[Show abstract][Hide abstract] 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 self-dual. 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 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motivated by a conjecture originally made by Shannon and proved by Ahlswede.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.
[Show abstract][Hide abstract] ABSTRACT: 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.
[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 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.
[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 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.
[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.
[Show abstract][Hide abstract] ABSTRACT: 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.
IEEE Transactions on Information Theory 08/2013; 60(3). DOI:10.1109/TIT.2013.2295356 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous ϑ function on general operator systems, as the norm-completion (or stabilization) of a “naive” generalization of ϑ. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to 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.
IEEE Transactions on Information Theory 02/2013; 59(2):1164-1174. DOI:10.1109/TIT.2012.2221677 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review A 01/2013; 88(1). DOI:10.1103/PhysRevA.88.010304 · 2.81 Impact Factor
[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’Hondt-Panangaden’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 Birkhoff-von 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
Operator-Valued 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 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 $\mathcal{O}(\frac{\log d}{d})$.
IEEE Transactions on Information Theory 09/2012; 60(4). DOI:10.1109/TIT.2014.2307575 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
[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.
Chinese Science Bulletin 06/2012; 57(16). DOI:10.1007/s11434-012-5147-6 · 1.58 Impact Factor
[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.
[Show abstract][Hide abstract] ABSTRACT: We prove a lower bound on the q-maximal fidelities between two quantum channels E0 and E1 and an upper bound on the q-maximal 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 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 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.
Quantum information & computation 01/2012; 12(1-2):138-148. · 1.39 Impact Factor
[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 environment-assisted classical capacity.
Physical Review A 07/2011; 84(1). DOI:10.1103/PhysRevA.84.012304 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper develops verification methodology for quantum programs, and the
contribution of the paper is two-fold: 1. Sharir, Pnueli and Hart [SIAM J.
Comput. 13(1984)292-314] 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 super-operators. It is shown that the Sharir-Pnueli-Hart
method can be elegantly generalized to quantum programs by exploiting the
Schr\"odinger-Heisenberg duality between quantum states and observables. In
particular, a completeness theorem for the Sharir-Pnueli-Hart verification
method of quantum programs is established. 2. As indicated by the completeness
theorem, the Sharir-Pnueli-Hart 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 finite-dimensional state spaces, we find a method for verification of
quantum programs much simpler than the Sharir-Pnueli-Hart method by employing
the matrix representation of super-operators and Jordan decomposition of
matrices. In particular, this method enables us to compute easily the average
running time and even to analyze some interesting long-run behaviors of quantum
programs in a finite-dimensional 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 one-shot zero-error local capacity is not optimal, but multi-use 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.
[Show abstract][Hide abstract] ABSTRACT: 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 Lovász' famous υ function, as the norm-completion (or stabilisation) of a “naive” generalisation of υ. 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 Lovász' original υ 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.