Abstract Multiple sequential observers, each independently obtaining preparation contextual outcome statistics through measurements on an ensemble, share the preparation contextuality (PC) enabled by the ensemble. We present a circuit model for high-dimensional weak measurements to explore PC sharing in high-dimensional systems using weak measurements with different pointers. We identify the relationship between the maximum number of sequential observers, system dimension, and pointer type, and find that PC can be shared by at most two sequential observers. For two observers to successfully share PC, the required minimum measurement strength increases with the dimension. Additionally, relaxing the equal-strength constraint on the first observer’s measurements expands the strength adjustable region from a linear interval to a planar area. Finally, we present a concrete implementation of our circuit model for unsharp measurement, with applications in PC sharing and d -level parity-oblivious random access code game. Our results provide theoretical guidance for experimental realization of high-dimensional weak measurements and deepen understanding of the relationship between PC and measurement in high-dimensional systems.
Multiple observers who independently harvest nonclassical correlations from a single physical system share the system's ability to enable quantum correlations. We show that any number of independent observers can share the preparation contextual outcome statistics enabled by state ensembles in quantum theory. Furthermore, we show that even in the presence of any amount of white noise, there exists quantum ensembles that enable such shared preparation contextuality. The findings are experimentally realised by applying sequential unsharp measurements to an optical qubit ensemble which reveals three shared demonstrations of preparation contextuality.
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.
In the work by Silva et al. [Phys. Rev. Lett. 114, 250401 (2015)], the sharing of nonlocality by multiple observers was demonstrated through the quantum violation of Clauser-Horne-Shimony-Halt inequality. We provide a scheme for the sharing of nonlocality and nontrivial preparation contextuality sequentially through the quantum violation of a family of Bell's inequalities, where Alice and Bob perform ${2}^{n\ensuremath{-}1}$ and
n numbers of measurements of dichotomic observables, respectively. For this, we consider that Alice always performs projective measurement and multiple Bobs sequentially perform unsharp measurement. We show that when Bob's choices of measurement settings are unbiased, a maximum of two Bobs can sequentially share the nonlocality through the violation of our inequalities. Further, we show that the local bound of the aforementioned family of inequalities gets reduced if nontrivial preparation noncontextuality assumptions are further imposed. Then there is a chance to share the nontrivial preparation contextuality for a greater number of Bobs than that of nonlocality. We demonstrate that the nontrivial preparation contextuality can be sequentially shared by arbitrary numbers of Bob for unbiased choices of his measurement settings.
A dimension witness provides a device-independent certification of the minimal dimension required to reproduce the observed data without imposing assumptions on the functioning of the devices used to generate the experimental statistics. In this paper, we provide a family of Bell expressions where Alice and Bob perform ${2}^{n\ensuremath{-}1}$ and
n number of dichotomic measurements, respectively, which serve as the device-independent dimension witnesses of Hilbert space of
2m dimensions with $m=1,2,...,{2}^{\ensuremath{\lfloor}n/2\ensuremath{\rfloor}}$. The family of Bell expressions considered here determines the success probability of a communication game known as the
n-bit parity-oblivious random access code. The parity obliviousness constraint is equivalent to the preparation noncontextuality assumption in an ontological model of an operational theory. For any given $n\ensuremath{\ge}3$, if such a constraint is imposed on the encoding scheme of the random access code, then the local bound of the Bell expression reduces to the preparation noncontextual bound. We provide explicit examples for the
n=4and5 case to demonstrate that the relevant Bell expressions certify the qubit and two-qubit system, and for the
n=6 case to demonstrate that the relevant Bell expression certifies the qubit, two-qubit, and three-qubit systems. We further demonstrate the sharing of quantum preparation contextuality by multiple Bobs sequentially to examine whether the number of Bobs sharing the preparation contextuality is dependent on the dimension of the system. We provide explicit examples of
n=5 and 6 to demonstrate that the number of Bobs sequentially sharing the contextuality remains the same for any of the
2m-dimensional systems.
Based on a sequential communication game, semi-device-independent certification of an unsharp instrument has recently been demonstrated [K. Mohan, A. Tavakoli, and N. Brunner, New J. Phys. 21, 083034 (2019)]. In this paper, we provide semi-device-independent self-testing protocols in the prepare-measure scenario to certify multiple unsharpness parameters along with the states and the measurement settings. This is achieved through the sequential quantum advantage shared by multiple independent observers in a suitable communication game known as the parity-oblivious random-access code. We demonstrate that in the three-bit parity-oblivious random-access code at most three independent observers can sequentially share quantum advantage. The optimal pair (triple) of quantum advantages enables us to uniquely certify the qubit states, the measurement settings, and the unsharpness parameter(s). The practical implementation of a given protocol involves inevitable losses. In a suboptimal scenario, we derive a certified interval within which a specific unsharpness parameter has to be confined. We extend our treatment to the four-bit case and show that at most two observers can share quantum advantage for the qubit system. Further, we provide an outline to argue that four sequential observers can share the quantum advantage for the two-qubit system, thereby enabling the certification of three unsharpness parameters.
Based on the quantum violation of the bipartite Bell inequality, it has been established that the sharing of nonlocality can be demonstrated for at most two sequential observers at one end and at most one pair of observers at both ends. In this work we study the sharing of nonlocality and preparation contextuality based on a bipartite Bell inequality, involving arbitrary
n measurements by one party and ${2}^{n\ensuremath{-}1}$ measurements by another party. Such a Bell inequality has two bounds, the local bound and the preparation noncontextual bound, which is smaller than the local bound. We show that while nonlocality can be shared only by the first pair of the sequential observers, the preparation contextuality can be shared by an arbitrary pair of independent sequential observers at both ends.