Local Hypothesis Testing Between a Pure Bipartite State and the White Noise State

In this paper, we treat a local discrimination problem in the framework of asymmetric hypothesis testing. We choose a known bipartite pure state |ψ) as an alternative hypothesis and the completely mixed state as a null hypothesis. As a result, we analytically derive an optimal type-2 error and an optimal positive operator valued measure (POVM) for one-way local operations and classical communication (LOCC) POVM and separable POVM. For two-way LOCC POVM, we study a family of simple three-step LOCC protocols, and show that the best protocol in this family has strictly better performance than any one-way LOCC protocol in low-dimensional systems when there may exist differences between two-way LOCC POVM and one-way LOCC POVM.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  C. Esseen.  Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law , 1945 .

[3]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[4]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

[5]  W. Hoeffding.  Asymptotically Optimal Tests for Multinomial Distributions , 1965 .

[6]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[7]  Horace P. Yuen,et al.  Multiple-parameter quantum estimation and measurement of nonselfadjoint observables , 1973, IEEE Trans. Inf. Theory.

[8]  Richard E. Blahut,et al.  Hypothesis testing and information theory , 1974, IEEE Trans. Inf. Theory.

[9]  Robert S. Kennedy,et al.  Optimum testing of multiple hypotheses in quantum detection theory , 1975, IEEE Trans. Inf. Theory.

[10]  Alexander Semenovich Holevo,et al.  Covariant measurements and uncertainty relations , 1979 .

[11]  A. S. Holevo,et al.  Capacity of a quantum communication channel , 1979 .

[12]  G. Wimmer.  Minimum mean square error estimation , 1979 .

[13]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[14]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .

[15]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[16]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[17]  W. Wootters,et al.  Optimal detection of quantum information. , 1991, Physical review letters.

[18]  Sergio Verdú,et al.  A general formula for channel capacity , 1994, IEEE Trans. Inf. Theory.

[19]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[20]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[21]  Charles H. Bennett,et al.  Purification of noisy entanglement and faithful teleportation via noisy channels. , 1995, Physical review letters.

[22]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .

[23]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[24]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[25]  G. Vidal,et al.  Robustness of entanglement , 1998, quant-ph/9806094.

[26]  E. Rains.  Bound on distillable entanglement , 1998, quant-ph/9809082.

[27]  H. Nagaoka,et al.  Strong converse theorems in the quantum information theory , 1999, 1999 Information Theory and Networking Workshop (Cat. No.99EX371).

[28]  C. H. Bennett,et al.  Quantum nonlocality without entanglement , 1998, quant-ph/9804053.

[29]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[30]  Tomohiro Ogawa,et al.  Strong converse and Stein's lemma in quantum hypothesis testing , 2000, IEEE Trans. Inf. Theory.

[31]  Guifre Vidal.  Entanglement monotones , 1998, quant-ph/9807077.

[32]  Vedral,et al.  Local distinguishability of multipartite orthogonal quantum states , 2000, Physical review letters.

[33]  Eric M. Rains.  A semidefinite program for distillable entanglement , 2001, IEEE Trans. Inf. Theory.

[34]  Oliver Rudolph.  The uniqueness theorem for entanglement measures , 2001, quant-ph/0105104.

[35]  A. Sen De,et al.  Distinguishability of Bell states. , 2001, Physical Review Letters.

[36]  S. Virmania,et al.  Optimal local discrimination of two multipartite pure states , 2001 .

[37]  L. Vaidman,et al.  Nonlocal variables with product-state eigenstates , 2001, quant-ph/0103084.

[38]  D. Leung,et al.  Hiding bits in bell states. , 2000, Physical Review Letters.

[39]  D. Markham,et al.  Optimal local discrimination of two multipartite pure states , 2001, quant-ph/0102073.

[40]  R. Werner,et al.  Entanglement measures under symmetry , 2000, quant-ph/0010095.

[41]  M. Hayashi.  Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing , 2002, quant-ph/0208020.

