Linking entanglement detection and state tomography via quantum 2-designs

Abstract

We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion for bipartite systems with equal dimensions is based on measurements from subsets of a quantum 2-design, e.g. mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios.For quantum information applications it is often more interesting to learn if multipartite quantum states are entangled than to identify quantum states themselves [1,2]. This is in fact what direct detection of entanglement executes, which utilizes an entanglement witness that works with individual measurements followed be postprocessing of the outcomes [3], to provide an experimentally feasible approach for this task [4]. Entanglement detection under less assumptions, for instance, when detectors are not trusted [5][6][7] or dimensions are unknown [8], is of practical significance for cryptographic applications.For the practical usefulness of entanglement detection, it is worth exploring the experimental resources. If a priori information about a quantum state is given, a set of entanglement witnesses may be constructed accordingly and exploited for entanglement detection. With no a priori information multiple entanglement witnesses may be required. One possible method is quantum state tomography which determines a ddimensional quantum state with O(d 2 ) measurements. Then, theoretical tools such as positive maps [9], e.g. partial transpose, or numerical tests involving semidefinite programming [10] can be applied. For entanglement witnesses, however, little is known about the minimal measurements for their realization. In fact, it may happen that repeating experiments for multiple witnesses may be less cost effective than state tomography [11], and quite possible that no useful information is obtained, neither for entanglement detection nor for quantum state identification. This raises questions on the usefulness of entanglement witnesses, in particular when a priori information about a particular state is not available.A useful experimental setup for entanglement detection may distinguish the largest collection of entangled states with as few measurements as possible. It is noteworthy that a tomographically complete measurement can ultimately identify a quantum state so that theoretical tools may completely determine whether it is entangled or separable. From a practical point of view, it would be therefore highly desirable that measurements for ...

Witnessing Entanglement In Compton Scattering Processes Via Mutually Unbiased Bases

“…Generally, witnesses based on MUBs have turned out to be powerful in detecting entanglement 22,23,41 and quite same considerations hold true if one uses symmetric informationally complete positive operator-valued measures (SICs), see ref. 42 , because in the case of complete sets MUBs or SICs are related by a quantum 2-design 43 .…”

Compounds of symmetric informationally complete measurements and their application in quantum key distribution

“…SIC POVMs are key tools for quantum state tomography [11][12][13], which has motivated their experimental realization in highdimensional Hilbert spaces [14][15][16]. Generally, SICs and SIC POVMs are used in a range of protocols: quantum key distribution (QKD) [17][18][19], entanglement detection [20][21][22], device-independent random number generation [23,24], dimension witnessing [25], and characterization of quantum devices [26][27][28][29][30]. Moreover, SICs have been studied in the context of quantum nonlocality [24,[31][32][33] and they have an interesting foundational role in QBism [34].…”

Entanglement criterion via general symmetric informationally complete measurements

“…In Chen et al and Xi et al [33,34], the authors presented separability criteria for both d-dimensional bipartite and multipartite systems based on general SIC-POVMs. Very recently, Bae et al [35] studied entanglement detection via quantum 2-designs, which includes SIC-POVMs as a special example. In Czartowki et al [36], the authors investigated the entanglement properties of multipartite systems with tight informationally complete measurements including SIC-POVMs.…”