Device-independent characterizations of a shared quantum state independent of any Bell inequalities

Abstract

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 that a quantum state with minimal size that is able to produce this correlation can always be pure. In this work, we first exhibit two deviceindependent 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 can be tight, and have various applications in quantum tasks. First, one of the characterizations allows us to bound the entanglement between Alice and Bob using Renyi entropies and also to bound the underlying Hilbert space dimension. Second, when the Hilbert space dimension bound is tight, the shared pure quantum state has to be maximally entangled. Third, the second characterization gives a sufficient condition that a Bell correlation cannot be generated by particular quantum states. We also show that our results can be generalized to the case of shared mixed states.Introduction.-In the study of quantum physics, frequently the internal workings of a quantum device are not exactly known. For example, it is often the case that we do not have sufficient knowledge of the internal physical structure, or the precision of the quantum controls is very limited, or even the devices we are using cannot be trusted. In these cases, it could be that the only reliable information available is the measurement statistics from observing the quantum system. However, sometimes we still want to draw nontrivial conclusions on the quantum properties of the involved system. This sounds like a challenging, or even impossible task, but it has been shown to be possible in many cases [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. These kinds of tasks are called device-independent as their application assumes only the correctness of quantum mechanics as a valid description of nature, and is independent of the internal workings of the devices used. Device-independence is a very valuable property in physical implementations of various quantum schemes. Typical examples of its usefulness include the transmission of information safely using untrusted devices, and easy monitoring of the overall performance of vulnerable quantum devices [11][12][13][14].We consider in this paper the setting of a Bell experiment, i.e., two spatially separated parties sharing a quantum state and performing local measurements on their subsystems. The corresponding statistics of the measurement outcomes is called a Bell correlation. It has been shown that the dimension and the entanglement of the underlying quantum state can be quantified in a device-independent way using only the Bell correlation data [5,[8][9][10]. In fact, some quantum states can even be pinned down completely by their violations of particular Bell inequalities, but this is only known to be possible for some special cases [1][2][...

Analytic semi-device-independent entanglement quantification for bipartite quantum states

“…Lastly, we would like to point out that E f (ρ) can also be lower bounded by the following alternative way. According to [44], E f (|ψ 1 ψ 1 |) can be lower bounded as the purity of Tr B (|ψ 1 ψ 1 |) can be upper bounded, where |ψ 1 is the principal component of ρ we have discussed above. Then by the continuous property of the entanglement of formation proved by [48,49], we can bound the gap between E f (ρ) and E f (|ψ 1 ψ 1 |).…”

“…But a major issue are raised in multipartite case and has to be addressed, which is the structure of multipartite entanglement is much more complicated. For example, because of the existence of Schmidt decompositions, entanglement quantification for bipartite pure states based on measurement statistics data can be achieved as addressed in [44], but Schmidt decompositions do not always exist for multipartite pure quantum states, thus this part has to be redeveloped carefully. Similarly, bounding coherent information or applying the continuous property of the entanglement of formation will be much more challenging in multipartite case.…”

“…Then according to[42], the Von Neumann entropy of ρ, denoted by S(ρ), can be upper bounded asS(ρ) ≤ −c i 9 i=1 log(c i ),where c 1 = 1 9 , andc 2 = • • • = c 9 = (1 − c 1 )/8.On the other hand, according to[43,44], the purity of ρ A = Tr B (ρ) (or ρ B = Tr A (ρ)) can also be upper bounded. Indeed, definef1(p) = min y 1 ,y 2 b 1 ,b 2 min ) = min x 1 ,x 2 a 1 ,a 2min y b p(a1b|x1y)p(a2b|x2y) 2 ,…”

Device-independent dimension test in a multiparty Bell experiment

“…As an example, suppose in addition to the correlation data, we are told that the shared quantum state can be transferred to a state of the form a i i m i n 1 å ñ = Ä | by local unitary operations. Then like in bipartite cases [20], we can give a nontrivial estimation for the amount of entanglement based on only the purity estimation of the reduced density matrices. It should be pointed out that because of the need of extra (quantum) information, this would no longer be fully device-independent, but still could be interesting nonetheless, as sometimes these assumptions may be reasonable.…”

Operational characterization of quantumness of unsteerable bipartite states

“…Since Eqs. (52) and (53) [40][41][42][43][44][45] of bipartite quantum correlations, which do not violate a Bell inequality, refers to the higher dimensionality of shared randomness needed to reproduce them in the classical simulation scenarios compared to that of the quantum systems producing the correlations. In this classical simulation scenario the local hidden variables are used by both the parties to generate the shared randomness.…”