Publications (74)200.18 Total impact

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ABSTRACT: We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $\alpha(G) \leq \vartheta(G)$ between the independence number and the Lov\'{a}sz number of $G$ can be made arbitrarily tight: Precisely, the inequality \[ \alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H) \] becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that \[ \alpha(G \times H) \leq \alpha^*(G)\,\alpha(H), \] with the fractional packing number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes the above an equality; conversely, for every graph $H$ there is a $G$ that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which $\alpha$ and $\alpha^*$ are dual to each other, and the Lov\'{a}sz number $\vartheta$ is selfdual. We also show results towards the duality of Schrijver's and Szegedy's variants $\vartheta^$ and $\vartheta^+$ of the Lov\'{a}sz number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.  [Show abstract] [Hide abstract]
ABSTRACT: We initiate the study of zeroerror communication via quantum channels assisted by noiseless feedback link of unlimited quantum capacity, generalizing Shannon's zeroerror 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 "noncommutative bipartite graph". We go on to show that the feedbackassisted capacity is nonzero (allowing for a constant amount of activating noiseless communication) if and only if the noncommutative bipartite graph is nontrivial, and give a number of equivalent characterizations. This result involves a farreaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nat Phys 8:475, 2012]. We then present an upper bound on the feedbackassisted zeroerror capacity, motivated by a conjecture originally made by Shannon and proved by Ahlswede.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 entanglementassisted capacity against an adversarially chosen channel from the set of all channels with the same Kraus span, which can also be interpreted as the feedbackassisted 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 feedbackassisted zeroerror capacity; however, we do not know whether they coincide in general. We illustrate our ideas with a number of examples, including classicalquantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.  [Show abstract] [Hide abstract]
ABSTRACT: We study the oneshot zeroerror classical capacity of a quantum channel assisted by quantum nosignalling correlations, and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one. Quantum nosignalling correlations are viewed as twoinput and twooutput 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 zeroerror classical simulation cost is precisely the conditional minentropy of the ChoiJamiolkowski matrix of the given channel. The zeroerror classical capacity is given by a similarlooking but different SDP; the asymptotic zeroerror 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 classicalquantum 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 zeroerror classical capacity of the graph assisted by quantum nosignalling correlations, the first information theoretic interpretation of the Lovasz number.  [Show abstract] [Hide abstract]
ABSTRACT: 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 threequbit 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 GreenbergHorneZeilinger (GHZ) state by SLOCC, asymptotically. We then apply similar techniques to obtain a lower bound on the entanglement transformation rates from an Npartite 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.  [Show abstract] [Hide abstract]
ABSTRACT: 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 threequbit 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 GreenbergHorneZeilinger (GHZ) state by SLOCC, asymptotically. We then apply similar techniques to obtain a lower bound on the entanglement transformation rates from an Npartite 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.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we study the entanglement transformation rate between multipartite states under stochastic local operations and classical communication (SLOCC). Firstly, we show that the entanglement transformation rate from $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$ to $\ket{W}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ is 1, that is, one can obtain 1 copy of $W$state, from 1 copy of $GHZ$state by SLOCC, asymptotically. We then generalize this result to a lower bound on the rate that from $N$partite $GHZ$state to Dicke states. For some special cases, the optimality of this bound is proved. We then discuss the tensor rank of matrix permanent by evaluating the the tensor rank of Dicke state. 
Article: When Do Local Operations and Classical Communication Suffice for TwoQubit State Discrimination?
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ABSTRACT: In this paper we consider the conditions under which a given ensemble of twoqubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the \emph{perfect} distinguishability problem of twoqubit 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 wellknown task of minimum error discrimination, it is shown that \textit{almost all} twoqubit 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 "nonlocality 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" singlecopy 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 minimumerror probability is considered. 
Article: ZeroError Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number
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ABSTRACT: We study the quantum channel version of Shannon's zeroerror capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (socalled operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zeroerror capacity of a quantum channel, as well as the quantum and entanglementassisted zeroerror capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous ϑ function on general operator systems, as the normcompletion (or stabilization) of a “naive” generalization of ϑ. We go on to show that this function upper bounds the number of entanglementassisted zeroerror messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lovász' original ϑ in the classical case, give several applications, and propose to study the operator systems associated with channels as “noncommutative graphs,” using the language of Hilbert modules.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we settle the longstanding open problem of the minimum cost of twoqubit gates for simulating a Toffoli gate. More precisely, we show that five twoqubit gates are necessary. Before our work, it is known that five gates are sufficient and only numerical evidences have been gathered, indicating that the fivegate implementation is necessary. The idea introduced here can also be used to solve the problem of optimal simulation of threequbit control phase introduced by Deutsch in 1989.  [Show abstract] [Hide abstract]
ABSTRACT: 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 Selinger’s suggestion of representing quantum programs by superoperators, and elucidates D’HondtPanangaden’s 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 Hoare’s induction rule is established in the quantum setting.  [Show abstract] [Hide abstract]
