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arXiv:1106.2337 · Quantum Information and Computation

Channel covariance, twirling, contraction and some upper bounds on the quantum capacity

Yingkai Ouyang

Published Sep 1, 2014• 23 citations
arXivHTMLPDF

Abstract

Evaluating the quantum capacity of quantum channels is an important but difficult problem, even for channels of low input and output dimension. Smith and Smolin showed that the quantum capacity of the Clifford-twirl of a qubit amplitude damping channel (a qubit depolarizing channel) has a quantum capacity that is at most the coherent information of the qubit amplitude damping channel evaluated on the maximally mixed input state. We restrict our attention to obtaining upper bounds on the quantum capacity using a generalization of Smith and Smolin's degradable extension technique. Given a degradable channel $\cN$ and a finite projective group of unitaries $\cV$, we show that the $\cV$-twirl of $\cN$ has a quantum capacity at most the coherent information of $\cN$ maximized over a $\cV$-contracted space of input states. As a consequence, degradable channels that are covariant with respect to diagonal Pauli matrices have quantum capacities that are their coherent information maximized over just the diagonal input states. As an application of our main result, we supply new upper bounds on the quantum capacity of some unital and non-unital channels -- $d$-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.

DOI
10.26421/qic14.11-12-2

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Evaluating the quantum capacity of quantum channels is an important but difficult problem, even for channels of low input and output dimension. Smith and Smolin showed that the quantum capacity of the Clifford-twirl of a qubit amplitude damping channel (a qubit depolarizing channel) has a quantum capacity that is at most the coherent information of the qubit amplitude damping channel evaluated on the maximally mixed input state. We restrict our attention to obtaining upper bounds on the quantum capacity using a generalization of Smith and Smolin's degradable extension technique. Given a degradable channel \cN\cN\cN and a finite projective group of unitaries \cV\cV\cV, we show that the \cV\cV\cV-twirl of \cN\cN\cN has a quantum capacity at most the coherent information of \cN\cN\cN maximized over a \cV\cV\cV-contracted space of input states. As a consequence, degradable channels that are covariant with respect to diagonal Pauli matrices have quantum capacities that are their coherent information maximized over just the diagonal input states. As an application of our main result, we supply new upper bounds on the quantum capacity of some unital and non-unital channels -- ddd-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.
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.
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 I will investigate the capacities of noisy quantum channels through a combined analytical and numerical approach. First, I introduce novel flagged extension techniques that embed a channel into a higher-dimensional space, enabling single-letter upper bounds on quantum and private capacities. My results refine previous bounds and clarify noise thresholds beyond which quantum transmission vanishes. Second, I present a simulation framework that uses coherent information to estimate channel capacities in practice, focusing on two canonical examples: the amplitude damping channel (which I confirm is degradable and thus single-letter) and the depolarizing channel (whose capacity requires multi-letter superadditivity). By parameterizing input qubit states on the Bloch sphere, I numerically pinpoint the maximum coherent information for each channel and validate the flagged extension bounds. Notably, I capture the abrupt transition to zero capacity at high noise and observe superadditivity for moderate noise levels.
We study four well-known capacities of a two-parameter family of qubit Pauli channels. These are the channels that are covariant under the SO(2)\text{SO}(2)SO(2) group and contain the depolarizing channel as a special case. We find exact expressions for the classical capacity and entanglement-assisted capacities, and analytically determine the regions where the quantum capacity of the channel vanishes. We then use a flag extension to find upper bound for the quantum capacity and private capacity of these channels in the entire region of parameter space and also obtain the lower bound for the quantum capacity by calculating the single-shot quantum capacity numerically. In conjunction with previous results on depolarizing channels, our result is one step forward for determining the capacities of the full Pauli channel.
In this article we consider flagged extensions of convex combination of quantum channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum Q and private P capacities of any channel being a mixture of a unitary map and another channel, with the probability associated to the unitary component being larger than 1/2 . We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.
The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.