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arXiv:1209.5643 · Physical Review Letters

Dimension Witnesses and Quantum State Discrimination

Nicolas Brunner, Miguel Navascués, Tamás Vértesi

Published Apr 8, 2013• 130 citations
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Abstract

Dimension witnesses allow one to test the dimension of an unknown physical system in a device-independent manner, that is, without placing assumptions about the functioning of the devices used in the experiment. Here we present simple and general dimension witnesses for quantum systems of arbitrary Hilbert space dimension. Our approach is deeply connected to the problem of quantum state discrimination, hence establishing a strong link between these two research topics. Finally, our dimension witnesses can distinguish between classical and quantum systems of the same dimension, making them potentially useful for quantum information processing.

DOI
10.1103/physrevlett.110.150501

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Dimension witnesses allow one to test the dimension of an unknown physical system in a device-independent manner, that is, without placing assumptions about the functioning of the devices used in the experiment. Here we present simple and general dimension witnesses for quantum systems of arbitrary Hilbert space dimension. Our approach is deeply connected to the problem of quantum state discrimination, hence establishing a strong link between these two research topics. Finally, our dimension witnesses can distinguish between classical and quantum systems of the same dimension, making them potentially useful for quantum information processing.
The dimension witness is an important concept in fundamental physics and quantum information processing which allows one to test the dimension of an unknown physical system in a device independent manner. Here, we report an experimental test of classical and quantum dimensions in a prepare and measure scenario through dimension witnesses based on quantum state discrimination. In our work, we have not only distinguished between quantum and classical systems of the same dimension (two, three, and four dimensions) but also distinguished between real and complex two-level quantum systems. We have also shown the strong link between dimension witnesses and quantum state discrimination which was introduced in N. Brunner, M. Navascu\'es, and T. V\'ertesi [Phys. Rev. Lett. 110, 150501 (2013)].
We address the problem of testing the dimensionality of classical and quantum systems in a "black-box" scenario. We develop a general formalism for tackling this problem. This allows us to derive lower bounds on the classical dimension necessary to reproduce given measurement data. Furthermore, we generalize the concept of quantum dimension witnesses to arbitrary quantum systems, allowing one to place a lower bound on the Hilbert space dimension necessary to reproduce certain data. Illustrating these ideas, we provide simple examples of classical and quantum dimension witnesses.
Analyzing the dimension of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. In this paper, we consider this problem in the prepare-and-measure scenario. Specifically, we provide a lower bound on the dimension of the prepared quantum systems, which is a function that only depends on the measurement statistics. Furthermore, we show that our bound performs well in several examples. In particular, we show that our bound provides insights into the notion of dimension witness, and we also use it to show that the sets of restricted-dimensional prepare-and-measure correlations are not always convex.
In quantum information processing, it is important to witness the entropy of the message in the device-independent way which was proposed recently [R. Chaves, J. B. Brask, and N. Brunner, Phys. Rev. Lett. 115, 110501 (2015)]. In this paper, we theoretically obtain the minimal quantum entropy for three widely used linear dimension witnesses, which is considered ``a difficult question.'' Then we experimentally test the classical and quantum entropy in a device-independent manner. The experimental results agree well with the theoretical analysis, demonstrating that entropy is needed in quantum systems that is lower than the entropy needed in classical systems with the given value of the dimension witness.
We report on an experimental test of classical and quantum dimension. We have used a dimension witness that can distinguish between quantum and classical systems of dimensions two, three, and four and performed the experiment for all five cases. The witness we have chosen is a base of semi-device-independent cryptographic and randomness expansion protocols. Therefore, the part of the experiment in which qubits were used is a realization of these protocols. In our work we also present an analytic method for finding the maximum quantum value of the witness along with corresponding measurements and preparations. This method is quite general and can be applied to any linear dimension witness.
Device-independent dimension witnesses provide a lower bound on the dimensionality of classical and quantum systems in a ``black box''scenario where only correlations between preparations, measurements, and outcomes are considered. We address the problem of the robustness of dimension witnesses, namely that to witness the dimension of a system or to discriminate between its quantum or classical nature, even in the presence of loss. We consider the case when shared randomness is allowed between preparations and measurements and provide a threshold in the detection efficiency such that dimension witnessing can still be performed.