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The communication cost of simulating POVMs over maximally entangled qubits

Ricard Ravell Rodríguez

Published Sep 9, 2019• 0 citations
OpenAlex

Abstract

In [Toner and Bacon, Phys. Rev. Lett. 91, 187904 (2003)], $1$ bit of communication was proven to be enough to simulate the statistics of local projective measurements over the maximally entangled state. Ever since then, the question of whether $1$ bit is also enough for the case of generalized measurements has been open. In this thesis, we retort to inefficiency-resistant Bell functionals, a powerful technique to prove lower bounds communication complexity, to numerically study this question. The results obtained suggest that, indeed, as is the case with projective measurements, $1$ bit of communication suffices to simulate POVMs over maximally entangled qubits.

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In [Toner and Bacon, Phys. Rev. Lett. 91, 187904 (2003)], 111 bit of communication was proven to be enough to simulate the statistics of local projective measurements over the maximally entangled state. Ever since then, the question of whether 111 bit is also enough for the case of generalized measurements has been open. In this thesis, we retort to inefficiency-resistant Bell functionals, a powerful technique to prove lower bounds communication complexity, to numerically study this question. The results obtained suggest that, indeed, as is the case with projective measurements, 111 bit of communication suffices to simulate POVMs over maximally entangled qubits.
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.
We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, |ΨAB⟩=p |00⟩+1−p |11⟩ (with 1/2≤p≤1 ). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when 2p(1−p)2p−1log⁡(p1−p)+2(1−p)≤1 , approximately 0.835≤p≤1 , we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero ( p→1 ). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.
Abstract According to Bell’s theorem, certain entangled states cannot be simulated classically using local hidden variables (LHV). Suppose that we can augment LHV by some amount of classical communication. The question then arises as to how many bits are needed to simulate entangled states? There is very strong evidence that a single bit of communication is powerful enough to simulate projective measurements on any two-qubit entangled state. However, the problem of simulating measurements on higher-dimensional systems remains largely unexplored. In this study, we present Bell-like scenarios, even with three inputs per party, in which bipartite correlations resulting from measurements on higher-dimensional states cannot be simulated with a single bit of communication. We consider the case where the communication direction is fixed and the case where it is bidirectional. To this end, we introduce constructions based on parallel repetition of pseudo-telepathy games and an original algorithm based on branch-and-bound technique to compute the one-bit classical bound. Two copies of emblematic Bell expressions, such as the Magic square pseudo-telepathy game, prove to be particularly powerful, requiring a 16 × 16 state to beat the bidirectional one-bit classical bound, and look a promising candidate for implementation on an optical platform.
Bell's theorem states that Local Hidden Variables (LHVs) cannot fully explain the statistics of measurements on some entangled quantum states. It is natural to ask how much supplementary classical communication would be needed to simulate them. We study two long-standing open questions in this field with neural network simulations and other tools. First, we present evidence that all projective measurements on partially entangled pure two-qubit states require only one bit of communication. We quantify the statistical distance between the exact quantum behaviour and the product of the trained network, or of a semianalytical model inspired by it. Second, while it is known on general grounds (and obvious) that one bit of communication cannot eventually reproduce all bipartite quantum correlation, explicit examples have proved evasive. Our search failed to find one for several bipartite Bell scenarios with up to 5 inputs and 4 outputs, highlighting the power of one bit of communication in reproducing quantum correlations.
We demonstrate how to universally simulate ensemble statistics of projective local measurements on any nnn-qubit state shared among nnn observers with classical communication and shared randomness. Our technique originates from protocols designed to simulate quantum nonlocality [in Horizons of the Mind (Springer, Cham, 2014)] and classical simulation of quantum circuits [Phys. Rev. Lett. 115, 070501 (2015)]. The protocol preserves three crucial aspects of the simulated quantum scenario in contrast to other approaches: no involvement of additional parties, none of the observers knows the global state of the system, and local measurement settings remain undisclosed. The simulation cost grows exponentially with the number of observers.
Dense coding is the seminal example of how entanglement can boost qubit communication, from sending one bit to sending two bits. This is made possible by projecting separate particles onto a maximally entangled basis. We investigate more general communication tasks, in both theory and experiment, and show that simpler measurements enable strong and sometimes even optimal entanglement-assisted qubit communication protocols. Using only partial Bell state analysers for two qubits, we demonstrate quantum correlations that cannot be simulated with two bits of classical communication. Then, we show that there exists an established and operationally meaningful task for which product measurements are sufficient for the strongest possible quantum predictions based on a maximally entangled two-qubit state. Our results reveal that there are scenarios in which the power of entanglement in enhancing quantum communication can be harvested in simple and scalable optical experiments.