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# Professor Runyao Duan

Future Fellow and Professor, A/DRsch Ctr Quantum Computat'n & Intelligent Systs
NA

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

## Chapters

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.
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 coujunctivity 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

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.
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
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.
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.
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.
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.
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.
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
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.
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.