Ultimate Data Hiding in Quantum Mechanics and Beyond

The phenomenon of data hiding, i.e. the existence of pairs of states of a bipartite system that are perfectly distinguishable via general entangled measurements yet almost indistinguishable under LOCC, is a distinctive signature of nonclassicality. The relevant figure of merit is the maximal ratio (...

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Bibliographic Details
Published in:Communications in mathematical physics Vol. 361; no. 2; pp. 661 - 708
Main Authors: Lami, Ludovico, Palazuelos, Carlos, Winter, Andreas
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2018
Springer Nature B.V
Subjects:
Banach spaces
Classical and Quantum Gravitation
Complex Systems
Figure of merit
Mathematical and Computational Physics
Mathematical Physics
Norms
Physics
Physics and Astronomy
Quantum mechanics
Quantum Physics
Quantum theory
Relativity Theory
Theoretical
Upper bounds
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary:The phenomenon of data hiding, i.e. the existence of pairs of states of a bipartite system that are perfectly distinguishable via general entangled measurements yet almost indistinguishable under LOCC, is a distinctive signature of nonclassicality. The relevant figure of merit is the maximal ratio (called data hiding ratio) between the distinguishability norms associated with the two sets of measurements we are comparing, typically all measurements vs LOCC protocols. For a bipartite n × n quantum system, it is known that the data hiding ratio scales as n , i.e. the square root of the real dimension of the local state space of density matrices. We show that for bipartite n A × n B systems the maximum data hiding ratio against LOCC protocols is Θ min { n A , n B } . This scaling is better than the previously obtained upper bounds O n A n B and min { n A 2 , n B 2 } , and moreover our intuitive argument yields constants close to optimal. In this paper, we investigate data hiding in the more general context of general probabilistic theories (GPTs), an axiomatic framework for physical theories encompassing only the most basic requirements about the predictive power of the theory. The main result of the paper is the determination of the maximal data hiding ratio obtainable in an arbitrary GPT, which is shown to scale linearly in the minimum of the local dimensions. We exhibit an explicit model achieving this bound up to additive constants, finding that quantum mechanical data hiding ratio is only of the order of the square root of the maximal one. Our proof rests crucially on an unexpected link between data hiding and the theory of projective and injective tensor products of Banach spaces. Finally, we develop a body of techniques to compute data hiding ratios for a variety of restricted classes of GPTs that support further symmetries.
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ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-018-3154-4