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# Professor Runyao Duan

Future Fellow and Professor, A/DRsch Ctr Quantum Computat'n & Intelligent Systs
Core Member, Centre for Quantum Computation and Intelligent Systems
NA

Phone
+61 2 9514 4619
Room
CB11.10.206
Can supervise: Yes

## Chapters

Ying, M., Duan, R., Duan, R. & Feng, Y. 2013, 'Predicate transformer semantics of quantum programs' in Semantic Techniques in Quantum Computation, Cambridge University Press, pp. 311-360.
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 Selingers suggestion of representing quantum programs by superoperators and elucidates DHondt-Panangadens theory of quantum weakest preconditions in detail. In the second part, we develop a quite complete predicate transformer semantics of quantum programs based on Birkhoffvon Neumann quantum logic by considering only quantum predicates expressed by projection operators. In particular, the universal conjunctivity and termination law of quantum programs are proved, and Hoares induction rule is established in the quantum setting.

## Conferences

Feng, Y., Duan, R. & Ying, M. 2012, 'Bisimulation for Quantum Processes.', ACM Trans. Program. Lang. Syst., pp. 17-17.
Duan, R., Severini, S. & Winter, A. 2011, 'Zero-error communication via quantum channels and a quantum Lovsz ?-function', 2011 IEEE International Symposium on Information Theory Proceedings (ISIT), IEEE, Piscataway, USA, pp. 64-68.
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 Lovasz' 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 Lovasz' 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', Proceedings 2010 IEEE International Symposium on Information Theory (ISIT 2010), Austin, USA, June 2010, pp. 2672-2676.
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 concatenated codes with quantum erasure-correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction.

## Journal articles

Duan, R., Severini, S. & Winter, A. 2015, 'On zero-error communication via quantum channels in the presence of noiseless feedback'.
We initiate the study of zero-error communication via quantum channels assisted by noiseless feedback link of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. This capacity depends only on the linear span of Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub "non-commutative bipartite graph". We go on to show that the feedback-assisted capacity is non-zero (allowing for a constant amount of 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, Nat Phys 8:475, 2012]. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved by Ahlswede. We demonstrate that this bound is additive and given by a nice minimax formula. We also prove a coding theorem showing that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" (de Finetti reduction) [Christandl/Koenig/Renner, PRL 102:020503, 2009] that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedback-assisted zero-error capacity; however, we do not know whether they coincide in general. 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.
Duan, R. & Winter, A. 2014, 'No-Signalling Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovasz Number'.
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 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.
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.
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.', Sci. Comput. Program., vol. 78, pp. 1679-1700.
Chitambar, E., Duan, R. & Hsieh, M.H. 2013, 'When do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination?', IEEE Trans. Inform. Theory, vol. 60, p. 3.
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 \emph{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 \textit{almost all} two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that \textit{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 $N\to\infty$, 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. 2013, 'Five Two-Qubit Gates Are Necessary for Implementing Toffoli Gate', Phys. Rev. A, vol. 88, p. 010304.
In this paper, we settle the long-standing open problem of the minimum cost of two-qubit gates for simulating a Toffoli gate. More precisely, we show that five two-qubit gates are necessary. Before our work, it is known that five gates are sufficient and only numerical evidences have been gathered, indicating that the five-gate implementation is necessary. The idea introduced here can also be used to solve the problem of optimal simulation of three-qubit control phase introduced by Deutsch in 1989.
Lu, C., Chen, J. & 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.
We prove a lower bound on the q-maximal fidelities between two quantum channels ? 0 and ? 1 and an upper bound on the q-maximal fidelities between a quantum channel ? and an identity I. Then we apply these two bounds to provide a simple sufficient and necessary condition for sequential perfect distinguish ability between ? and I and provide both a lower bound and an upper bound on the minimum number of queries required to sequentially perfectly discriminating ? and I. Interestingly, in the 2-dimensional case, both bounds coincide. Based on the optimal perfect discrimination protocol presented in [20], we can further generalize the lower bound and upper bound to the minimum number of queries to perfectly discriminating ? and I over all possible discrimination schemes. Finally the two lower bounds are shown remain working for perfectly discriminating general two quantum channels ? 0 and ? 1 in sequential scheme and over all possible discrimination schemes respectively. Rinton Press.
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.
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.', Phys Rev Lett, vol. 109, no. 2, p. 020506.
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.
Yu, N., Duan, R. & Xu, Q. 2012, 'Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture'.
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.
