POVMs and hidden variables

J. M. Jauch, C. Piron

Daniel Cavalcanti, Leonardo Guerini, Rafael Rabelo +1 more

We present a general criterion for deciding the existence of local-hidden-state (LHS) models and consequently local-hidden-variable (LHV) models for arbitrary states that can be implemented by semidefinite programming. This method can be applied to any given state and general (POVM) measurements and can also be adapted to generate random states with LHS models. As applications we provide an estimate of the fraction of two-qubit entangled states that have LHS models, and present new explicit examples of such states, including Bell-diagonal states, and noisy GHZ and W states. We also generate a list of 750000 new entangled two-qubit states with LHS model for projective measurements, 1500 for POVM measurements, and 150 genuine multipartite entangled three-qubit local states.

Jeffrey Bub

Andrzej Grudka, Paweł Kurzyński

Cabello and Nakamura have shown [A. Cabello, Phys. Rev. Lett. 90, 190401 (2003)10.1103/PhysRevLett.90.190401] that the Kochen-Specker theorem can be applied to two-dimensional systems if one uses positive operator-valued measures (POVM). We show that the contextuality in their models is not of the Kochen-Specker type, but it is rather a result of not keeping track of the whole system on which the measurement is performed. This is connected to the fact that there is no one-to-one correspondence between the POVM elements and projectors on the extended Hilbert space, and the same POVM element has to originate from two different projectors when used in Cabello-Nakamura models. Moreover, we propose a hidden-variable formulation of the above models.

James E. Troupe, Jacob Farinholt

In this article we present a new prepare and measure quantum key distribution protocol that uses an experimentally accessible measure of single qubit contextuality to warranty the security of the quantum channel. The definition of contextuality used is that due to Spekkens in which any underlying hidden variable model of the physical system is noncontextual if the probability distribution of the model's hidden variables is independent of the preparation and measurement context. Under this more general definition of noncontextuality the measurement outcomes of a single qubit can be shown to require a contextual model in order to reproduce the results of some non-projective, positive operator valued measurements (POVMs). The proposed protocol utilizes a particular set of POVMs to exhibit the degree of contextuality of the qubits exiting the QKD system's quantum channel. The contextuality measure is defined in terms of the weak values of a set of projective measurements that are implemented as POVMs using weak measurements. The resulting QKD protocol should have a key rate that is equal to or greater than BB84 since none of the sifted key is used to test for the presence of an eavesdropper. Additionally, the new protocol is shown to be immune to detector based attacks.

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