R, Feng, Y, Li, X & Ying, M 2005, 'Trade-Off Between Multiple-Copy Transformation And Entanglement Catalysis', *Physical Review A*, vol. 71, no. 6, pp. 1-7.

#### View description

We demonstrate that multiple copies of a bipartite entangled pure state may serve as a catalyst for certain entanglement transformations while a single copy cannot. Such a state is termed a

Feng, Y, Duan, R & Ying, M 2005, 'Catalyst-Assisted Probabilistic Entanglement Transformation', *IEEE Transactions On Information Theory*, vol. 51, no. 3, pp. 1090-1101.View/Download from: Publisher's site

#### View description

We are concerned with catalyst-assisted probabilistic entanglement transformations. A necessary and sufficient condition is presented under which there exist partial catalysts that can increase the maximal transforming probability of a given entanglement

Feng, Y, Duan, R, Ji, Z-F & Ying, M 2005, 'Proof rules for purely quantum programs', *CoRR*, vol. abs/cs/0507043.

Sun, X, Duan, R & Ying, M 2005, 'The Existence Of Quantum Entanglement Catalysts', *IEEE Transactions On Information Theory*, vol. 51, no. 1, pp. 75-80.View/Download from: Publisher's site

#### View description

Without additional resources, it is often impossible to transform one entangled quantum state into another with local quantum operations and classical communication. Jonathan and Plenio (Phys. Rev. Lett., vol. 83, p. 3566, 1999) presented an interesting

Duan, R, Ji, Z, Feng, Y & Ying, M 2004, 'Quantum Operation Quantum Fourier Transform And Semi-Definite Programming', *Physics Letters A*, vol. 323, no. 1-2, pp. 48-56.View/Download from: Publisher's site

#### View description

We analyze a class of quantum operations based on a geometrical representation of d-level quantum system (or qudit for short). A sufficient and necessary condition of complete positivity, expressed in terms of the quantum Fourier transform, is found for

Ji, Z, Duan, R & Ying, M 2004, 'Comparability Of Multipartite Entanglement', *Physics Letters A*, vol. 330, no. 6, pp. 418-423.View/Download from: Publisher's site

#### View description

We prove, in a multipartite setting, that it is always feasible to exactly transform a genuinely m-partite entangled pure state with sufficient many copies to any other m-partite state via local quantum operation and classical communication. This result

Feng, Y, Duan, R & Ying, M 2004, 'When Catalysis Is Useful For Probabilistic Entanglement Transformation', *Physical Review A*, vol. 69, no. 6, pp. 1-5.

#### View description

We determine all 2x2 quantum states that can serve as useful catalysts for a given probabilistic entanglement transformation, in the sense that they can increase the maximal transformation probability. When higher-dimensional catalysts are considered, a

Feng, YA, Duan, RY & Ying, MS 2004, 'Unambiguous discrimination between mixed quantum states', *PHYSICAL REVIEW A*, vol. 70, no. 1.View/Download from: Publisher's site

Feng, Y, Zhang, S, Duan, R & Ying, M 2002, 'Lower Bound On Inconclusive Probability Of Unambiguous Discrimination', *Physical Review A*, vol. 66, no. 6, pp. 1-4.

#### View description

We derive a lower bound on the inconclusive probability of unambiguous discrimination among n linearly independent quantum states by using the constraint of no signaling. It improves the bound presented in the. paper of Zhang, Feng, Sun, and Ying [Phys.

Feng, Y, Duan, R & Ji, Z, 'Condition and capability of quantum state separation', *Physical Review A*, vol. 72, no. 1.View/Download from: Publisher's site

Feng, Y, Duan, R & Ji, Z, 'Optimal dense coding with arbitrary pure entangled states', *Physical Review A*, vol. 74, no. 1.View/Download from: Publisher's site

Duan, R, Feng, Y & Ying, M, 'An Equivalence of Entanglement-Assisted Transformation and Multiple-Copy Entanglement Transformation'.

#### View description

We examine the powers of entanglement-assisted transformation and

multiple-copy entanglement transformation. First, we find a sufficient

condition of when a given catalyst is useful in producing another specific

target state. As an application of this condition, for any non-maximally

entangled bipartite pure state and any integer $n$ not less than 4, we are able

to explicitly construct a set of $n\times n$ quantum states which can be

produced by using the given state as a catalyst. Second, we prove that for any

positive integer $k$, entanglement-assisted transformation with $k\times

k$-dimensional catalysts is useful in producing a target state if and only if

multiple-copy entanglement transformation with $k$ copies of state is useful in

producing the same target. Moreover, a necessary and sufficient condition for

both of them is obtained in terms of the Schmidt coefficients of the target.

