Adaptive Advantage in Entanglement-Assisted Communications

Entanglement is known to boost the efficiency of classical communication. In distributed computation, for instance, exploiting entanglement can reduce the number of communicated bits or increase the probability to obtain a correct answer. Entanglement-assisted classical communication protocols usually consist of two successive rounds: first, a Bell test round, in which the parties measure their local shares of the entangled state, and then a communication round, where they exchange classical messages. Here, we go beyond this standard approach and investigate adaptive uses of entanglement: we allow the receiver to wait for the arrival of the sender’s message before measuring their share of the entangled state. We first show that such adaptive protocols improve the success probability in random access codes. Second, we show that once adaptive measurements are used, an entanglement-assisted bit becomes a strictly stronger resource than a qubit in prepare-and-measure scenarios. We briefly discuss the extension of these ideas to scenarios involving quantum communication and we identify resource inequalities.

Figure 1

(a) Nonadaptive scenario for classical communication. Alice and Bob use their inputs $(x,y)$ to perform local measurements on a shared state $\psi$. Alice’s outcome $m$ is relayed to Bob who uses it to map his measurement output $b^{\prime }$ into the final output $b$. (b) Adaptive scenario for classical communication. Alice uses $x$ to measure her share and relays the output $m$ to Bob who uses it to select the quantum measurement which produces the final output $b$.

Figure 2

RAC strategies based on EA bits or trits when the shared state is $|{\varphi }^{+}⟩$. Alice measures the observable $\{[(-1{)}^{x_1}Z+(-1{)}^{x_2}X]/\sqrt{2}\}$, obtains $a=\pm 1$, and sends a message $m$ to Bob. The black arrows illustrate the Bloch vectors of Bob’s local qubit after Alice’s measurement for each value of $(x_1,x_2)$ and $m$. Red lines represent Bob’s measurements. (a) Optimal (nonadaptive) EA bit strategy, where $m=a$. (b) A natural nonadaptive EA trit strategy, where $m=+1$ if $a=+1$ and $m=(-1,x_1)$ if $a=-1$. (c) Optimal (adaptive) EA trit strategy, where $m=+1$ if $a=+1$ and $m=(-1,x_1\oplus x_2)$ if $a=-1$.

Figure 3

Relations between various adaptive and nonadaptive resources.