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Published November 2009 | public
Journal Article

Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding

Abstract

We consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements. Let the bias of a POVM be defined as the total variational distance between the outcome distributions for the two states to be distinguished. The performance of a set of measurements can then be defined as the ratio of the bias of this POVM and the largest bias achievable by any measurements. We first provide lower bounds on the performance of various POVMs acting on a single system such as the isotropic POVM, and spherical 2 and 4-designs, and show how these bounds can lead to certainty relations. Furthermore, we prove lower bounds for several interesting POVMs acting on multipartite systems, such as the set of local POVMS, POVMs which can be implemented using local operations and classical communication (LOCC), separable POVMs, and finally POVMs for which every bipartition results in a measurement having positive partial transpose (PPT). In particular, our results show that a scheme of Terhal et. al. for hiding data against local operations and classical communication [31] has the best possible dimensional dependence.

Additional Information

© 2009 Springer. Received: 31 October 2008; Accepted: 4 June 2009; Published online: 13 August 2009. AW thanks the members of the Pavia Quantum Information group for an enjoyable afternoon in October 2007, where he had occasion to discuss some of the questions of the present paper, when they were still in a nascent state. In particular the feedback of G. M. D'Ariano, G. Chiribella and M. F. Sacchi, and their suggestions regarding the use of symmetry, are gratefully acknowledged. Ashley Montanaro provided the pointer to the paper by Ambainis and Emerson, and provided the example mentioned in Appendix A. WM would like to thank Dan Shepherd for a useful discussion about groups and diagrams. WM was supported by the U.K. EPSRC. SW was supported by NSF grant number PHY-04056720. AW was supported by the U.K. EPSRC through the "QIP IRC" and an Advanced Fellowship, by a Royal Society Wolfson Merit Award and by the European Commission through IP "QAP". The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.

Additional details

Created:
August 19, 2023
Modified:
October 19, 2023