If a measurement is made on one half of a bipartite system, then, conditioned on the outcome, the other half has a new reduced state. If these reduced states defy classical explanation—that is, if shared randomness cannot produce these reduced states for all possible measurements—the bipartite state is said to be steerable. Determining which states are steerable is a challenging problem even for low dimensions. In the case of two-qubit systems, a criterion is known for T-states (that is, those with maximally mixed marginals) under projective measurements. In the current work, we introduce the concept of keyring models—a special class of local hidden state models. When the measurements made correspond to real projectors, these allow us to study steerability beyond T-states. Using keyring models, we completely solve the steering problem for real projective measurements when the state arises from mixing a pure two-qubit state with uniform noise. We also give a partial solution in the case when the uniform noise is replaced by independent depolarizing channels.
We are grateful to Kim Winick for numerous helpful discussions, to Emanuel Knill, Sania Jevtic, Stephen Jordan, and Chau Nguyen for useful feedback on an earlier version of the manuscript, and to Nicholas Brunner, Daniel Cavalcanti, and Ivan Supic for pointers to the literature. R.C. is supported by the EPSRC’s Quantum Communications Hub (Grant No. EP/M013472/1) and by an EPSRC First (Grant No. EP/P016588/1). C.A.M. and Y.S. were supported in part by US NSF Grant Nos. 1500095, 1526928, and 1717523. Y.S. was also supported in part by University of Michigan.
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