Communication Enhancement through Quantum Coherent Control of N Channels in an Indefinite Causal-Order Scenario

Abstract

In quantum Shannon theory, transmission of information is enhanced by quantum features. Up to very recently, the trajectories of transmission remained fully classical. Recently, a new paradigm was proposed by playing quantum tricks on two completely depolarizing quantum channels i.e. using coherent control in space or time of the two quantum channels. We extend here this control to the transmission of information through a network of an arbitrary number N of channels with arbitrary individual capacity i.e. information preservation characteristics in the case of indefinite causal order. We propose a formalism to assess information transmission in the most general case of N channels in an indefinite causal order scenario yielding the output of such transmission. Then we explicitly derive the quantum switch output and the associated Holevo limit of the information transmission for N = 2, N = 3 as a function of all involved parameters. We find in the case N = 3 that the transmission of information for three channels is twice of transmission of the two channel case when a full superposition of all possible causal orders is used.

Sending classical information via three noisy channels in superposition of causal orders

“…Remarkably, when the 3-channel resources are fully exploited, for the single equally weighted combination of the m = 6 case, the Holevo information is approximately two times that of the two channel configuration up to d = 10 as it was found in Ref. [13].…”

“…Following the procedure described in Ref. [13], we found the quantum 3-switch matrix is a 6 × 6 blocksymmetry matrix…”

“…where S ≡ S(N 1 , N 2 , N 3 ) (ρ ⊗ ρ c ), diagonal and offdiagonal elements are d × d matrices, linear combinations of the identity matrix 1 t and the target system is the density matrix ρ, see Appendix A of Ref. [13]. This quantum 3-switch matrix (5) provides the output state of the quantum switch as a function of all the parameters involved: the dimension of the target system d, the depolarization strengths q i 's, the probabilities P k and the density matrix ρ.…”

“…Indeed, this wealth of combinations of causal orders has been not yet exploited as it does not appear when only two channels are used in a quantum 2-switch. In a recent work [13], we presented a general procedure to study the transmission of classical information through N quantum noisy channels using the quantum N -switch. We use it here the transmission of information beyond the two-channel paradigm resulting from fine quantum control of three noisy channels.…”

“…We follow the formalism described in Ref. [13] and obtain the quantum 3-switch matrix to calculate the Holevo information, which quantifies transmission of information [21]. In Section III, we discuss the possible combinations of causal order that one can use to transmit information with three channels, and we analyze the effects on the transmission of information by superimposing m causal orders, where m = 1, 2, .…”