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arXiv:1806.01816 · Physical Review A

Witnessing bipartite entanglement sequentially by multiple observers

Anindita Bera, Shiladitya Mal, Aditi Sen, Ujjwal Sen

Published Dec 3, 2018• 76 citations
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Abstract

We investigate sharing of bipartite entanglement in a scenario where half of an entangled pair is possessed and projectively measured by one observer, called Alice, while the other half is subjected to measurements performed sequentially, independently, and unsharply, by multiple observers, called Bobs. We find that there is a limit on the number of observers in this entanglement distribution scenario. In particular, for a two-qubit maximally entangled initial shared state, no more than 12 Bobs can detect entanglement with a single Alice for arbitrary, possibly unequal, sharpness parameters of the measurements by the Bobs. The number of Bobs remains unaltered for a finite range of near-maximal pure initial entanglement, a feature that also occurs in the case of equal sharpness parameters at the Bobs. Furthermore, we show that for nonmaximally entangled shared pure states, the number of Bobs decreases with the amount of initial entanglement, providing a coarse-grained but operational measure of entanglement.

DOI
10.1103/physreva.98.062304

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We investigate sharing of bipartite entanglement in a scenario where half of an entangled pair is possessed and projectively measured by one observer, called Alice, while the other half is subjected to measurements performed sequentially, independently, and unsharply, by multiple observers, called Bobs. We find that there is a limit on the number of observers in this entanglement distribution scenario. In particular, for a two-qubit maximally entangled initial shared state, no more than 12 Bobs can detect entanglement with a single Alice for arbitrary, possibly unequal, sharpness parameters of the measurements by the Bobs. The number of Bobs remains unaltered for a finite range of near-maximal pure initial entanglement, a feature that also occurs in the case of equal sharpness parameters at the Bobs. Furthermore, we show that for nonmaximally entangled shared pure states, the number of Bobs decreases with the amount of initial entanglement, providing a coarse-grained but operational measure of entanglement.
The problem of sharing quantum correlations is an interesting problem in the study of quantum information theory. Silva et al. proposed the study of sharing quantum nonlocality at first. They studied the fundamental limits on nonlocality, asking whether a single pair of entangled qubits could generate a long sequence of nonlocal correlations. At the same time, the sequential scenario was also introduced first, in which Alice and Bob each have half of a pair of entangled qubit states. The first Bob measures his half and then passes his part to a second Bob who measures again and so on. Obviously, even partial preservation of entanglement in a shared state in spite of a few sequences of local operations performed by the sharing parties can be important for information processing schemes in which entanglement is utilized as a resource. Thus, the problem of sharing quantum entanglement has also been extensively investigated. Recently, Srivastava et al. proved that there exist a class of T-states whose entanglement can be shared by arbitrarily many independent observers in [ Phys. Rev. A 2022 105 062413 ]. Here, we want to find whether there are other entangled states that can be shared entanglement arbitrarily many times. In this paper, we consider the problem of sharing quantum entanglement when the initial shared state is a two-qubit entangled Werner state. The goal is to maximize the number of Bobs that can share entanglement with a single Alice. By suitably choosing the entanglement witness operator and the unsharp measurement settings by the Bobs, we prove that there exist two-qubit entangled initial shared Werner states whose entanglement can be detected by arbitrarily many sequential observers Bobs with a single Alice. Then, we also consider the special case of the Werner state, that is, the maximally entangled state as the initial shared state. In this case, its entanglement can also be witnessed arbitrarily many times, and the number of Bobs increases with the decrease of parameter.
We investigate the scenario where an observer, Alice, shares a two-qubit state with an arbitrary number of observers, Bobs, via sequentially and independently recycling the qubit in possession of the first Bob. It is known that there exist entangled states which can be used to have an arbitrarily long sequence of Bobs who can violate the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. We show that there exist entangled states that do not violate the Bell inequality and whose entanglement can be detected by an arbitrary number of Bobs by suitably choosing the entanglement witness operator and the unsharp measurement settings by the Bobs. This proves that the set of states that can be used to witness entanglement sequentially is larger than those that can witness sequential violation of local realism. There exist, therefore, two-party quantum correlations that are Bell ``classical'', but whose entanglement ``nonclassicality'' can be witnessed sequentially and independently by an arbitrarily large number of observers at one end of the shared state with the single observer at the other end.
