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arXiv:1208.2550 · Physical Review A

Quantum lost property: A possible operational meaning for the Hilbert-Schmidt product

Matthew F. Pusey, Terry Rudolph

Published Oct 18, 2012• 1 citation
arXivHTMLPDF

Abstract

Minimum-error state discrimination between two mixed states $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ can be aided by the receipt of ``classical side information'' specifying which states from some convex decompositions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ apply in each run. We quantify this phenomena by the average trace distance and give lower and upper bounds on this quantity as functions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$. The lower bound is simply the trace distance between $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, trivially seen to be tight. The upper bound is $\sqrt{1\ensuremath{-}\mathrm{tr}(\ensuremath{\rho}\ensuremath{\sigma})}$, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of ``unbiased decompositions,'' which may be of independent interest, and prove it for a few special cases. Finally, we point towards a link with a notion of nonclassicality known as preparation contextuality.

DOI
10.1103/physreva.86.044301

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Minimum-error state discrimination between two mixed states \ensuremathρ\ensuremath{\rho}\ensuremathρ and \ensuremathσ\ensuremath{\sigma}\ensuremathσ can be aided by the receipt of classical side information'' specifying which states from some convex decompositions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ apply in each run. We quantify this phenomena by the average trace distance and give lower and upper bounds on this quantity as functions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$. The lower bound is simply the trace distance between $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, trivially seen to be tight. The upper bound is $\sqrt{1\ensuremath{-}\mathrm{tr}(\ensuremath{\rho}\ensuremath{\sigma})}$, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of unbiased decompositions,'' which may be of independent interest, and prove it for a few special cases. Finally, we point towards a link with a notion of nonclassicality known as preparation contextuality.
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The trace distance between two quantum states, $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $\ensuremath{\rho}$ or $\ensuremath{\sigma}$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.
We introduce the concept of boundariness capturing the most efficient way of expressing a given element of a convex set as a probability mixture of its boundary elements. In other words, this number measures (without the need of any explicit topology) how far the given element is from the boundary. It is shown that one of the elements from the boundary can be always chosen to be an extremal element. We focus on evaluation of this quantity for quantum sets of states, channels, and observables. We show that boundariness is intimately related to (semi)norms that provide an operational interpretation of this quantity. In particular, the minimum error probability for discrimination of a pair of quantum devices is lower bounded by the boundariness of each of them. We proved that for states and observables this bound is saturated and conjectured this feature for channels. The boundariness is zero for infinite-dimensional quantum objects as in this case all the elements are boundary elements.
We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.
We derive a general lower bound on the minimum-error probability for ambiguous discrimination between arbitrary mmm mixed quantum states with given prior probabilities. When m=2m=2m=2, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out an upper bound on the minimum-error probability for ambiguous discrimination of mixed quantum states. Therefore, some sufficient conditions are obtained for the minimum-error probability attaining this bound. Finally, under the condition of the minimum-error probability attaining this bound, we compare the minimum-error probability for ambiguously discriminating arbitrary mmm mixed quantum states with the optimal failure probability for unambiguously discriminating the same states.
For the optimal success probability under minimum-error discrimination between $r\ensuremath{\ge}2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the known general bounds, lower and upper. We also present the example where the values of the new general lower and upper bounds on the optimal success probability are tighter than the values of most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.