[42]  L. Hardy,et al.  Nonlocality, asymmetry, and distinguishing bipartite states. , 2002, Physical review letters.

[43]  Debbie W. Leung,et al.  Quantum data hiding , 2002, IEEE Trans. Inf. Theory.

[44]  R F Werner,et al.  Hiding classical data in multipartite quantum states. , 2002, Physical review letters.

[45]  Dong Yang,et al.  Optimally conclusive discrimination of nonorthogonal entangled states by local operations and classical communications , 2002 .

[46]  Ujjwal Sen,et al.  Locally accessible information: how much can the parties gain by cooperating? , 2003, Physical review letters.

[47]  Jihane Mimih,et al.  Distinguishing two-qubit states using local measurements and restricted classical communication , 2003 .

[48]  A. Harrow,et al.  Robustness of quantum gates in the presence of noise , 2003, quant-ph/0301108.

[49]  P. Goldbart,et al.  Geometric measure of entanglement and applications to bipartite and multipartite quantum states , 2003, quant-ph/0307219.

[50]  S. Virmani,et al.  Construction of extremal local positive-operator-valued measures under symmetry , 2002, quant-ph/0212020.

[51]  Masahito Hayashi,et al.  General formulas for capacity of classical-quantum channels , 2003, IEEE Transactions on Information Theory.

[52]  Masahito Hayashi,et al.  On error exponents in quantum hypothesis testing , 2004, IEEE Transactions on Information Theory.

[53]  M. Hayashi,et al.  Finding a maximally correlated state: Simultaneous Schmidt decomposition of bipartite pure states , 2004, quant-ph/0405107.

[54]  Anthony Chefles.  Condition for unambiguous state discrimination using local operations and classical communication , 2004 .

[55]  H. Fan.  Distinguishability and indistinguishability by local operations and classical communication. , 2004, Physical review letters.

[56]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[57]  Christopher King,et al.  Correcting quantum channels by measuring the environment , 2005, Quantum Inf. Comput..

[58]  John Watrous,et al.  Bipartite subspaces having no bases distinguishable by local operations and classical communication. , 2005, Physical review letters.

[59]  Michael Nathanson.  Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases , 2005 .

[60]  F. Brandão.  Quantifying entanglement with witness operators , 2005, quant-ph/0503152.

[61]  M. Ying,et al.  Optimal conclusive discrimination of two states can be achieved locally , 2004, quant-ph/0407120.

[62]  J. Deuschel,et al.  A Quantum Version of Sanov's Theorem , 2004, quant-ph/0412157.

[63]  Masahito Hayashi.  Asymptotic theory of quantum statistical inference : selected papers , 2005 .

[64]  Somshubhro Bandyopadhyay,et al.  Local Distinguishability of Any Three Quantum States , 2006 .

[65]  Masahito Hayashi,et al.  Local copying and local discrimination as a study for nonlocality of a set of states , 2006 .

[66]  Masahito Hayashi,et al.  A study of LOCC-detection of a maximally entangled state using hypothesis testing , 2006 .

[67]  M. Murao,et al.  Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication. , 2005, Physical review letters.

[68]  Yoshiko Ogata.  Local distinguishability of quantum states in infinite-dimensional systems , 2006 .

[69]  Hypothesis testing for an entangled state produced by spontaneous parametric down-conversion , 2006, quant-ph/0603254.

[70]  H. Nagaoka.  The Converse Part of The Theorem for Quantum Hoeffding Bound , 2006, quant-ph/0611289.

[71]  Masato Koashi,et al.  ‘Quantum Nonlocality without Entanglement’ in a Pair of Qubits , 2007, OSA Workshop on Entanglement and Quantum Decoherence.

[72]  Yuan Feng,et al.  Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication. , 2007, Physical review letters.

[73]  F. Hiai,et al.  Large deviations and Chernoff bound for certain correlated states on a spin chain , 2007, 0706.2141.

