Quantum channel discrimination plays a crucial role in quantum information theory. Of particular interest is the case in which the channels can be discriminated perfectly. In this work, we focus on the perfect quantum channel discrimination task in a parallel scheme. We develop an SDP formulation combined with a bisection procedure to compute a quantum state for perfect discrimination in time linear in the number of copies. In addition, we obtain the minimal number of copies of quantum channels to achieve perfect discrimination. Thanks to that, we settle in the affirmative Conjecture 1 of Duan, Guo, Li and Li [arXiv:1605.02294, IEEE ISIT 2016], which characterizes the number of parallel uses needed to discriminate perfectly a distinguished family of operator subspaces. All our results are obtained using the notion and basic properties of the numerical range. In particular, the key fact that we prove and use is that the minimal angle of the numerical range of a tensor product of matrix subspaces equals the sum of the minimal angles of the numerical ranges of the individual subspaces.
Quantum channel discrimination plays a crucial role in quantum information theory. Of particular interest is the case in which the channels can be discriminated perfectly. In this work, we focus on the perfect quantum channel discrimination task in a parallel scheme. We develop an SDP formulation combined with a bisection procedure to compute a quantum state for perfect discrimination in time linear in the number of copies. In addition, we obtain the minimal number of copies of quantum channels to achieve perfect discrimination. Thanks to that, we settle in the affirmative Conjecture 1 of Duan, Guo, Li and Li [arXiv:1605.02294, IEEE ISIT 2016], which characterizes the number of parallel uses needed to discriminate perfectly a distinguished family of operator subspaces. All our results are obtained using the notion and basic properties of the numerical range. In particular, the key fact that we prove and use is that the minimal angle of the numerical range of a tensor product of matrix subspaces equals the sum of the minimal angles of the numerical ranges of the individual subspaces.
This thesis explores the applications of matrix spaces in quantum information and computational complexity, specifically in the areas of quantum state/channel discrimination, entanglement transformation and isomorphism testing.We achieve the following contributions:• We derive a necessary condition which determines whether a set of orthogonal bipartite states can be distinguished by positive-partial-transpose (PPT) operations, in the many-copy scenario.We reduce the discrimination task to the following problem:Decide whether there exist a nonzero bipartite matrix with positive partial-transpose, such that all its eigenvectors with nonzero eigenvalues is orthogonal to a given bipartite vector space.As applications, we reprove and extend a variety of existing results, including the local indistinguishability of the bipartite maximally entangled state and its orthogonal complement [Yu, Duan and Ying, IEEE transaction on Information Theory, 2014] and the maximum dimension of non-positive-partialtranspose subspaces [Johnston, Physical Review A, 2013].• We show that determining the parallel distinguishability of quantum channels is equivalent to determining whether the orthogonal complement of a given matrix space contains nonzero positive semidefinite matrices.Our characterization immediately implies a necessary condition to decide the parallel distinguishability.We further prove that our condition is also sufficient for two large families of quantum channels, which leads to an alternating proof for the parallel distinguishability of vi Abstract unitary operations [Acín, Physical Review Letters, 2001].We also present an illustrative example showing our necessary condition cannot determine the parallel distinguishability.• We systematically study the tripartite-to-bipartite entanglement transformations by stochastic local operations and classical communication (SLOCC).Such a problem is equivalent to computing the maximal rank of a matrix space [Chitambar, Duan and Shi, Physical Review A, 2010].We analyze the SLOCC convertibility in both the finite-copy and asymptotic regimes.In particular, we derive explicit formulas which compute asymptotic entanglement transformation rates for two families of tripartite states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants.Significantly, we show that the problem of deciding the asymptotic SLOCC convertibility of tripartite pure states to the bipartite maximally entangled states and the non-commutative symbolic determinant identity testing problem is algorithmically equivalent, which builds a new connection between problems in algebraic complexity and problems in asymptotic SLOCC entanglement transformations.