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arXiv:2304.05791 · Physical Review A

Sequential sharing of two-qudit entanglement based on the entropic uncertainty relation

Ming‐Liang Hu, Heng Fan

Published Jul 20, 2023• 23 citations
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Abstract

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 $d$ of each subsystem, while the required minimum measurement precision to achieve entanglement sharing decreases to its asymptotic value with the increase of $d$. The maximum number of observers remain unaltered even when the state is not maximally entangled but has strong-enough entanglement.

DOI
10.1103/physreva.108.012423

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Entanglement and uncertainty relation are two focuses of quantum theory. We relate entanglement sharing to the entropic uncertainty relation in a (d\ifmmode×\else\texttimes\fid)(d\ifmmode\times\else\texttimes\fi{}d)(d\ifmmode×\else\texttimes\fid)-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.
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.
Nonlocality and quantum measurement are two fundamental topics in quantum theory and their interplay attracts intensive focuses since the discovery of Bell theorem. Recently, nonlocality sharing among multiple observers with one entangled pair has been predicted and experimentally observed by generalized quantum measurement---weak measurement. However, only the one-sided sequential case, i.e., one Alice and multiple Bobs, is widely discussed and little is known about the two-sided case. Here, we theoretically and experimentally explore the nonlocality sharing in the two-sided sequential measurements case in which one entangled pair is distributed to multiple Alices and Bobs. We experimentally observed double Einstein-Podolsky-Rosen (EPR) steering among four observers in a photonic system. In the case that all observers adopt the same measurement strength of the weak measurement, it is observed that double EPR steering can be demonstrated simultaneously. The results not only deepen our understanding of relation between sequential measurements and nonlocality but also may find important applications in many quantum information tasks, such as randomness certification.
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.
It was showed in [Phys. Rev. Lett. 125 090401 (2020)] that there exist unbounded number of independent Bobs who can share quantum nonlocality with a single Alice by performing sequentially measurements on the Bob’s half of the maximally entangled pure two-qubit state. However, from practical perspectives, errors in entanglement generation and noises in quantum measurements will result in the decay of nonlocality in the scenario. In this paper, we analyze the persistency and termination of sharing nonlocality in the noisy scenario. We first obtain the two sufficient conditions under which there exist n independent Bobs who can share nonlocality with a single Alice under noisy measurements and the noisy initial two qubit entangled state. Analyzing the two conditions, we find that the influences on persistency under different kinds of noises can cancel each other out. Furthermore, we describe the change patterns of the maximal nonlocality-sharing number under the influence of different noises. Finally, we extend our investigation to the case of arbitrary finite-dimensional systems.
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.