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arXiv:1407.2146

Device independent Schmidt rank witness by using Hardy paradox

Amit Mukherjee, Arup Roy, Some Sankar Bhattacharya, Subhadipa Das, Md. Rajjak Gazi, Manik Banik

Published Jul 8, 2014• 2 citations
arXivHTMLPDF

Abstract

Schmidt rank of bipartite pure state serves as a testimony of entanglement. It is a monotone under local operation + classical communications (LOCC) and puts restrictions in LOCC convertibility of quantum states. Identifying the Schmidt rank of an unknown quantum state therefore seek importance from information theoretic perspective. In this work it is shown that a modified version of Hardy's argument, which reveals the contradiction of quantum theory with local realism, turns out to be useful for inspecting the minimal Schmidt rank of the unknown state and hence also the minimal dimension of the system. Use of Hardy's test in such task provides a practical advantage: the Schmidt rank can be determined without knowing the detailed functioning of the experimental devices i.e., Hardy's test suffices to be a device independent Schmidt rank witness.

DOI
10.48550/arxiv.1407.2146

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Schmidt rank of bipartite pure state serves as a testimony of entanglement. It is a monotone under local operation + classical communications (LOCC) and puts restrictions in LOCC convertibility of quantum states. Identifying the Schmidt rank of an unknown quantum state therefore seek importance from information theoretic perspective. In this work it is shown that a modified version of Hardy's argument, which reveals the contradiction of quantum theory with local realism, turns out to be useful for inspecting the minimal Schmidt rank of the unknown state and hence also the minimal dimension of the system. Use of Hardy's test in such task provides a practical advantage: the Schmidt rank can be determined without knowing the detailed functioning of the experimental devices i.e., Hardy's test suffices to be a device independent Schmidt rank witness.
Abstract We investigate the dimensionality of bipartite quantum systems by construction of a device-independent null witness test. This test assesses whether a given bipartite state conforms with the expected quantum dimension, Schmidt number, and distinguishes between real and complex spaces. By employing local measurements on each party, the proposed method aims to determine the minimal rank. By performing an experimental demonstration on IBM Quantum devices, we prove the exceptional accuracy of the test and its usefulness in diagnostics beyond routine calibrations. One of the tests shows agreement with theoretical expectations within statistical errors. However, the second test failed by more than 6 standard deviations, indicating unspecified parasitic entanglements, with no known simple origin.
Knowing the dimension of an unknown physical system has practical relevance, as dimensionality plays an important role in various information theoretic tasks. In this work we show that a modified version of Hardy's argument, which reveals the contradiction of quantum theory with local realism, turns out to be useful for inspecting the minimal subsystem dimension of an unknown correlated quantum system. The use of Hardy's test in this task has a novel advantage: the subsystem dimension can be determined without knowing the detailed functioning of the experimental devices; i.e., Hardy's test suffices to be a device-independent dimension witness.
Abstract In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: the vacuum in quantum field theory, Araki–Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.
Bipartite quantum states with higher Schmidt numbers have been shown to outperform those with lower Schmidt numbers in various quantum information-processing tasks, highlighting the operational advantage of entanglement dimensionality. Certifying the Schmidt number of such states is therefore crucial for efficient resource utilization. Ideally, this certification should rely as little as possible on the certifying devices to ensure robustness against their potential imperfections. Fully device-independent certification via Bell-nonlocal games offers strong robustness but suffers from fundamental limitations: It cannot certify the Schmidt number of all entangled states. We demonstrate that this insufficiency of Bell-nonlocal games is not limited to entangled states that do not exhibit Bell nonlocality. Specifically, we prove the existence of Bell-nonlocal states whose Schmidt number cannot be certified by any Bell-nonlocal game when the parties are restricted to local projective measurements. To overcome this, we develop a measurement-device-independent certification method based on semiquantum nonlocal games, which assume trusted preparation devices but treat measurement devices as black boxes. We prove that for any bipartite state with a Schmidt number exceeding rrr, a semiquantum nonlocal game exists that can certify its Schmidt number. Finally, we provide an explicit construction of such a semiquantum nonlocal game based on an optimal Schmidt number witness operator.
In a Bell experiment two parties share a quantum state and perform local measurements on their subsystems separately, and the statistics of the measurement outcomes are recorded as a Bell correlation. For any Bell correlation, it turns out a quantum state with minimal size that is able to produce this correlation can always be pure. In this work, we first exhibit two characterizations for the pure state that Alice and Bob share using only the correlation data. Specifically, we give two conditions that the Schmidt coefficients must satisfy, which have very broad applications in many quantum tasks. First, one of the characterizations allows us to bound the entanglement between Alice and Bob using Renyi entropies and also to estimate the underlying Hilbert space dimension. Second, we provide examples to show that our results can be tight, where the shared quantum state is characterized completely. In this situation, our result serves as a general theoretical tool to self-test bipartite pure quantum states in a device-independent manner. Third, the second characterization gives a sufficient condition that a Bell correlation cannot be generated by maximally entangled states of certain dimension. We also show that our results can be generalized to the case of shared mixed states.
The Schmidt number is an important kind of characterization of quantum entanglement. Quantum states with higher Schmidt numbers demonstrate significant advantages in various quantum information processing tasks. By deriving a class of k-positive linear maps based on symmetric measurements, we present new Schmidt-number witnesses of class (k + 1). By detailed example, we show that our Schmidt number witnesses identify better the Schmidt number of quantum states in high-dimensional systems. Furthermore, we note that the Fedorov ratio, which coincides with the Schmidt number for pure Gaussian states and provides a close approximation in non-Gaussian cases such as spontaneous parametric down-conversion, serves as an experimentally accessible tool for validating the proposed (k +1)-class Schmidt-number witnesses.