Entanglement and uncertainty relation are two focuses of quantum theory. We relate entanglement sharing to the entropic uncertainty relation in a $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional system via weak measurements with different pointers. We consider both the scenarios of one-sided sequential measurements in which the entangled pair is distributed to multiple Alices and one Bob and two-sided sequential measurements in which the entangled pair is distributed to multiple Alices and Bobs. It is found that the maximum number of observers sharing the entanglement strongly depends on the measurement scenarios, the pointer states of the apparatus, and the local dimension ddd of each subsystem, while the required minimum measurement precision to achieve entanglement sharing decreases to its asymptotic value with the increase of ddd. The maximum number of observers remain unaltered even when the state is not maximally entangled but has strong-enough entanglement.
Violation of a Bell inequality certifies that the underlying state must be entangled in a device-independent way, although there exist some entangled states which do not violate such an inequality. However, for every entangled state, it is possible to find a Hermitian operator called an entanglement witness that can detect entanglement through some local measurements in a device-dependent method, but implementation of wrong measurements may lead to fake detection of entanglement. To avoid such difficulties, measurement-device-independent entanglement witness (MDI-EW) based on a semiquantum nonlocal game was proposed, which is not only robust against wrong measurements but also against a specific kind of lossy detectors. We employ here a measurement-device-independent entanglement witness to detect entanglement in a scenario where half of an entangled pair is possessed by a single observer while the other half is with multiple observers performing unsharp measurements, sequentially, independently, and preserving entanglement as much as possible. Interestingly, we find that the numbers of successful observers who can detect entanglement, measurement-device-independently, both with equal and unequal sharpness parameters of the noisy measurements, are greater than that obtained with standard and Bell-inequality-based entanglement detection methods, reflecting its robustness. The entanglement contents of the sequentially shared states are also analyzed. Unlike other scenarios, our investigations also reveal that in this measurement-device-independent situation, states having entanglement in proximity to maximal remain entangled until there are two sequential observers, even if they measure sharply.
Two spatially separated observers, Alice and Bob, share a bipartite two-qubit entangled state and perform measurements to witness entanglement. After their measurements, they pass their qubits to a subsequent pair of observers who try to perform the same task independently, and so on. Here we ask: what is the maximum number of such pairs that can perform this task successfully? It has previously been conjectured that not more than one pair of observers can detect Clauser-Horne-Shimony-Holt ``Bell-nonlocal'' correlations in this setup. We prove that, on the contrary, entanglement can be witnessed by arbitrarily many pairs of observers. The dissimilar nature between entanglement and Bell-nonlocal correlations is therefore uncovered in a rather radical way, when considering sequentially acting pairs of observers. We prove this statement to be true when the initial shared state is any pure entangled state, a class of Bell-nonlocal mixed states, or a class of Bell-local entangled states.
Recently, it has been shown that at most two observers (Bobs) can sequentially demonstrate bipartite nonlocality with a spatially separated single observer (Alice) invoking a scenario where an entangled system of two spin-12\frac{1}{2}21​ particles are shared between a single Alice in one wing and several Bobs on the other wing, who act sequentially and independently of each other [Phys. Rev. Lett. 114, 250401 (2015)]. This has been probed through the quantum violations of CHSH inequality, i.e., when each observer performs two dichotomic measurements. In the present study we investigate how many Bobs can sequentially demonstrate bipartite nonlocality with a single Alice in the above scenario when the number of measurement settings per observer is increased. It is shown that at most two Bobs can exhibit bipartite nonlocality with a single Alice using local realist inequalities with three as well as four dichotomic measurements per observer. We then conjecture that the above feature remains unchanged contingent upon using local realist inequalities with nnn dichotomic measurements per observer, where nnn is arbitrary. We further present the robustness of bipartite nonlocality sharing in the above scenario against the entanglement and mixedness of the shared state.