• We devise algorithms which test isometry between alternating matrix spaces over finite field.Algorithms for such a problem in time polynomial in the underlying vector space size resolves the believed bottleneck case of the group isomorphism problem, i.e. testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order.Our approach is to view it as a linear algebraic analog of the graph isomorphism problem.We devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters.in a random model of alternating matrix spaces in vein of the Erdős-Rényi model of random graphs.To devise our algorithm, we developed linear algebraic individualization and refinement techniques, which are crucial in the first average-case efficient algorithm for graph isomorphism, devised by Babai, Erdős, and Selkow in the 1970s [Babai, Erdős and Selkow, SIAM Journal on Computing, 1980].We also adapt Luks' dynamic programming technique for graph isomorphism [Luks, STOC, 1999] to slightly improve the worst-case time complexity of the alternating matrix space isometry problem.Very little of the work in this thesis would have been possible without the support of my supervisors, collaborators, colleagues, friends and family.First and foremost, I would like to thank my supervisor, Runyao Duan.Without his guidance and insight, I would not have complete my PhD study.I am greatly indebted to him for leading me into the quantum world.Dating back to 2013 in Wuhan University, when I was still a third-year undergraduate student, I was so inspired by his talk about quantum computation.He encouraged me to investigate the parallel distinguishability of quantum channels, which becomes the departure point of my research career.Furthermore, I am grateful for Runyao's spiritual support for my life.He is not only my academic supervisor but also a spiritual mentor, sharing his experience and scope in both research and life.Thanks to my co-supervisor, Youming Qiao, my research interests have been developed and extended.I would like to thank him for his inspirational discussions and conversations from which I really benefit a lot.I have learned about many interesting topics from him, especially for introducing me to the beauty of computational complexity theory.
In the problem of quantum channel discrimination, one distinguishes between a given number of quantum channels, which is done by sending an input state through a channel and measuring the output state. This work studies applications of variational quantum circuits and machine-learning techniques for discriminating such channels. In particular, we explore (i) the practical implementation of embedding this task into the framework of variational quantum computing, (ii) training a quantum classifier based on variational quantum circuits, and (iii) applying the quantum kernel estimation technique. For testing these three channel discrimination approaches, we considered a pair of entanglement-breaking channels and the depolarizing channel with two different depolarization factors. For the approach (i), we address solving the quantum channel discrimination problem using widely discussed parallel and sequential strategies. We show the advantage of the latter in terms of better convergence with less quantum resources. Quantum channel discrimination with a variational quantum classifier (ii) allows one to operate even with random and mixed input states and simple variational circuits. The kernel-based classification approach (iii) is also found effective as it allows one to discriminate depolarizing channels associated not with just fixed values of the depolarization factor, but with ranges of it. Additionally, we discovered that a simple modification of one of the commonly used kernels significantly increases the efficiency of this approach. Finally, our numerical findings reveal that the performance of variational methods of channel discrimination depends on the trace of the product of the output states. These findings demonstrate that quantum machine learning can be used to discriminate channels, such as those representing physical noise processes.
Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank k of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce k-injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of k-injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers.
For every k, we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.
We prove a lower bound on the q-maximal fidelities between two quantum channels $\E_0$ and $\E_1$ 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 \cite{DFY09}, 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 $\E_0$ and $\E_1$ in sequential scheme and over all possible discrimination schemes respectively.
We find that the perfect distinguishability of two quantum operations by a parallel scheme depends only on an operator subspace generated from their Choi-Kraus operators. We further show that any operator subspace can be obtained from two quantum operations in such a way. This connection enables us to study the parallel distinguishability of operator subspaces directly without explicitly referring to the underlining quantum operations. We obtain a necessary and sufficient condition for the parallel distinguishability of an operator subspace that is either one-dimensional or Hermitian. In both cases the condition is equivalent to the non-existence of positive definite operator in the subspace, and an optimal discrimination protocol is obtained. Finally, we provide more examples to show that the non-existence of positive definite operator is sufficient for many other cases, but in general it is only a necessary condition.