Entanglement-assisted classical communication can simulate classical communication without causal order

Phenomena induced by the existence of entanglement, such as nonlocal correlations, exhibit characteristic properties of quantum mechanics distinguishing from classical theories. When entanglement is accompanied by classical communication, it enhances the power of quantum operations jointly performed by two spatially separated parties. Such a power has been analyzed by the gap between the performances of joint quantum operations implementable by local operations at each party connected by classical communication with and without the assistance of entanglement. In this work, we present a formulation for joint quantum operations connected by classical communication beyond special relativistic causal order but without entanglement and still within quantum mechanics. Using the formulation, we show that entanglement-assisting classical communication necessary for implementing a class of joint quantum operations called separable operations can be interpreted to simulate “classical communication” that does not respect causal order. Our results reveal a counterintuitive aspect of entanglement related to space-time.

Figure 1

Graphical representations of four types of quantum operations, ${\left\{{\mathcal{M}}_{o|i}\right\}}_o,{\left\{{\mathcal{M}}_o\right\}}_o,{\mathcal{M}}_{|i}$, and $\mathcal{M}$. A box represents a quantum operation. The upper terminals of a box represent inputs and the lower ones represent outputs. Dotted lines represent classical input or classical output and solid lines represent quantum input or output. (a) Probabilistic operations have a classical output corresponding to the outcome of the measurement. (b) Deterministic operations do not provide a classical output.

Figure 2

Joint quantum operations of two parties, Alice and Bob, consisting of local operations. Dotted arrows represent classical communication between local operations and solid arrows represent quantum communications between local operations. In an LOCC protocol, local operations are totally ordered. Local operations performed by different parties are connected only by classical communication, whereas local operations within each party are connected by quantum communication.

Figure 3

Deterministic joint quantum operation consisting of local operations connected by the $\infty$-shaped loop. Any element of LOCC* can be represented by this joint quantum operation and vice versa.

Figure 4

Inclusion relation between the classes of deterministic joint quantum operations CPTP, SEP, LOCC*, and LOCC. LOCC* is equivalent to the set of separable operations (SEP). LOCC is strictly smaller than SEP. LOCC* is strictly smaller than the set of CPTP.

Figure 5

Inclusion relation between the classes of linear CP maps. LOCC* is equivalent to the set of separable maps (SEP). LOCC is strictly smaller than SEP. SEP is strictly smaller than the set of CPTP. The intersection of SLOCC and the set of CPTP maps (TP SLOCC elements) is LOCC since any TP SLOCC element can be implemented by a LOCC protocol. The intersection of SLOCC* and the set of CPTP maps (TP SLOCC* elements) is LOCC*. Note that SEP is not included by SLOCC in this inclusion relation while every map in SEP can be simulated by SLOCC. A SLOCC map whose success probability is strictly less than 1 is defined by a linear CPTD map and thus not included in the set of CPTP maps. The success probability of simulating maps in SEP but not in LOCC by SLOCC is strictly less than 1.