Yu, N., Duan, R. & Ying, M. 2012, 'Distinguishability of Quantum States by Positive Operator-Valued Measures with Positive Partial Transpose', IEEE Trans. Inform. Theory, vol. 60, p. 4.
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 $\mathcal{O}(\frac{\log d}{d})$.
Feng, Y., Duan, R. & Ying, M. 2011, 'Locally undetermined states, generalized schmidt decomposition, and application in deistributed comuting.', Quantum Information & Computation, vol. 9, pp. 997-1012.
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.
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.
Chen, X., Duan, R., Ji, Z. & Zeng, B. 2010, 'Quantum state reduction for universal measurement based computation.', Phys Rev Lett, vol. 105, no. 2, p. 020502.
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 quantum computers. Despite considerable progress in the past decade, it remains a great challenge to search for new universal resource states with naturally occurring Hamiltonians and to better understand the entanglement structure of these kinds of states. Here we show that most of the resource states currently known can be reduced to the cluster state, the first known universal resource state, via adaptive local measurements at a constant cost. This new quantum state reduction scheme provides simpler proofs of universality of resource states and opens up plenty of space to the search of new resource states.
Yu, N., Duan, R. & Ying, M. 2010, 'Optimal Simulation Of A Perfect Entangler', Physical Review A, vol. 81, no. 3, pp. 1-4.
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
Yu, N., Chitambar, E., Guo, C. & Duan, R. 2010, 'Tensor rank of the tripartite state vertical bar W >(circle times n)', PHYSICAL REVIEW A, vol. 81, no. 1.
Chen, L., Chitambar, E.A., 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.
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
Chitambar, E.A., Duan, R. & Shi, Y. 2010, 'Multipartite-To-Bipartite Entanglement Transformations And Polynomial Identity Testing', Physical Review A, vol. 81, no. 5, pp. 1-4.
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
Yu, N., Duan, R. & Ying, M. 2010, 'Any $2\otimes n$ subspace is locally distinguishable', Phys. Rev. A, vol. 84, p. 012304.
A subspace of a multipartite Hilbert space is called \textit{locally indistinguishable} if any orthogonal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any $m\otimes n$ bipartite system such that $m>2$ 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 by showing that any $2\otimes n$ bipartite subspace is locally distinguishable in the sense it contains a basis perfectly distinguishable by LOCC. As an interesting application, we show that any quantum channel with two Kraus operations has optimal environment-assisted classical capacity.
Feng, Y., Duan, R. & Ying, M. 2010, 'Bisimulation for quantum processes', CoRR, vol. abs/1007.2584.
Duan, R., Severini, S. & Winter, A. 2010, 'Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function', IEEE Trans. Inf. Theory 59(2):1164-1174, 2013.
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 operator space 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 Lovasz' famous theta function, as the norm-completion (or stabilisation) of a "naive" generalisation of theta. 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 Lovasz' original theta in the classical case, give several applications, and propose to study the operator spaces associated to channels as "non-commutative graphs", using the language of Hilbert modules.
Lu, C., Chen, J. & Duan, R. 2010, 'Optimal Perfect Distinguishability between Unitaries and Quantum Operations'.
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.
Chen, J., Chen, X., Duan, R., Ji, Z. & Zeng, B. 2010, 'No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems', Phys. Rev. A, vol. 83, p. 050301.
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.
Chen, L., Chitambar, E., Duan, R., Ji, Z. & Winter, A. 2010, 'Tensor rank and stochastic entanglement catalysis for multipartite pure states.', Phys Rev Lett, vol. 105, no. 20, p. 200501.
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.
Duan, R., Feng, Y., Xin, Y. & Ying, M. 2009, 'Distinguishability of Quantum States by Separable Operations.', IEEE Transactions on Information Theory, vol. 55, pp. 1320-1330.
Ying, M., Feng, Y., Duan, R. & Ji, Z. 2009, 'An algebra of quantum processes', ACM Transactions on Computational Logic, vol. 10, no. 3.
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. 2009 ACM.
Chitambar, E.A. & Duan, R. 2009, 'Nonlocal Entanglement Transformations Achievable by Separable Operations', Physical Review Letters, vol. 103, no. 11, pp. 1-4.
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.
Duan, R. & Shi, Y. 2009, 'When is there a multipartite maximum entangled state?'.
For a multipartite quantum system of the dimension $d_1\otimes d_2\otimes... d_n$, $d_1\ge d_2\ge...\ge d_n$, is there an entangled state {\em 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 $d_1\ge\Pi_{i=2}^n d_i$, such state exists. We show that this condition is also necessary. Our proof, somewhat surprisingly, uses results from algebraic complexity theory.