[74]  Yong-Sheng Zhang,et al.  Local distinguishability of orthogonal quantum states and generators of SU( N ) , 2006, quant-ph/0608040.

[75]  Heng Fan,et al.  Distinguishing bipartite states by local operations and classical communication , 2007 .

[76]  Masahito Hayashi,et al.  An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses , 2007, IEEE Transactions on Information Theory.

[77]  Masahito Hayashi,et al.  Two-way classical communication remarkably improves local distinguishability , 2007, 0708.3154.

[78]  Scott M. Cohen.  Local distinguishability with preservation of entanglement , 2007 .

[79]  K. Audenaert,et al.  Discriminating States: the quantum Chernoff bound. , 2006, Physical review letters.

[80]  Martin B. Plenio,et al.  An introduction to entanglement measures , 2005, Quantum Inf. Comput..

[81]  Masahito Hayashi.  Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding , 2006, quant-ph/0611013.

[82]  F. Hiai,et al.  Error exponents in hypothesis testing for correlated states on a spin chain , 2007, 0707.2020.

[83]  M. Nussbaum,et al.  Asymptotic Error Rates in Quantum Hypothesis Testing , 2007, Communications in Mathematical Physics.

[84]  Samuel L. Braunstein,et al.  ε-convertibility of entangled states and extension of Schmidt rank in infinite-dimensional systems , 2008, Quantum Inf. Comput..

[85]  J. Deuschel,et al.  Typical Support and Sanov Large Deviations of Correlated States , 2007, math/0703772.

[86]  Andreas J. Winter,et al.  On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states , 2008, 2008 IEEE Information Theory Workshop.

[87]  A. Hayashi,et al.  State discrimination with error margin and its locality , 2008, 0804.4349.

[88]  Runyao Duan,et al.  Local distinguishability of orthogonal 2 ⊗ 3 pure states , 2008 .

[89]  Nilanjana Datta,et al.  Max- Relative Entropy of Entanglement, alias Log Robustness , 2008, 0807.2536.

[90]  F. Hiai,et al.  Asymptotic distinguishability measures for shift-invariant quasifree states of fermionic lattice systems , 2008 .

[91]  Locality and nonlocality in quantum pure-state identification problems , 2008, 0807.1364.

[92]  Masahito Hayashi,et al.  Group theoretical study of LOCC-detection of maximally entangled states using hypothesis testing , 2008, 0810.3380.

[93]  Milan Mosonyi.  Hypothesis testing for Gaussian states on bosonic lattices , 2009 .

[94]  Masahito Hayashi,et al.  Discrimination of Two Channels by Adaptive Methods and Its Application to Quantum System , 2008, IEEE Transactions on Information Theory.

[95]  A. Hayashi,et al.  Discrimination with error margin between two states: Case of general occurrence probabilities , 2009, 0906.4884.

[96]  M. Nussbaum,et al.  THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING , 2006, quant-ph/0607216.

[97]  Yuan Feng,et al.  Distinguishability of Quantum States by Separable Operations , 2007, IEEE Transactions on Information Theory.

[98]  G. Guo,et al.  Subspaces without locally distinguishable orthonormal bases , 2009 .

[99]  M. B. Plenio,et al.  Entanglement of multiparty stabilizer, symmetric, and antisymmetric states , 2007, CLEO/Europe - EQEC 2009 - European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference.

[100]  Masahito Hayashi,et al.  Institute for Mathematical Physics Quantum Hypothesis Testing with Group Symmetry Quantum Hypothesis Testing with Group Symmetry , 2022 .

[101]  H. Vincent Poor,et al.  Channel Coding Rate in the Finite Blocklength Regime , 2010, IEEE Transactions on Information Theory.

[102]  Guang-Can Guo,et al.  A sufficient and necessary condition for 2n − 1 orthogonal states to be locally distinguishable in a C2⊗Cn system , 2010 .

[103]  F. Brandão,et al.  A Generalization of Quantum Stein’s Lemma , 2009, 0904.0281.