This equivalence of entanglement-assisted transformation and multiple-copy

entanglement transformation implies many interesting properties of entanglement

transformation. Furthermore, these results are generalized to the case of

probabilistic entanglement transformations.

Chen, J, Chen, X, Duan, R, Ji, Z & Zeng, B, 'No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems', *Phys. Rev. A*, vol. 83, p. 050301.View/Download from: Publisher's site

#### View description

One-way quantum computing achieves the full power of quantum computation by

performing single particle measurements on some many-body entangled state,

known as the resource state. As single particle measurements are relatively

easy to implement, the preparation of the resource state becomes a crucial

task. An appealing approach is simply to cool a strongly correlated quantum

many-body system to its ground state. In addition to requiring the ground state

of the system to be universal for one-way quantum computing, we also want the

Hamiltonian to have non-degenerate ground state protected by a fixed energy

gap, to involve only two-body interactions, and to be frustration-free so that

measurements in the course of the computation leave the remaining particles in

the ground space. Recently, significant efforts have been made to the search of

resource states that appear naturally as ground states in spin lattice systems.

The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet,

it remains an open question whether there could be such a natural resource

state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to

this question by proving that it is impossible for a genuinely entangled qubit

states to be a non-degenerate ground state of any two-body frustration-free

Hamiltonian. What is more, we prove that every spin-1/2 frustration-free

Hamiltonian with two-body interaction always has a ground state that is a

product of single- or two-qubit states, a stronger result that is interesting

independent of the context of one-way quantum computing.

Chitambar, E, Duan, R & Shi, Y, 'Tripartite to Bipartite Entanglement Transformations and Polynomial Identity Testing', *Phys. Rev. A*, vol. 81, p. 052310.View/Download from: Publisher's site

#### View description

We consider the problem of deciding if a given three-party entangled pure

state can be converted, with a non-zero success probability, into a given

two-party pure state through local quantum operations and classical

communication. We show that this question is equivalent to the well-known

computational problem of deciding if a multivariate polynomial is identically

zero. Efficient randomized algorithms developed to study the latter can thus be

applied to the question of tripartite to bipartite entanglement

transformations.

Duan, R, 'Super-Activation of Zero-Error Capacity of Noisy Quantum Channels'.

#### View description

We study various super-activation effects in the following zero-error

communication scenario: One sender wants to send classical or quantum

information through a noisy quantum channel to one receiver with zero

probability of error. First we show that there are quantum channels of which a

single use is not able to transmit classical information perfectly yet two uses

can. This is achieved by employing entangled input states between different

uses of the given channel and thus cannot happen for classical channels. Second

we exhibit a class of quantum channel with vanishing zero-error classical

capacity such that when a noiseless qubit channel or one ebit shared

entanglement are available, it can be used to transmit $\log_2 d$ noiseless

qubits, where 2d is the dimension of input state space. Third we further

construct quantum channels with vanishing zero-error classical capacity when

assisted with classical feedback can be used to transmit both classical and

quantum information perfectly. These striking findings not only indicate both

the zero-error quantum and classical capacities of quantum channels satisfy a

strong super-additivity beyond any classical channels, but also highlight the

activation power of auxiliary physical resources in zero-error communication.

Duan, R & Shi, Y, 'Entanglement between Two Uses of a Noisy Multipartite Quantum Channel Enables Perfect Transmission of Classical Information', *Phys. Rev. Lett.*, vol. 101, p. 020501.View/Download from: Publisher's site

#### View description

Suppose that $m$ senders want to transmit classical information to $n$

receivers with zero probability of error using a noisy multipartite

communication channel. The senders are allowed to exchange classical, but not

quantum, messages among themselves, and the same holds for the receivers. If

the channel is classical, a single use can transmit information if and only if

multiple uses can. In sharp contrast, we exhibit, for each $m$ and $n$ with

$m\ge 2$ or $n\ge 2$, a quantum channel of which a single use is not able to

transmit information yet two uses can. This latter property requires and is

enabled by quantum entanglement.

Lu, C, Chen, J & Duan, R, 'Optimal Perfect Distinguishability between Unitaries and Quantum Operations'.

