Certifying the Classical Simulation Cost of a Quantum Channel

TL;DR

This paper develops device-independent tests to determine the classical communication cost of simulating quantum channels, providing bounds and characterizations for various channel types.

Contribution

It introduces new tests and Bell inequalities to bound and characterize the signaling dimension of quantum channels in a device-independent manner.

Findings

Single tests determine when noisy classical channels can be simulated with less communication.

Eight Bell inequalities fully characterize simulation conditions for four-outcome measurement channels.

Tight bounds on signaling dimension for partial replacer channels, especially the erasure channel.

Abstract

A fundamental objective in quantum information science is to determine the cost in classical resources of simulating a particular quantum system. The classical simulation cost is quantified by the signaling dimension which specifies the minimum amount of classical communication needed to perfectly simulate a channel's input-output correlations when unlimited shared randomness is held between encoder and decoder. This paper provides a collection of device-independent tests that place lower and upper bounds on the signaling dimension of a channel. Among them, a single family of tests is shown to determine when a noisy classical channel can be simulated using an amount of communication strictly less than either its input or its output alphabet size. In addition, a family of eight Bell inequalities is presented that completely characterize when any four-outcome measurement channel, such as a Bell measurement, can be simulated using one communication bit and shared randomness. Finally, we bound the signaling dimension for all partial replacer channels in $d$ dimensions. The bounds are found to be tight for the special case of the erasure channel.