Using postmeasurement information in state discrimination

We consider a special form of state discrimination in which after the measurement we are given additional information that may help us identify the state. This task plays a central role in the analysis of quantum cryptographic protocols in the noisy-storage model, where the identity of the state corresponds to a certain bit string, and the additional information is typically a choice of encoding that is initially unknown to the cheating party. We first provide simple optimality conditions for measurements for any such problem and show upper and lower bounds on the success probability. For a certain class of problems, we furthermore provide tight bounds on how useful postmeasurement information can be. In particular, we show that for this class finding the optimal measurement for the task of state discrimination with postmeasurement information does in fact reduce to solving a different problem of state discrimination without such information. However, we show that for the corresponding classical state discrimination problems with postmeasurement information such a reduction is impossible, by relating the success probability to the violation of Bell inequalities. This suggests the usefulness of postmeasurement information as another feature that distinguishes the classical from a quantum world.

[1]  J. Preskill,et al.  Quantum Information and Computation (QUIC) , 2001 .

[2]  R F Werner,et al.  Iterative optimization of quantum error correcting codes. , 2005, Physical review letters.

[3]  Dave Cliff,et al.  In/Proceedings of the 15th IEEE International Conference on the Engineering of Complex Computer Systems/ (ICECCS 2010), Oxford , 2010 .

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

[5]  W. Marsden.  I and J , 2012 .

[6]  Tomohiro Ogawa,et al.  Strong converse to the quantum channel coding theorem , 1999, IEEE Trans. Inf. Theory.

[7]  Beth Kelley,et al.  Advances in optics and photonics make wind energy more viable , 2011 .

[8]  J. Fiurášek,et al.  Quantum inference of states and processes , 2002, quant-ph/0210146.

[9]  Dave Bacon,et al.  Optimal single-copy measurement for the hidden-subgroup problem , 2007, 0706.4478.

[10]  Jon Tyson.  Error rates of Belavkin weighted quantum measurements and a converse to Holevo's asymptotic optimality theorem , 2009, 0907.1884.

[11]  J. Fiurášek,et al.  Finding optimal strategies for minimum-error quantum-state discrimination , 2002, quant-ph/0201109.

[12]  Gilles Brassard,et al.  Quantum Cryptography , 2005, Encyclopedia of Cryptography and Security.

[13]  Yonina C. Eldar.  A semidefinite programming approach to optimal unambiguous discrimination of quantumstates , 2003, IEEE Trans. Inf. Theory.

[14]  E. Wigner,et al.  About the Pauli exclusion principle , 1928 .

[15]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[16]  V. Belavkin.  Optimal multiple quantum statistical hypothesis testing , 1975 .

[17]  E. Wigner,et al.  Über das Paulische Äquivalenzverbot , 1928 .

[18]  Carlos Mochon.  Hamiltonian Oracles , 2006 .

[19]  S. Barnett,et al.  Quantum state discrimination , 2008, 0810.1970.

[20]  Jon Tyson.  Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo–Curlander bounds , 2009, 0907.2094.

[21]  William K. Wootters,et al.  A ‘Pretty Good’ Measurement for Distinguishing Quantum States , 1994 .

[22]  A. Holevo.  Statistical decision theory for quantum systems , 1973 .

[23]  Andreas J. Winter,et al.  State Discrimination With Post-Measurement Information , 2008, IEEE Transactions on Information Theory.

[24]  A. Winter,et al.  Higher entropic uncertainty relations for anti-commuting observables , 2007, 0710.1185.

[25]  Alexander Russell,et al.  For distinguishing conjugate hidden subgroups, the pretty good measurement is as good as it gets , 2007, Quantum Inf. Comput..

[26]  Christian Schaffner,et al.  Cryptography from noisy storage. , 2007, Physical review letters.

[27]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.