Classical Cost of Transmitting a Qubit

Abstract

We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.

Almost Qudits in the Prepare-and-Measure Scenario

“…Indeed, even the fundamental unit of quantum information, namely, the qubit, is expressed in terms of the (minimal meaningful) quantum dimension. It is natural that much research has been devoted to the quantum dimension: the deviceindependent certification of it [1][2][3][4][5], investigating the cost of classically simulating qubits [6][7][8], the advantage of using d-dimensional quantum systems (qudits) over classical systems (dits) in useful tasks [9][10][11], and performing quantum protocols in experiments where nothing but the dimension is assumed [12][13][14][15]. A large number of experiments have followed (see, e.g., [16][17][18][19][20][21][22][23]).…”