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arXiv:2012.13876 · Physical Review A

Device-independent certification of the Hilbert-space dimension using a family of Bell expressions

A. K. Pan, Shyam Sundar Mahato

Published Nov 24, 2020• 29 citations
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Abstract

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 ${2}^{m}$ 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 $n=4\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}5$ 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 ${2}^{m}$-dimensional systems.

DOI
10.1103/physreva.102.052221

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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 2n\ensuremath−1{2}^{n\ensuremath{-}1}2n\ensuremath−1 and nnn number of dichotomic measurements, respectively, which serve as the device-independent dimension witnesses of Hilbert space of 2m{2}^{m}2m dimensions with m=1,2,...,2\ensuremath⌊n/2\ensuremath⌋m=1,2,...,{2}^{\ensuremath{\lfloor}n/2\ensuremath{\rfloor}}m=1,2,...,2\ensuremath⌊n/2\ensuremath⌋. The family of Bell expressions considered here determines the success probability of a communication game known as the nnn-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≥3n\ensuremath{\ge}3n\ensuremath≥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=4and5n=4\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}5n=4and5 case to demonstrate that the relevant Bell expressions certify the qubit and two-qubit system, and for the n=6n=6n=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=5n=5n=5 and 6 to demonstrate that the number of Bobs sequentially sharing the contextuality remains the same for any of the 2m{2}^{m}2m-dimensional systems.
We provide an interesting two-party parity-oblivious communication game whose success probability is solely determined by the Bell expression. The parity-oblivious condition in an operational quantum theory implies the preparation noncontextuality in an ontological model of it. We find that the aforementioned Bell expression has two upper bounds in an ontological model: the usual local bound and a nontrivial preparation noncontextual bound arising from the nontrivial parity-oblivious condition, which is smaller that the local bound. We first demonstrate the communication game when both Alice and Bob perform three measurements of dichotomic observables in their respective sites. The optimal quantum value of the Bell expression in this scenario enables us to device-independently self-test the maximally entangled state and trine set of observables, three-outcome qubit positive-operator-valued measures and 1.58 bits of local randomness. Further, we generalize the above communication game in that both Alice and Bob perform the same but arbitrary (odd) number (n>3n>3n>3) of measurements. Based on the optimal quantum value of the relevant Bell expression for any arbitrary nnn, we also demonstrate device-independent self-testing of the state and measurements.
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 nnn 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.
Abstract We present a self-testing protocol in a constrained prepare-measure scenario, based on a communication game known as parity-oblivious multiplexing (POM) task. In this scenario, a parity-oblivious constraint is imposed on the preparations, allowing us to define a classical bound derived from a preparation noncontextual ontological model. We demonstrate that the optimal quantum value exceeds preparation noncontextual bound and therefore enables self-testing of the preparations and the measurement devices. Contrasting the prevailing consensus that the self-testing in prepare-measure scenario inevitably requires an upper bound on the dimension of the quantum system, we derive the optimal quantum success probability of the POM task devoid of assuming the dimension of the quantum system. Furthermore, by proving the existence of a unitary, we show that the optimal preparations and measurements in an unknown but finite dimensional Hilbert space, responsible for the observed input–output correlations, can be mapped, via a unitary, onto a known finite-dimensional quantum system. Our results thus pave the way for scalable, single system based certification protocols in the prepare-measure scenario.
The quantum ontological feature of contextuality apart from being central to foundations of quantum theory forms the basis of quantum advantage in a multitude of information processing tasks. In particular, the contextuality of preparation procedures was shown to power a particular two-party information processing task “parity oblivious multiplexing” (Spekkens et al. Phys Rev Lett 102:010401 (2009). Specifically, it was shown that there exists a limit to how well any preparation noncontextual theory can perform in this task. This limit constitutes a noncontextuality inequality. Moreover, the authors demonstrated quantum violation of this inequality along with preparation contextuality associated with the ontic description underlying two-level completely mixed quantum state. In this work, we extend these arguments to apply to arbitrary dimensions by introducing a class of two-party information processing tasks, namely d-level parity oblivious random access codes. We analytically obtain classical (or equivalently preparation noncontextual) bounds on the success probability for these tasks for arbitrary d. For each value of d, this bound constitutes a unique noncontextuality inequality. Remarkably, these bounds are independent of the amount of communication. Furthermore, we find a classical protocol utilizing a dc=dd_\mathrm{c}=ddc​=d -dimensional classical message which saturates this bound. In order to establish nontriviality of these inequalities, we provide evidence of significant quantum violations. Specifically, by numerical techniques, we show that for d=3,…,10d=3,\ldots ,10d=3,…,10 , the noncontextuality bound is violated by quantum theory. (1) We provide explicit quantum protocols which violate the associated noncontextuality inequality for d=3,4,5d=3,4,5d=3,4,5 employing dq=dd_\mathrm{q}=ddq​=d -leveled quantum systems. (2) Using see-saw semi-definite programming (SDP) technique, we find evidence (lower bounds) of significant quantum violation of these inequalities for d=3,…,10d=3,\ldots ,10d=3,…,10 . (3) With the help of state-of-the-art (NPA-hierarchy like) SDP technique, we provide upper bounds (independent of the dimension of the involved quantum systems) on quantum violation for d=3,…,10d=3,\ldots ,10d=3,…,10 . The introduced class of information tasks, thus, provides for operational depiction of preparation contextuality of the ontic description underlying mixed higher dimensional quantum systems.
We investigate the prepare-and-broadcast scenario, a multipartite extension of dimension-constrained prepare-and-measure experiments in which a quantum system is distributed to multiple receivers. We derive fundamental trade-offs between prepare-and-measure witnesses, Bell nonlocality, and quantum random access code performance. We further develop a semi-device-independent randomness certification framework based on prepare-and-broadcast witnesses, showing that the maximal quantum violation certifies two bits of joint randomness, exceeding the limit achievable from the CHSH inequality while remaining robust to noise. Finally, we show that the prepare-and-broadcast scenario naturally accommodates stronger adversarial models in which the eavesdropper retains a quantum system correlated with the measurement device, providing a natural framework for semi-device-independent randomness certification against quantum side information.
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.