[104]  E Bagan,et al.  Local discrimination of mixed States. , 2010, Physical review letters.

[105]  Michael Nathanson.  Testing for a pure state with local operations and classical communication , 2009, 0906.2382.

[106]  Masahito Hayashi,et al.  Quantum Information: An Introduction , 2010 .

[107]  S. Bandyopadhyay.  Entanglement and perfect discrimination of a class of multiqubit states by local operations and classical communication , 2010 .

[108]  Imre Csiszár,et al.  Information Theory - Coding Theorems for Discrete Memoryless Systems, Second Edition , 2011 .

[109]  Hermann Kampermann,et al.  Asymptotically perfect discrimination in the local-operation-and-classical-communication paradigm , 2011 .

[110]  M. Owari,et al.  Asymptotic local hypothesis testing between a pure bipartite state and the completely mixed state , 2011, 1105.3789.

[111]  Eric Chitambar,et al.  Local quantum transformations requiring infinite rounds of classical communication. , 2011, Physical review letters.

[112]  Ke Li,et al.  Second Order Asymptotics for Quantum Hypothesis Testing , 2012, ArXiv.

[113]  Laura Mančinska,et al.  Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask) , 2012, 1210.4583.

[114]  R. Renner,et al.  One-shot classical-quantum capacity and hypothesis testing. , 2010, Physical review letters.

[115]  Hoi-Kwong Lo,et al.  Increasing entanglement monotones by separable operations. , 2012, Physical review letters.

[116]  Masahito Hayashi,et al.  A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks , 2012, IEEE Transactions on Information Theory.

[117]  Yury Polyanskiy,et al.  Saddle Point in the Minimax Converse for Channel Coding , 2013, IEEE Transactions on Information Theory.

[118]  Janis Noetzel,et al.  Hypothesis testing on invariant subspaces of the symmetric group: part I. Quantum Sanov's theorem and arbitrarily varying sources , 2014, ArXiv.

[119]  Milán Mosonyi,et al.  Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies , 2013, ArXiv.

[120]  Pierre Moulin,et al.  The Log-Volume of Optimal Codes for Memoryless Channels, Within A Few Nats , 2013, ArXiv.

[121]  Te Sun Han,et al.  Second-Order Resolvability, Intrinsic Randomness, and Fixed-Length Source Coding for Mixed Sources: Information Spectrum Approach , 2011, IEEE Transactions on Information Theory.

[122]  Laura Mančinska,et al.  A Framework for Bounding Nonlocality of State Discrimination , 2012, Communications in Mathematical Physics.

[123]  Min-Hsiu Hsieh,et al.  Revisiting the optimal detection of quantum information , 2013, 1304.1555.

[124]  D. Leung,et al.  When the asymptotic limit offers no advantage in the local-operations-and-classical-communication paradigm , 2014 .

[125]  Andreas Winter,et al.  Relative Entropy and Squashed Entanglement , 2012, 1210.3181.

[126]  Mark M. Wilde,et al.  Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication , 2014, Communications in Mathematical Physics.

[127]  Yuval Peres,et al.  Adversarial Hypothesis Testing and a Quantum Stein’s Lemma for Restricted Measurements , 2013, IEEE Transactions on Information Theory.

[128]  Samuel L. Braunstein,et al.  Asymmetric quantum hypothesis testing with Gaussian states , 2014, 1407.0884.

[129]  Ke Li.  Discriminating quantum states: the multiple Chernoff distance , 2015, ArXiv.

[130]  Masahito Hayashi,et al.  Correlation detection and an operational interpretation of the Rényi mutual information , 2014, 2015 IEEE International Symposium on Information Theory (ISIT).

[131]  Masahito Hayashi,et al.  Tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state , 2014, 2015 IEEE International Symposium on Information Theory (ISIT).

[132]  Pierre Moulin,et al.  The Log-Volume of Optimal Codes for Memoryless Channels, Asymptotically Within a Few Nats , 2013, IEEE Transactions on Information Theory.

[133]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.