ABSTRACT: We study the distinguishability of bipartite quantum states by Positive OperatorValued Measures with positive partial transpose (PPT POVM). 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 previously known ones. (3). We study the entanglement cost of distinguishing quantum states. It is proved that entanglement $\sqrt{2/3}\ket{00}+\sqrt{1/3}\ket{11}$ is sufficient and necessary for distinguishing three Bell states by PPT POVMs. An upper bound of entanglement cost for 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 twoqubit 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 $\mathcal{O}(\frac{\log d}{d})$.  [Show abstract] [Hide abstract]
ABSTRACT: We explicitly exhibit a set of four ququadququad 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 zeroerror classical capacity can achieve the full dimension of the input space but only with a multishot protocol.  [Show abstract] [Hide abstract]
ABSTRACT: 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 quantum programs and communication protocols are also reviewed. The potential applications of programming techniques and related formal methods in quantum engineering are pointed out.  [Show abstract] [Hide abstract]
ABSTRACT: 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. 
Article: Some bounds on the minimum number of queries required for quantum channel perfect discrimination
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ABSTRACT: We prove a lower bound on the qmaximal fidelities between two quantum channels E0 and E1 and an upper bound on the qmaximal fidelities between a quantum channel E and an identity I. Then we apply these two bounds to provide a simple sufficient and necessary condition for sequential perfect distinguishability between E and I and provide both a lower bound and an upper bound on the minimum number of queries required to sequentially perfectly discriminating E and I. Interestingly, in the 2dimensional 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 E and I over all possible discrimination schemes. Finally the two lower bounds are shown remain working for perfectly discriminating general two quantum channels E0 and E1 in sequential scheme and over all possible discrimination schemes respectively.  [Show abstract] [Hide abstract]
ABSTRACT: A subspace of a multipartite Hilbert space is said to be locally indistinguishable if any orthonormal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any m⊗n bipartite system with m>2 and n>2 has a locally indistinguishable subspace. However, it has been an open problem since 2005 whether there is a locally indistinguishable bipartite subspace with a qubit subsystem. We settle this problem in negative by showing that any 2⊗n bipartite subspace contains a basis that is locally distinguishable. As an interesting application, we show that any quantum channel with two Kraus operators has optimal environmentassisted classical capacity.  [Show abstract] [Hide abstract]
ABSTRACT: We study the quantum channel version of Shannon's zeroerror 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 entanglementassisted capacity can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous υ function, as the normcompletion (or stabilisation) of a “naive” generalisation of υ. We go on to show that this function upper bounds the number of entanglementassisted zeroerror 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 Lovász' original υ in the classical case, give several applications, and propose to study the linear spaces of operators associated to channels as “noncommutative graphs”, using the language of operator systems and Hilbert modules. 
Article: Verification of Quantum Programs
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ABSTRACT: This paper develops verification methodology for quantum programs, and the contribution of the paper is twofold: 1. Sharir, Pnueli and Hart [SIAM J. Comput. 13(1984)292314] presented a general method for proving properties of probabilistic programs, in which a probabilistic program is modeled by a Markov chain and an assertion on the output distribution is extended into an invariant assertion on all intermediate distributions. Their method is essentially a probabilistic generalization of the classical Floyd inductive assertion method. In this paper, we consider quantum programs modeled by quantum Markov chains which are defined by superoperators. It is shown that the SharirPnueliHart method can be elegantly generalized to quantum programs by exploiting the Schr\"odingerHeisenberg duality between quantum states and observables. In particular, a completeness theorem for the SharirPnueliHart verification method of quantum programs is established. 2. As indicated by the completeness theorem, the SharirPnueliHart method is in principle effective for verifying all properties of quantum programs that can be expressed in terms of Hermitian operators (observables). But it is not feasible for many practical applications because of the complicated calculation involved in the verification. For the case of finitedimensional state spaces, we find a method for verification of quantum programs much simpler than the SharirPnueliHart method by employing the matrix representation of superoperators and Jordan decomposition of matrices. In particular, this method enables us to compute easily the average running time and even to analyze some interesting longrun behaviors of quantum programs in a finitedimensional state space.  [Show abstract] [Hide abstract]
ABSTRACT: In order to better understand the class of quantum operations that preserve the positivity of partial transpose (PPT operations) and its relation to the widely used class of local operations and classical communication (LOCC), we study the problem of distinguishing orthogonal maximally entangled states (MES) by PPT operations. Firstly, we outline a rather simple proof to show that the number of $d\otimes d$ PPT distinguishable MES is at most $d$, which slightly generalizes existing results on this problem. Secondly, we construct 4 MES in $4\otimes 4$ state space that cannot be distinguished by PPT operations. Before our work, it was unknown whether there exists $d$ MES in $d\otimes d$ state space that are locally indistinguishable. This example leads us to a novel phenomenon of "Entanglement Catalysis Discrimination". Moreover, we find there exists a set of locally indistinguishable states $K$ such that $K^{\otimes m}$ is locally distinguishable for some finite $m$. As an interesting application, we exhibit a quantum channel with one sender and two receivers, whose oneshot zeroerror local capacity is not optimal, but multiuse would enhance the capacity to achieve the full output dimension even without entangled input. Finally, we consider the entanglement cost of distinguishing three Bell states and a $2\otimes 2$ entangled basis. In the former case a bipartite pure entangled state with the largest Schmidt coefficient at most 2/3 is necessary and sufficient, while in the latter case an additional Bell state, or one ebit, should be supplied.
Publication Stats
864  Citations  
200.18  Total Impact Points  
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Institutions

20032015

Tsinghua University
 Department of Computer Science and Technology
Peping, Beijing, China


20092014

University of Technology Sydney
 Centre for Quantum Computation and Intelligent Systems (QCIS)
Sydney, New South Wales, Australia


2010

University of Waterloo
 Department of Combinatorics & Optimization
Ватерлоо, Ontario, Canada


2005

National Tsing Hua University
Hsinchuhsien, Taiwan, Taiwan
