Equivalence between simulability of high-dimensional measurements and high-dimensional steering

The effect of quantum steering arises from the judicious combination of an entangled state with a set of incompatible measurements. Recently, it was shown that this form of quantum correlations can be quantified in terms of a dimension, leading to the notion of genuine high-dimensional steering. While this naturally connects to the dimensionality of entanglement (Schmidt number), we show that this effect also directly connects to a notion of dimension for measurement incompatibility. More generally, we present a general connection between the concepts of steering and measurement incompatibility, when quantified in terms of dimension. From this connection, we propose an alternative twist on the problem of simulating quantum correlations. Specifically, we show how the correlations of certain high-dimensional entangled states can be exactly recovered using only shared randomness and lower-dimensional entanglement. Finally, we derive criteria for testing the dimension of measurement incompatibility and discuss the extension of these ideas to quantum channels.

Figure 1

Concepts and connections that appear in this work. (a) Quantum steering scenario. (b) A set of measurements is $n$ simulable if they can be replaced by an $n$-partially entanglement breaking channel ($n$-PEB) followed by some measurements. (c) Illustration of the Schmidt number (SN) of a bipartite state: the state of two five-level systems is a combination of states with only qubit entanglement, hence the overall state has SN of at most 2.

Figure 1

Concepts and connections that appear in this work. (a) Quantum steering scenario. (b) A set of measurements is $n$ simulable if they can be replaced by an $n$-partially entanglement breaking channel ($n$-PEB) followed by some measurements. (c) Illustration of the Schmidt number (SN) of a bipartite state: the state of two five-level systems is a combination of states with only qubit entanglement, hence the overall state has SN of at most 2.

Figure 2

Here $|{\psi }^{\lambda }\rangle$ represents some state that depends on the shared variable $\lambda$. (a) Local hidden-state model: One aims at simulating the assemblage with a separable state. (b) $n$ preparability: Simulation of an assemblage using states with low-dimensional entanglement. For a concrete realization see [13] and Sec. 4. (c) High-dimensional entanglement and steering properties of the isotropic state with local dimension $d=4$. The SN bounds can be found in [22] and in this work we translate known values on the $n$ simulability of all PVMs from [13] to thresholds on states being such that they can only lead to $n$-preparable assemblages under all projective measurements. In the figure this is referred to as the one-sided semi-device-independent Schmidt number (1-SDI-SN) under all PVMs. The bound for LHS models for all POVMs is from [31, 32].

Figure 2

Here $|{\psi }^{\lambda }\rangle$ represents some state that depends on the shared variable $\lambda$. (a) Local hidden-state model: One aims at simulating the assemblage with a separable state. (b) $n$ preparability: Simulation of an assemblage using states with low-dimensional entanglement. For a concrete realization see [13] and Sec. 4. (c) High-dimensional entanglement and steering properties of the isotropic state with local dimension $d=4$. The SN bounds can be found in [22] and in this work we translate known values on the $n$ simulability of all PVMs from [13] to thresholds on states being such that they can only lead to $n$-preparable assemblages under all projective measurements. In the figure this is referred to as the one-sided semi-device-independent Schmidt number (1-SDI-SN) under all PVMs. The bound for LHS models for all POVMs is from [31, 32].