Higher-order transformations of bidirectional quantum processes

TL;DR

This work develops a comprehensive framework for higher-order quantum theory focused on bistochastic channels, capturing processes where input-output direction can be indefinite. By extending the Choi-operator formalism to partially bistochastic maps and building a recursive type system, it characterizes a hierarchy of admissible transformations, including those with fixed global order and local indefinite directions. It then introduces bistochastic generalizations of quantum combs and process matrices, proving that these objects form a richer class than their standard counterparts and can exhibit stronger signaling capabilities. The framework enables constructions such as the quantum time flip and flippable quantum SWITCH, and it points to practical implications for quantum communication, thermodynamics, and causal modeling under time-symmetric quantum theory.

Abstract

Bidirectional devices are devices for which the roles of the input and output ports can be exchanged. Mathematically, these devices are described by bistochastic quantum channels, namely completely positive linear maps that are both trace-preserving and identity-preserving. Recently, it has been shown that bidirectional quantum devices can, in principle, be used in ways that are incompatible with a definite input-output direction, giving rise to a new phenomenon called input-output indefiniteness. Here we characterize the most general forms of input-output indefiniteness, associated with a hierarchy of higher-order transformations built from transformations of bistochastic quantum channels. Some levels of the hierarchy correspond to transformations that combine bistochastic channels in a definite causal order, while generally using each channel in an indefinite input-output direction. For other levels of the hierarchy, the indefiniteness can involve both the local input-output direction of each process and the global causal order among the processes. On the foundational side, the hierarchy of higher-order transformations characterized here can be regarded as the largest set of physical processes compatible with a time-symmetric variant of quantum theory, where the possible state transformations are restricted to bistochastic channels.

Figures (4)

• Figure 1: A network of higher-order maps allows local agents to implement more general quantum operations (namely, higher-order maps), thereby inducing an indefinite causal structure among the local quantum systems, while the corresponding local laboratories remain causally ordered with respect to one another.
• Figure 2: A bi-tooth quantum comb can be realized as a quantum circuit with open slots via sequential composition of partially bistochastic channels.
• Figure 3: A bi-slot quantum comb can be realized as a sequential composition of bistochastic supermaps.
• Figure 4: A standard process matrix and a bistochastic process matrix for $n=2$.

Theorems & Definitions (13)

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