Li, Y, Qiao, Y, Wang, X & Duan, R 2018, 'Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces', Communications in Mathematical Physics, vol. 358, no. 2, pp. 791-814.View/Download from: UTS OPUS or Publisher's site
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings, utilizing powerful results from the structure of matrix spaces. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation when multiple copies are provided. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate. Computing this rate turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipa...
© 2017 Elsevier B.V. A matrix space of size m×n is a linear subspace of the linear space of m×n matrices over a field F. The rank of a matrix space is defined as the maximal rank over matrices in this space. A matrix space A is called rank-critical, if any matrix space which properly contains it has rank strictly greater than that of A. In this note, we first exhibit a necessary and sufficient condition for a matrix space A to be rank-critical, when F is large enough. This immediately implies the sufficient condition for a matrix space to be rank-critical by Draisma (2006) , albeit requiring the field to be slightly larger. We then study rank-critical spaces in the context of compression and primitive matrix spaces. We first show that every rank-critical matrix space can be decomposed into a rank-critical compression matrix space and a rank-critical primitive matrix space. We then prove, using our necessary and sufficient condition, that the block-diagonal direct sum of two rank-critical matrix spaces is rank-critical if and only if both matrix spaces are primitive, when the field is large enough.
Liu, S, Wang, X, Zhou, L, Guan, J, Li, Y, He, Y, Duan, R & Ying, M 2018, 'Q| SI>: A Quantum Programing Environment' in Symposium on Real-Time and Hybrid Systems (LVCS 11180), Springer, Switzerland, pp. 133-164.View/Download from: UTS OPUS or Publisher's site
© Springer Nature Switzerland AG 2018. This paper describes a quantum programming environment, named Q| SI⟩, to support quantum programming using a quantum extension of the while -language. Embedded in the.Net framework, the Q| SI⟩ platform includes a quantum while -language compiler and a suite of tools to simulate quantum computation, optimize quantum circuits, analyze and verify quantum programs. This paper demonstrates Q| SI⟩ in use. Quantum behaviors are simulated on classical platforms with a combination of components and the compilation procedures for different back-ends are described in detail. Q| SI⟩ bridges the gap between quantum hardware and software. As a scalable framework, this platform allows users to code and simulate customized functions, optimize them for a range of quantum circuits, analyze the termination of a quantum program, and verify the program’s correctness (The software of Q| SI⟩ is available at http://www.qcompiler.com.).
Du, Y, Liu, T, Li, Y, Duan, R & Tao, D 2018, 'Quantum divide-and-conquer anchoring for separable non-negative matrix factorization', IJCAI International Joint Conference on Artificial Intelligence, International Joint Conference on Artificial Intelligence, IJCAI-ECAI, Stockholm, Sweden, pp. 2093-2099.View/Download from: UTS OPUS
© 2018 International Joint Conferences on Artificial Intelligence. All right reserved. It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X − WH τ || 2F . Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems.
Li, Y & Qiao, Y 2017, 'Linear algebraic analogues of the graph isomorphism problem and the erds-rényi model', Annual Symposium on Foundations of Computer Science - Proceedings, IEEE Annual Symposium on Foundations of Computer Science, IEEE, Berkeley, CA, USA, pp. 463-474.View/Download from: UTS OPUS or Publisher's site
© 2017 IEEE. A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as a linear algebraic analogue of the graph isomorphism problem. This viewpointleads us to explore the possibility of transferring techniques for graph isomorphism to this long-believed bottleneck case of group isomorphism.In 1970s, Babai, Erds, and Selkow presented the first average-case efficient graph isomorphism testing algorithm (SIAM J Computing, 1980). Inspired by that algorithm, 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 Erdos-R4;enyi model of random graphs. For this, we develop a linear algebraic analogue of the classical individualisation technique, a technique belonging to a set of combinatorial techniques that has been critical for the progress on the worst-case time complexity for graph isomorphism, but was missing in the group isomorphism context. This algorithm also enables us to improve Higmans 57-year-old lower bound on the number of p-groups (Proc. of the LMS, 1960). We finally show that Luks dynamic programming technique for graph isomorphism (STOC 1999) can be adapted to slightly improve the worst-case time complexity of the alternating matrix space isometry problem in a certain range of parameters.Most notable progress on the worst-case time complexity of graph isomorphism, including Babais recent breakthrough (STOC 2016) and Babai and Luks previous record (STOC 1983), has relied on both group theoretic and combinatorial techniques. By developing a linear algebraic analogue of the individualisation ...
Duan, R, Guo, C, Li, CK & Li, Y 2016, 'Parallel distinguishability of quantum operations', Proceedings of the IEEE International Symposium on Information Theory (ISIT), IEEE International Symposium on Information Theory, IEEE, Barcelona, Spain, pp. 2259-2263.View/Download from: UTS OPUS or Publisher's site
© 2016 IEEE.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.