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Physical Review A

Hardy's test as a device-independent dimension witness

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

Published Aug 3, 2015• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.92.022302

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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.
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.
In this Letter, we compute an analogue of Tsirelson's bound for Hardy's test of nonlocality, that is, the maximum violation of locality constraints allowed by the quantum formalism, irrespective of the dimension of the system. The value is found to be the same as the one achievable already with two-qubit systems, and we show that only a very specific class of states can lead to such maximal value, thus highlighting Hardy's test as a device-independent self-test protocol for such states. By considering realistic constraints in Hardy's test, we also compute device-independent upper bounds on this violation and show that these bounds are saturated by two-qubit systems, thus showing that there is no advantage in using higher-dimensional systems in experimental implementations of such a test.
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.
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.
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.
Recent years have witnessed a rapidly growing interest in high-dimensional quantum entanglement for fundamental studies as well as towards novel applications. Therefore, the ability to verify entanglement between physical qudits, ddd-dimensional quantum systems, is of crucial importance. To show nonclassicality, Hardy's paradox represents ``the best version of Bell's theorem'' without using inequalities. However, so far it has only been tested experimentally for bidimensional vector spaces. Here, we formulate a theoretical framework to demonstrate the ladder proof of Hardy's paradox for arbitrary high-dimensional systems. Furthermore, we experimentally demonstrate the ladder proof by taking advantage of the orbital angular momentum of high-dimensionally entangled photon pairs. We perform the ladder proof of Hardy's paradox for dimensions 3 and 4, both with the ladder up to the third step. Our paper paves the way towards a deeper understanding of the nature of high-dimensionally entangled quantum states and may find applications in quantum information science.