Li, Y., Duan, R. & Ying, M. 2009, 'Local Unambiguous Discrimination with Remaining Entanglement'.
A bipartite state which is secretly chosen from a finite set of known entangled pure states cannot be immediately useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a new way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whatever orthogonal or not, can be locally distinguished in this way, which further implies that pure entanglement can be deterministically extracted from unknown pure entanglement. These results make it clear why a large class of entangled bipartite quantum operations including unitary operations and measurements that are globally distinguishable can also be locally distinguishable: they can generate pure entanglement consistently.
Duan, R. 2009, 'Super-Activation of Zero-Error Capacity of Noisy Quantum Channels'.
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.
Chitambar, E. & Duan, R. 2009, 'Nonlocal entanglement transformations achievable by separable operations.', Phys Rev Lett, vol. 103, no. 11, p. 110502.
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.
Yu, N., Duan, R. & Ying, M. 2009, 'Optimal Simulation of a Perfect Entangler', Phys. Rev. A, vol. 81, p. 032328.
A $2\otimes 2$ 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 is required to simulate some perfect entangler with one-qubit unitary operations as free resources? We completely solve this problem by presenting an analytical formula for the optimal number of runs of the entangling operation. Our result reveals an entanglement strength of two-qubit unitary operations.
Duan, R., Feng, Y. & Ying, M. 2009, 'Perfect distinguishability of quantum operations.', Phys Rev Lett, vol. 103, no. 21, p. 210501.
We provide a feasible necessary and sufficient condition for when an unknown quantum operation (quantum device) secretly selected from a set of known quantum operations can be identified perfectly within a finite number of queries, and thus complete the characterization of the perfect distinguishability of quantum operations. We further design an optimal protocol which can achieve the perfect discrimination between two quantum operations by a minimal number of queries. Interestingly, we find that an optimal perfect discrimination between two isometries is always achievable without auxiliary systems or entanglement.
Chitambar, E., Duan, R. & Shi, Y. 2009, 'Tripartite to Bipartite Entanglement Transformations and Polynomial Identity Testing', Phys. Rev. A, vol. 81, p. 052310.
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.
Yu, N., Chitambar, E., Guo, C. & Duan, R. 2009, 'The Tensor Rank of the Tripartite State $\ket{W}^{\otimes n}$}', Phys. Rev. A, vol. 81, p. 014301.
Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its non-additivity as an entanglement measure has recently been observed. In this note, we estimate the tensor rank of multiple copies of the tripartite state $\ket{W}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$. Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the tensor rank of $\ket{W}^{\otimes 2}$ is seven, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between $\ket{W}^{\otimes n}$ and multiple copies of the state $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$.
Ji, Z.-.F., Wang, G., Duan, R., Feng, Y. & Ying, M. 2008, 'Parameter Estimation of Quantum Channels.', IEEE Transactions on Information Theory, vol. 54, pp. 5172-5185.
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.
Duan, R., Feng, Y. & Ying, M. 2008, 'Local Distinguishability Of Multipartite Unitary Operations', Physical Review Letters, vol. 100, no. 2, pp. 1-4.
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.A., Duan, R. & Shi, Y. 2008, 'Tripartite Entanglement Transformations And Tensor Rank', Physical Review Letters, vol. 101, no. 14, pp. 1-4.
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
Xin, Y. & Duan, R. 2008, 'Local distinguishability of orthogonal 2 circle times 3 pure states', PHYSICAL REVIEW A, vol. 77, no. 1.
Duan, R., Feng, Y. & Ying, M. 2008, 'Local distinguishability of multipartite unitary operations.', Phys Rev Lett, vol. 100, no. 2, p. 020503.
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.
Chitambar, E., Duan, R. & Shi, Y. 2008, 'Tripartite entanglement transformations and tensor rank.', Phys Rev Lett, vol. 101, no. 14, p. 140502.
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.
Ying, M., Chen, J., Feng, Y. & Duan, R. 2007, 'Commutativity of quantum weakest preconditions.', Inf. Process. Lett., vol. 104, pp. 152-158.
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.
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.
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. & Ying, M. 2007, 'Entanglement Is Not Necessary For Perfect Discrimination Between Unitary Operations', Physical Review Letters, vol. 98, no. 10, pp. 1-4.
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
Chen, J., Duan, R., Ji, Z., Ying, M. & Yu, J. 2007, 'Existence of Universal Entangler'.