#### View description

We study optimal perfect distinguishability between a unitary and a general

quantum operation. In 2-dimensional case we provide a simple sufficient and

necessary condition for sequential perfect distinguishability and an analytical

formula of optimal query time. We extend the sequential condition to general

d-dimensional case. Meanwhile, we provide an upper bound and a lower bound for

optimal sequential query time. In the process a new iterative method is given,

the most notable innovation of which is its independence to auxiliary systems

or entanglement. Following the idea, we further obtain an upper bound and a

lower bound of (entanglement-assisted) q-maximal fidelities between a unitary

and a quantum operation. Thus by the recursion in [1] an upper bound and a

lower bound for optimal general perfect discrimination are achieved. Finally

our lower bound result can be extended to the case of arbitrary two quantum

operations.

Xin, Y & Duan, R, 'Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions', *Phys.Rev.A*, vol. 76, p. 044301.View/Download from: Publisher's site

#### View description

In this note we generalize Nielsen's marjoization criterion for the

convertibility of bipartite pure states [Phys. Rev. Lett \textbf{83},

436(1999)] to a special class of multipartite pure states which have

generalized Schmidt decompositions.

Xin, Y & Duan, R, 'Local distinguishability of orthogonal2⊗3pure states', *Physical Review A*, vol. 77, no. 1.View/Download from: Publisher's site

Yu, N, Chitambar, E, Guo, C & Duan, R, 'Tensor rank of the tripartite state|W〉⊗n', *Physical Review A*, vol. 81, no. 1.View/Download from: Publisher's site

Liu, S, Wang, X, Zhou, L, Guan, J, Li, Y, He, Y, Duan, R & Ying, M 2018, 'Q| SI>: A Quantum Programing Environment' in *Symposium on Real-Time and Hybrid Systems (LVCS 11180)*, Springer, Switzerland, pp. 133-164.View/Download from: Publisher's site

#### View description

© Springer Nature Switzerland AG 2018. This paper describes a quantum programming environment, named Q| SI⟩, to support quantum programming using a quantum extension of the while -language. Embedded in the.Net framework, the Q| SI⟩ platform includes a quantum while -language compiler and a suite of tools to simulate quantum computation, optimize quantum circuits, analyze and verify quantum programs. This paper demonstrates Q| SI⟩ in use. Quantum behaviors are simulated on classical platforms with a combination of components and the compilation procedures for different back-ends are described in detail. Q| SI⟩ bridges the gap between quantum hardware and software. As a scalable framework, this platform allows users to code and simulate customized functions, optimize them for a range of quantum circuits, analyze the termination of a quantum program, and verify the program’s correctness (The software of Q| SI⟩ is available at http://www.qcompiler.com.).

Ying, M, Duan, R, Feng, Y & Ji, Z 2010, 'Predicate Transformer Semantics of Quantum Programs' in Gay, S & Mackie, I (eds), *Semantic Techniques in Quantum Computation*, Cambridge University Press, Cambridge, pp. 311-360.

#### View description

This chapter presents a systematic exposition of predicate transformer semantics for quantum programs. It is divided into two parts: The first part reviews the state transformer (forward) semantics of quantum programs according to Selingerâs suggestion of representing quantum programs by superoperators and elucidates DâHondt-Panangadenâs theory of quantum weakest preconditions in detail. In the second part, we develop a quite complete predicate transformer semantics of quantum programs based on Birkhoffâvon Neumann quantum logic by considering only quantum predicates expressed by projection operators. In particular, the universal coujunctivity and termination law of quantum programs are proved, and Hoareâs induction rule is established in the quantum setting.

Du, Y, Liu, T, Li, Y, Duan, R & Tao, D 2018, 'Quantum divide-and-conquer anchoring for separable non-negative matrix factorization', *IJCAI International Joint Conference on Artificial Intelligence*, International Joint Conference on Artificial Intelligence, IJCAI-ECAI, Stockholm, Sweden, pp. 2093-2099.

#### View description

© 2018 International Joint Conferences on Artificial Intelligence. All right reserved. It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X − WH τ || 2F . Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems.

