Causal Limit on Quantum Communication

The capacity of a channel is known to be equivalent to the highest rate at which it can generate entanglement. Analogous to entanglement, the notion of a causality measure characterizes the temporal aspect of quantum correlations. Despite holding an equally fundamental role in physics, temporal quantum correlations have yet to find their operational significance in quantum communication. Here we uncover a connection between quantum causality and channel capacity. We show the amount of temporal correlations between two ends of the noisy quantum channel, as quantified by a causality measure, implies a general upper bound on its channel capacity. The expression of this new bound is simpler to evaluate than most previously known bounds. We demonstrate the utility of this bound by applying it to a class of shifted depolarizing channels, which results in improvement over previously known bounds for this class of channels.

Figure 1

A quantum state of a collection of $k$ qubits is encoded into a larger Hilbert space. The encoded quantum information is sent through $n$ parallel copies of the resource channel $\mathcal{N}$ after which it decoded. In general, the dimensions of the input and the output of channel $\mathcal{N}$ need not be the same. As encoding and decoding are both physical processes, they are completely positive trace preserving maps.

Figure 2

Difference between the HW and causality bounds on quantum channel capacity of a shifted depolarizing channel. Notice that the two bounds coincide when there is no shift (standard depolarizing channel) but the causality bound is tighter when the shift $\gamma$ increases.

Figure 3

Difference between the previously known bound from Ref. [39] and the causality bound on quantum channel capacity of a shifted depolarizing channel. The causality bound is tighter for almost all values of $\gamma$ and $p$. Only in the region of small shift $\gamma$ and small probability $p$, which corresponds to the bottom left corner of the diagram, the causality bound is less tight.