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Register a new userDistinguishability measures between ensembles of quantum states
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A quantum ensemble ${(p_x, rho_x)}$ is a set of quantum states each occurring randomly with a given probability. Quantum ensembles are necessary to describe situations with incomplete a priori information, such as the output of a stochastic quantum channel (generalized measurement), and play a central role in quantum communication. In this paper, we propose measures of distance and fidelity between two quantum ensembles. We consider two approaches: the first one is based on the ability to mimic one ensemble given the other one as a resource and is closely related to the Monge-Kantorovich optimal transportation problem, while the second one uses the idea of extended-Hilbert-space (EHS) representations which introduce auxiliary pointer (or flag) states. Both types of measures enjoy a number of desirable properties. The Kantorovich measures, albeit monotonic under deterministic quantum operations, are not monotonic under generalized measurements. In contrast, the EHS measures are. We present operational interpretations for both types of measures. We also show that the EHS fidelity between ensembles provides a novel interpretation of the fidelity between mixed states--the latter is equal to the maximum of the fidelity between all pure-state ensembles whose averages are equal to the mixed states being compared. We finally use the new measures to define distance and fidelity for stochastic quantum channels and positive operator-valued measures (POVMs). These quantities may be useful in the context of tomography of stochastic quantum channels and quantum detectors.
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The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper, we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity, and we evaluate their performance using simulators of quantum computers. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as quantum combs). The fidelity-based algorithms offer novel physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate these algorithms by using a variational approach with parameterized quantum circuits and find that they converge well for the examples that we consider.
Quantifying how distinguishable two stochastic processes are lies at the heart of many fields, such as machine learning and quantitative finance. While several measures have been proposed for this task, none have universal applicability and ease of use. In this Letter, we suggest a set of requirements for a well-behaved measure of process distinguishability. Moreover, we propose a family of measures, called divergence rates, that satisfy all of these requirements. Focussing on a particular member of this family -- the co-emission divergence rate -- we show that it can be computed efficiently, behaves qualitatively similar to other commonly-used measures in their regimes of applicability, and remains well-behaved in scenarios where other measures break down.
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In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in $m_1otimes m_2otimesldotsotimes m_n$ , where the quantum system $m_1$ belongs to Alice, $m_2$ belongs to Bob, ldots and $m_n$ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in $m_1otimes m_2otimes m_3$. With the conclusion in $m_1otimes m_2otimes m_3$, we prove distinguishability or indistinguishability of some quantum states. At last, we give the $n$-party distinguishable quantum states in $m_1otimes m_2otimescdotsotimes m_n$. Our study further reveals quantum nonlocality in multipartite high-dimensional.
In this article, we show a sufficient and necessary condition for locally distinguishable bipartite states via one-way local operations and classical communication (LOCC). With this condition, we present some minimal structures of one-way LOCC indistinguishable quantum state sets. As long as an indistinguishable subset exists in a state set, the set is not distinguishable. We also list several distinguishable sets as instances.
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