A gate is called 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 generating entanglements, and thus provides important physical resources for accomplishing many tasks in quantum computing and quantum information. This Letter demonstrates that a universal entangler always exists except for a degenerated case. Nevertheless, the problem how to find a universal entangler remains open.
Feng, Y., Duan, R., Ji, Z. & Ying, M. 2007, 'Probabilistic bisimulations for quantum processes', INFORMATION AND COMPUTATION, vol. 205, no. 11, pp. 1608-1639.
Feng, Y., Duan, R., Ji, Z.-.F. & Ying, M. 2007, 'Proof rules for the correctness of quantum programs.', Theor. Comput. Sci., vol. 386, pp. 151-166.
Duan, R., Xin, Y. & Ying, M. 2007, 'Locally Indistinguishable Subspaces Spanned by Three-Qubit Unextendible Product Bases'.
We study the local distinguishability of general multi-qubit 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 the local distinguishability of multi-qubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension $2^n$. When $n=2$ such a condition is also sufficient. Employing these results, we show that any orthonormal basis of a subspace spanned by arbitrary three-qubit orthogonal unextendible product bases (UPB) cannot be exactly distinguishable by local operations and classical communication. This not only reveals another intrinsic property of three-qubit orthogonal UPB, but also provides a class of locally indistinguishable subspaces with dimension 4. We also explicitly construct locally indistinguishable subspaces with dimensions 3 and 5, respectively. In particular, 3 is the minimal possible dimension of locally indistinguishable subspaces. Combining with the previous results, we conclude that any positive integer between 3 and 7 is the possible dimension of some three-qubit locally indistinguishable subspace.
Wu, X. & Duan, R. 2007, 'Exact Quantum Search by Parallel Unitary Discrimination Schemes'.
We study the unsorted database search problem with items $N$ from the viewpoint of unitary discrimination. Instead of considering the famous $O(\sqrt{N})$ Grover's the bounded-error algorithm for the original problem, we seek for the results about the exact algorithms, i.e. the ones succeed with certainty. Under the standard oracle model $\sum_j (-1)^{\delta_{\tau j}}|j>< j|$, we demonstrate a tight lower bound ${2/3}N+o(N)$ of the number of queries for any parallel scheme with unentangled input states. With the assistance of entanglement, we obtain a general lower bound ${1/2}(N-\sqrt{N})$. We provide concrete examples to illustrate our results. In particular, we show that the case of N=6 can be solved exactly with only two queries by using a bipartite entangled input state. Our results indicate that in the standard oracle model the complexity of exact quantum search with one unique solution can be strictly less than that of the calculation of OR function.
Xin, Y. & Duan, R. 2007, 'Local distinguishability of orthogonal 2\otimes3 pure states', Phys.Rev.A, vol. 77, p. 012315.
We present a complete characterization for the local distinguishability of orthogonal $2\otimes 3$ 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 indistinguishable by local projective measurements and classical communication (LPCC) can be perfectly distinguishable by LOCC. That indicates the ability of LOCC for discriminating $2\otimes 3$ states is strictly more powerful than that of LPCC, which is strikingly different from the case of multi-qubit states. We also show that classical communication plays a crucial role for local distinguishability by constructing a class of $m\otimes n$ states which require at least $2\min\{m,n\}-2$ rounds of classical communication in order to achieve a perfect local discrimination.
Xin, Y. & Duan, R. 2007, 'Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions', Phys.Rev.A, vol. 76, p. 044301.
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.
Duan, R. & Shi, Y. 2007, 'Entanglement between Two Uses of a Noisy Multipartite Quantum Channel Enables Perfect Transmission of Classical Information', Phys. Rev. Lett., vol. 101, p. 020501.
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.
Duan, R., Feng, Y., Ji, Z. & Ying, M. 2007, 'Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication.', Phys Rev Lett, vol. 98, no. 23, p. 230502.
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 dk always contains at least N= Sigma(k=1)(K) (dk-1)+1 members that are unambiguously distinguishable using local operations and classical communication (LOCC). We further show that this lower bound is optimal by analytically constructing a special product basis having only N members unambiguously distinguishable by LOCC. Interestingly, such a special product basis not only gives a stronger form of the weird phenomenon "nonlocality without entanglement," but also implies the existence of a locally distinguishable entangled basis.
Duan, R., Feng, Y. & Ying, M. 2007, 'Entanglement is not necessary for perfect discrimination between unitary operations.', Phys Rev Lett, vol. 98, no. 10, p. 100503.
