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arXiv:2407.02726 · Physical Review A

General communication enhancement via the quantum switch

Zhen Wu, James Fullwood, Zhihao Ma, Si-Qi Zhou, Qi Zhao, Giulio Chiribella

Published Jan 3, 2025• 9 citations
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Abstract

Recent studies have shown that quantum information may be effectively transmitted by a finite collection of completely depolarizing channels in a coherent superposition of different orders, via an operation known as the quantum $\mathtt{SWITCH}$. Such results are quite remarkable, as completely depolarizing channels taken in isolation and in a definite order can only output white noise. For general channels, however, little is known about the potential communication enhancement provided by the quantum $\mathtt{SWITCH}$. In this work, we define an easily computable quantity ${\mathcal{P}}_{n}$ associated with the quantum $\mathtt{SWITCH}$ of $n$ copies of a fixed channel, and we conjecture that ${\mathcal{P}}_{n}>0$ is both a necessary and sufficient condition for communication enhancement via the quantum $\mathtt{SWITCH}$. In support of our conjecture, we derive a simple analytic expression for the classical capacity of the quantum $\mathtt{SWITCH}$ of $n$ copies of an arbitrary Pauli channel in terms of the quantity ${\mathcal{P}}_{n}$, which we then use to show that our conjecture indeed holds in the space of all Pauli channels. Utilizing such results, we then formulate a communication protocol involving the quantum $\mathtt{SWITCH}$ which enhances the private capacity of the BB84 channel.

DOI
10.1103/physreva.111.012605

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Recent studies have shown that quantum information may be effectively transmitted by a finite collection of completely depolarizing channels in a coherent superposition of different orders, via an operation known as the quantum SWITCH\mathtt{SWITCH}SWITCH. Such results are quite remarkable, as completely depolarizing channels taken in isolation and in a definite order can only output white noise. For general channels, however, little is known about the potential communication enhancement provided by the quantum SWITCH\mathtt{SWITCH}SWITCH. In this work, we define an easily computable quantity Pn{\mathcal{P}}_{n}Pn​ associated with the quantum SWITCH\mathtt{SWITCH}SWITCH of nnn copies of a fixed channel, and we conjecture that Pn>0{\mathcal{P}}_{n}>0Pn​>0 is both a necessary and sufficient condition for communication enhancement via the quantum SWITCH\mathtt{SWITCH}SWITCH. In support of our conjecture, we derive a simple analytic expression for the classical capacity of the quantum SWITCH\mathtt{SWITCH}SWITCH of nnn copies of an arbitrary Pauli channel in terms of the quantity Pn{\mathcal{P}}_{n}Pn​, which we then use to show that our conjecture indeed holds in the space of all Pauli channels. Utilizing such results, we then formulate a communication protocol involving the quantum SWITCH\mathtt{SWITCH}SWITCH which enhances the private capacity of the BB84 channel.
If two identical copies of a completely depolarizing channel are put into a superposition of their possible causal orders, they can transmit nonzero classical information. Here we study how well we can transmit classical information with NNN depolarizing channels put in superposition of MMM causal orders via a quantum switch. We calculate the Holevo quantity if the superposition uses only cyclic permutations of channels and find that it increases with MMM and it is independent of NNN. For a qubit it never reaches 1 if we are increasing MMM. On the other hand, the classical capacity decreases with the dimension ddd of the message system. Further, for N=3N=3N=3 and 4 we studied the superposition of all causal orders and uniformly superposed causal orders belonging to different cosets created by a cyclic permutation subgroup.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.
Abstract Superdense coding is a fundamental protocol in quantum information theory that facilitates the transmission of classical information at capacities exceeding classical limits by exploiting quantum entanglement. However, the presence of noise in communication channels, particularly in the case of Pauli channels, poses a significant challenge to the efficiency of superdense coding. In this paper, we explore an active protocol designed for scenarios involving two distinct, addressable noisy channels. We investigate how a quantum switch, by actively composing these two channels in an indefinite causal order, can construct a new composite channel with significantly enhanced capacity. Our findings indicate that, when confronted with the same type of Pauli channels which are commutative, the quantum switch offers no advantage over classical strategies constrained by definite causal order. Remarkably, when two different channels are composed, the quantum switch can exploit the non-commutativity of their underlying operators to coherently cancel their noise effects. This vividly illustrates the ‘different is more’ principle: using the quantum switch to actively compose two different non-commuting channels yields a significant capacity gain over composing two identical commutative channels. This underscores the potential of the quantum switch as a tool for active noise-aware protocol design, where the difference itself is a resource, rather than as a tool for passive noise mitigation.
Abstract The quantum switch is known to remove noise probabilistically, but whether this constitutes an advantage over causally ordered processing has remained contentious: postselection, weak classical benchmarks, and task-dependent comparisons have all drawn criticism. Here we give an exact analysis of the question for a qubit under Pauli noise. For any two qubit Pauli channels, measuring the switch’s control qubit implements a commutation syndrome : it sorts each noise realization by whether the two Kraus errors commuted, syndrome information that an unencoded qubit does not otherwise possess, and the optimal measure-and-feed-forward protocol has a closed form. The analytical results have two levels of generality. For the exactly solvable pair, an isotropic depolarizing channel and a flip channel along an arbitrary axis, we prove a complete ceiling theorem: every definite-order protocol using each channel once, with arbitrary memoryless intermediate operations and receiver-side post-processing, deterministic or heralded, has entanglement fidelity at most ma x P c P , and the switch exceeds this ceiling deterministically , by p q / 3 over most of parameter space. For arbitrary qubit Pauli pairs, the ceiling is proven against arbitrary receiver-side post-processing, and the gain is strictly positive if and only if the two syndrome branches call for different Pauli corrections, confining the exceptions to error-dominated noise. Side systems are covered whenever they are non-signaling about the message. Beyond these theorems the evidence is numerical: strict gain in 99.8 % of 5000 random Pauli pairs, and no case, within a 200 -pair sample, in which an optimized midpoint echo overturns the gain. The state-resolved gain obeys an exact geometric law, Δ F = 2 p q / 3 | r ^ ⋅ n ^ | 2 : alignment is the condition under which the degenerate two-error syndrome becomes perfectly correctable. The separation survives control dephasing down to a threshold visibility that falls with noise strength, and at strong noise it outperforms coherent control of the two orderings even with twice the channel uses.
A central challenge in quantum information theory is to determine whether entanglement provides an advantage for information processing when coherent operations across multiple channel uses are allowed. For depolarizing channels, King's proof of the additivity of the minimum output entropy rules out such an advantage for classical communication at leading asymptotic order, while leaving open the possibility of an advantage for higher orders. In this work, we conjecture a majorization relation for the outputs of depolarizing channels and prove an approximate version of this conjecture. Our proof combines an exact ordering of shifted log-determinants with an anti-concentration bound for the spectrum of the product-state output, refining King's analysis based on Schatten norms. As a direct application, we obtain a tight third-order expansion for the classical capacity of quantum depolarizing channels. Consequently, entangled codewords and collective decoding can improve the message size over the product-coding strategy by at most O(log⁡log⁡n)O(\log \log n)O(loglogn) bits, sharpening our understanding of the role of entanglement in the finite-blocklength regime.
Abstract Determining whether a noisy quantum channel can be used to reliably transmit quantum information is a challenging problem in quantum information theory. This is because it requires computation of the channel’s coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel’s ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels’ input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.