Xie, W, Wang, X & Duan, R 2018, 'Converse Bounds for Classical Communication over Quantum Broadcast Channels and Quantum Multi-Access Channels', *2018 IEEE International Symposium on Information Theory (ISIT)*, IEEE International Symposium on Information Theory, IEEE, Vail, CO, USA, pp. 2341-2345.View/Download from: Publisher's site

#### View description

© 2018 IEEE. We explore the classical communication over quantum channels with one sender and two receivers, or with two senders and one receiver, in both one-shot and asymptotic regimes. First, for the quantum broadcast channel (QBC) and the quantum multi-access channel (QMAC), we study the classical communication assisted by no-signalling and positive-partial-transpose-preserving codes, and obtain efficiently computable one-shot bounds to assess the performance of classical communication. Second, we consider the asymptotic communication capability of communication over the QBC and QMAC. We derive an efficiently computable strong converse bound for the capacity region, which behaves better than the previous semidefinite programming strong converse bound for point-to-point channels. Third, we obtain a converse bound on the one-shot capacity region based on the hypothesis testing divergence between the given channel and a certain class of subchannels. As applications, we analyze the communication performance for some basic network channels, including the classical broadcast channels and a specific class of quantum broadcast channels.

Wang, X, Xie, W & Duan, R 2017, 'Semidefinite programming converse bounds for classical communication over quantum channels', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Aachen, Germany, pp. 1728-1732.View/Download from: Publisher's site

#### View description

© 2017 IEEE. We study the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes. We first show that both the optimal success probability of a given transmission rate and one-shot-error capacity can be formalized as semidefinite programs (SDPs) when assisted by NS or NS∩PPT codes. Based on this, we derive SDP finite blocklength converse bounds for general quantum channels, which also reduce to the converse bound of Polyanskiy, Poor, and Verdii for classical channels. Furthermore, we derive an SDP strong converse bound for the classical capacity of a general quantum channel: for any code with a rate exceeding this bound, the optimal success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bound, we derive improved upper bounds to the classical capacity of the amplitude damping channels and also establish the strong converse property for a new class of quantum channels.

Duan, R, Guo, C, Li, CK & Li, Y 2016, 'Parallel distinguishability of quantum operations', *Proceedings of the IEEE International Symposium on Information Theory (ISIT)*, IEEE International Symposium on Information Theory, IEEE, Barcelona, Spain, pp. 2259-2263.View/Download from: Publisher's site

#### View description

© 2016 IEEE.We find that the perfect distinguishability of two quantum operations by a parallel scheme depends only on an operator subspace generated from their Choi-Kraus operators. We further show that any operator subspace can be obtained from two quantum operations in such a way. This connection enables us to study the parallel distinguishability of operator subspaces directly without explicitly referring to the underlining quantum operations. We obtain a necessary and sufficient condition for the parallel distinguishability of an operator subspace that is either one-dimensional or Hermitian. In both cases the condition is equivalent to the non-existence of positive definite operator in the subspace, and an optimal discrimination protocol is obtained. Finally, we provide more examples to show that the non-existence of positive definite operator is sufficient for many other cases, but in general it is only a necessary condition.

Wang, X & Duan, R 2016, 'A semidefinite programming upper bound of quantum capacity', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Barcelona, Spain, pp. 1690-1694.View/Download from: Publisher's site

#### View description

© 2016 IEEE.Recently the power of positive partial transpose preserving (PPTp) and no-signalling (NS) codes in quantum communication has been studied. We continue with this line of research and show that the NS/PPTp/NS∩PPTp codes assisted zero-error quantum capacity depends only on the non-commutative bipartite graph of the channel and the one-shot case can be computed efficiently by semidefinite programming (SDP). As an example, the activated PPTp codes assisted zero-error quantum capacity is carefully studied. We then present a general SDP upper bound QΓ of quantum capacity and show it is always smaller than or equal to the 'Partial transposition bound' introduced by Holevo and Werner, and the inequality could be strict. This upper bound is found to be additive, and thus is an upper bound of the potential PPTp assisted quantum capacity as well. We further demonstrate that QΓ is strictly better than several previously known upper bounds for an explicit class of quantum channels. Finally, we show that QΓ can be used to bound the super-activation of quantum capacity.

Wang, X & Duan, R 2016, 'On the quantum no-signalling assisted zero-error classical simulation cost of non-commutative bipartite graphs', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Barcelona, Spain, pp. 2244-2248.View/Download from: Publisher's site

#### View description

© 2016 IEEE.Using one channel to simulate another exactly with the aid of quantum no-signalling correlations has been studied recently. The one-shot no-signalling assisted classical zero-error simulation cost of non-commutative bipartite graphs has been formulated as semidefinite programms [Duan and Winter, IEEE Trans. Inf. Theory 62, 891 (2016)]. Before our work, it was unknown whether the one-shot (or asymptotic) no-signalling assisted zero-error classical simulation cost for general non-commutative graphs is multiplicative (resp. additive) or not. In this paper we address these issues and give a general sufficient condition for the multiplicativity of the one-shot simulation cost and the additivity of the asymptotic simulation cost of non-commutative bipartite graphs, which include all known cases such as extremal graphs and classical-quantum graphs. Applying this condition, we exhibit a large class of so-called cheapest-full-rank graphs whose asymptotic zero-error simulation cost is given by the one-shot simulation cost. Finally, we disprove the multiplicativity of one-shot simulation cost by explicitly constructing a special class of qubit-qutrit non-commutative bipartite graphs.