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.
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.
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.
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
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.
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
Duan, R., Feng, Y. & Ying, M. 2006, 'Partial Recovery of Quantum Entanglement.', IEEE Transactions on Information Theory, vol. 52, pp. 3080-3104.
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.
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., Ji, Z.-.F., Feng, Y. & Ying, M. 2006, 'Some Issues in Quantum Information Theory.', J. Comput. Sci. Technol., vol. 21, pp. 776-789.
Ji, Z., Feng, Y., Duan, R. & Ying, M. 2006, 'Identification and distance measures of measurement apparatus.', Phys Rev Lett, vol. 96, no. 20, p. 200401.
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. Based on these results, a brief discussion on the problem of how to appropriately define distance measures of measurements is also provided.
Ji, Z., Feng, Y., Duan, R. & Ying, M. 2006, 'Boundary effect of deterministic dense coding', Phys. Rev. A, vol. 73, p. 034307.
We present a rigorous proof of an interesting boundary effect of deterministic dense coding first observed by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Namely, it is shown that $d^2-1$ cannot be the maximal alphabet size of any isometric deterministic dense coding schemes utilizing $d$-level partial entanglement.
Duan, R., Feng, Y. & Ying, M. 2006, 'Entanglement Is Not Necessary for Perfect Discrimination between Unitary Operations', Physical Review Letters, vol. 98, p. 10.
We show that a unitary operation (quantum circuit) secretely 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 is 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.
Feng, Y., Duan, R. & Ji, Z. 2006, 'Optimal dense coding with arbitrary pure entangled states', Phys. Rev. A, vol. 74, p. 012310.
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 coding are derived. We also construct the optimal protocol which saturates the upper bound in each case.
Feng, Y., Duan, R., Ji, Z.-.F. & Ying, M. 2006, 'Probabilistic bisimilarities between quantum processes', CoRR, vol. abs/cs/0601014.
Feng, Y., Duan, R. & Ying, M. 2005, 'Catalyst-assisted probabilistic entanglement transformation.', IEEE Transactions on Information Theory, vol. 51, pp. 1090-1101.
Feng, Y., Duan, R. & Ji, Z. 2005, 'Condition And Capability Of Quantum State Separation', Physical Review A, vol. 72, no. 1, pp. 1-6.
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., Ji, Z. & Ying, M. 2005, 'Efficiency Of Deterministic Entanglement Transformation', Physical Review A, vol. 71, no. 2, pp. 1-7.
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
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.
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
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.
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., 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.
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
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.
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
Feng, Y., Duan, R. & Ji, Z. 2005, 'Condition and capability of quantum state separation', Phys. Rev. A, vol. 72, p. 012313.
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 we show that in probabilistic manner, linearity is in fact the only one that restricts the physically realizable tasks. To be specific, if a system is prepared in a state secretly chosen from a linearly independent pure state set, then any quantum state separation can be physically realized with a positive probability. Furthermore, we derive a lower bound on the average failure probability of any quantum state separation.
Feng, Y., Duan, R., Ji, Z.-.F. & Ying, M. 2005, 'Proof rules for purely quantum programs', CoRR, vol. abs/cs/0507043.
Ji, Z., Duan, R. & Ying, M. 2004, 'Comparability of multipartite entanglement', Phys. Lett. A, vol. 330, pp. 418-423.
We prove, in a multipartite setting, that it's always feasible to exactly transform a genuinely $m$-partite entangled state with sufficient many copies to any other $m$-partite state via local quantum operation and classical communication. This result affirms the comparability of multipartite pure entangled states.
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.
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, Y., Duan, R. & Ying, M. 2004, 'Unambiguous discrimination between quantum mixed states', Phys. Rev. A, vol. 70, p. 012308.
We prove that the states secretly chosen from a mixed state set can be perfectly discriminated if and only if these states are orthogonal. The sufficient and necessary condition when nonorthogonal quantum mixed states can be unambiguously discriminated is also presented. Furthermore, we derive a series of lower bounds on the inconclusive probability of unambiguous discrimination of states from a mixed state set with \textit{a prior} probabilities.
Feng, Y., Duan, R. & Ying, M. 2004, 'When Catalysis is Useful for Probabilistic Entanglement Transformation', Phys. Rev. A, vol. 69, p. 062310.
We determine all $2\times 2$ 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 sufficient and necessary condition is derived under which a certain probabilistic transformation has useful catalysts.
Duan, R., Feng, Y. & Ying, M. 2004, 'An Equivalence of Entanglement-Assisted Transformation and Multiple-Copy Entanglement Transformation'.
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.
Duan, R., Feng, Y. & Ying, M. 2004, 'Partial Recovery of Quantum Entanglement', IEEE Trans. Inform. Theory, vol. 52, p. 7.
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 of transformation. However, an interesting phenomenon called partial entanglement recovery shows that it is possible to recover some amount of entanglement by adding another entangled state and transforming the two entangled states collectively. In this paper we are mainly concerned with the feasibility of partial entanglement recovery. The basic problem we address is whether a given state is useful in recovering entanglement lost in a specified transformation. In the case where the source and target states of the original transformation satisfy the strict majorization relation, a necessary and sufficient condition for partial entanglement recovery is obtained. For the general case we give two sufficient conditions. We also give an efficient algorithm for the feasibility of partial entanglement recovery in polynomial time. As applications, we establish some interesting connections between partial entanglement recovery and the generation of maximally entangled states, quantum catalysis, mutual catalysis, and multiple-copy entanglement transformation.
Duan, R., Feng, Y., Li, X. & Ying, M. 2004, 'Multiple-copy entanglement transformation and entanglement catalysis', Phys. Rev. A, vol. 71, p. 042319.
We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys. Rev. A \textbf{65}, 052315 (2002)] can be implemented by a suitable entanglement-assisted local transformation [D. Jonathan and M. B. Plenio, Phys. Rev. Lett. \textbf{83}, 3566 (1999)]. Furthermore, we show that the combination of multiple-copy entanglement transformation and the entanglement-assisted one is still equivalent to the pure entanglement-assisted one. The mathematical structure of multiple-copy entanglement transformations then is carefully investigated. Many interesting properties of multiple-copy entanglement transformations are presented, which exactly coincide with those satisfied by the entanglement-assisted ones. Most interestingly, we show that an arbitrarily large number of copies of state should be considered in multiple-copy entanglement transformations.
Feng, Y., Duan, R. & Ying, M. 2004, 'Catalyst-assisted Probabilistic Entanglement Transformation', IEEE Trans. Inform. Theory, vol. 51, p. 3.
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 transformation. We also design an algorithm which leads to an efficient method for finding the most economical partial catalysts with minimal dimension. The mathematical structure of catalyst-assisted probabilistic transformation is carefully investigated.
Duan, R., Feng, Y., Ji, Z. & Ying, M. 2004, 'Efficiency of Deterministic Entanglement Transformation', Phys. Rev. A, vol. 71, p. 022305.
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 characterized by deterministic entanglement exchange rate, and it is proved to be always positive and bounded from top by the infimum of the ratios of Renyi's entropies of source state and target state. A careful analysis shows that the deterministic entanglement exchange rate cannot be increased even in the presence of catalysts. As an application, we show that there can be two incomparable states with deterministic entanglement exchange rate strictly exceeding 1.
Duan, R., feng, Y. & Ying, M. 2004, 'Entanglement-assisted transformation is asymptotically equivalent to multiple-copy transformation', Phys. Rev. A, vol. 72, p. 024306.
We show that two ways of manipulation of quantum entanglement, namely, entanglement-assisted local transformation [D. Jonathan and M. B. Plenio, Phys. Rev. Lett. {\bf 83}, 3566 (1999)] and multiple-copy transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys. Rev. A {\bf 65}, 052315 (2002)], are equivalent in the sense that they can asymptotically simulate each other's ability to implement a desired transformation from a given source state to another given target state with the same optimal success probability. As a consequence, this yields a feasible method to evaluate the optimal conversion probability of an entanglement-assisted transformation.
Duan, R., Ji, Z., Feng, Y. & Ying, M. 2003, 'Quantum operation, quantum Fourier transform and semi-definite programming', Phys. Lett. A 323(2004)48.
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 this class of operations. A more general class of operations on qudits is also considered and its completely positive condition is reduced to the well-known semi-definite programming problem.
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. 83, 3566(1999)] presented an interesting example showing that the presence of another state, called a catalyst, enables such a transformation without changing the catalyst. They also pointed out that in general it is very hard to find an analytical condition under which a catalyst exists. In this paper we study the existence of catalysts for two incomparable quantum states. For the simplest case of $2\times 2$ catalysts for transformations from one $4\times 4$ state to another, a necessary and sufficient condition for existence is found. For the general case, we give an efficient polynomial time algorithm to decide whether a $k\times k$ catalyst exists for two $n\times n$ incomparable states, where $k$ is treated as a constant.