Feng, Y, Duan, R & Ying, M 2011, 'Bisimulation for Quantum Processes', *Proceedings of the 38th annual ACM SIGPLAN-SIGACT symposium on Principles of programming language*, ACM-SIGACT Symposium on Principles of Programming Languages, ACM, Austin, Texas, USA, pp. 523-534.View/Download from: Publisher's site

#### View description

Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit much more faults than classical protocol designers since human intuition is much better adapted to the classical world than the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. One of the most serious issues in quantum process algebra is to discover a quantum generalization of the notion of bisimulation, which lies in a central position in process algebra, preserved by parallel composition in the presence of quantum entanglement, which has no counterpart in classical computation. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communications are involved. Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communications) scheme, where classical communications must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communications is crucial for process algebra approach to verification of quantum cryptographic protocols. In this paper we introduce a novel notion of bisimulation for quantum processes and prove that it is congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communications are present.We also establish some basic algebraic laws for this bisimulation.

Duan, R, Severini, S & Winter, A 2011, 'Zero-error communication via quantum channels and a quantum Lovász Θ-function', *2011 IEEE International Symposium on Information Theory Proceedings (ISIT)*, IEEE International Symposium on Information Theory, IEEE, St Petersburg, Russia, pp. 64-68.View/Download from: Publisher's site

#### View description

We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain linear space operators as the quantum generalisation of the adjacency matrix, in terms of which the plain, quantum and entanglement-assisted capacity can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lova´sz' famous ? function, as the norm-completion (or stabilisation) of a naive generalisation of ?. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite programme, whose dual we write down explicitly, and that it is multiplicative with respect to the natural (strong) graph product. We explore various other properties of the new quantity, which reduces to Lova´sz' original ? in the classical case, give several applications, and propose to study the linear spaces of operators associated to channels as non-commutative graphs, using the language of operator systems and Hilbert modules.

Duan, R, Grassl, M, Ji, Z & Zeng, B 2010, 'Multi-error-correcting amplitude damping codes', *IEEE International Symposium on Information Theory - Proceedings*, International Symposium on Information Theory, IEEE, Austin, USA, pp. 2672-2676.View/Download from: Publisher's site

#### View description

We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use co

Lai, C-Y & Duan, R 2016, 'On the One-Shot Zero-Error Classical Capacity of Classical-Quantum Channels Assisted by Quantum Non-signalling Correlations'.

#### View description

Duan and Winter studied the one-shot zero-error classical capacity of a

quantum channel assisted by quantum non-signalling correlations, and formulated

this problem as a semidefinite program depending only on the Kraus operator

space of the channel. For the class of classical-quantum channels, they showed

that the asymptotic zero-error classical capacity assisted by quantum

non-signalling correlations, minimized over all classical-quantum channels with

a confusability graph $G$, is exactly $\log \vartheta(G)$, where $\vartheta(G)$

is the celebrated Lov\'{a}sz theta function. In this paper, we show that the

same result holds in the one-shot setting for a class of circulant graphs

defined by equal-sized cyclotomic cosets, which include the cycle graphs of odd

length, the Paley graphs of prime vertices, and the cubit residue graphs of

prime vertices. Examples of other graphs are also discussed. This endows the

Lov\'{a}sz $\theta$ function with a more straightforward operational meaning.

Yu, N, Duan, R & Xu, Q, 'Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture'.

#### View description

Motivated by the recent resolution of Asymptotic Quantum Birkhoff Conjecture

(AQBC), we attempt to estimate the distance between a given unital quantum

channel and the convex hull of unitary channels. We provide two lower bounds on

this distance by employing techniques from quantum information and operator

algebras, respectively. We then show how to apply these results to construct

some explicit counterexamples to AQBC. We also point out an interesting

connection between the Grothendieck's inequality and AQBC.

#### Projects

**Selected projects**