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arXiv:1611.09248 · IEEE Information Theory Workshop (ITW)

An upper bound on quantum capacity of unital quantum channels

Anurag Anshu

Published Nov 1, 2017• 1 citation
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Abstract

We analyze the quantum capacity of a unital quantum channel, using ideas from the proof of near-optimality of Petz recovery map [Barnum and Knill 2000] and give an upper bound on the quantum capacity in terms of regularized output 2-norm of the channel. We also show that any code attempting to exceed this upper bound must incur large error in decoding, which can be viewed as a weaker version of the strong converse results for quantum capacity. As an application, we find nearly matching upper and lower bounds (up to an additive constant) on the quantum capacity of quantum expander channels. Using these techniques, we further conclude that the `mixture of random unitaries' channels arising in the construction of quantum expanders in [Hastings 2007] show a trend in multiplicativity of output 2-norm similar to that exhibited in [Montanaro 2013] for output œ-norm of random quantum channels.

DOI
10.1109/itw.2017.8277947

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We analyze the quantum capacity of a unital quantum channel, using ideas from the proof of near-optimality of Petz recovery map [Barnum and Knill 2000] and give an upper bound on the quantum capacity in terms of regularized output 2-norm of the channel. We also show that any code attempting to exceed this upper bound must incur large error in decoding, which can be viewed as a weaker version of the strong converse results for quantum capacity. As an application, we find nearly matching upper and lower bounds (up to an additive constant) on the quantum capacity of quantum expander channels. Using these techniques, we further conclude that the `mixture of random unitaries' channels arising in the construction of quantum expanders in [Hastings 2007] show a trend in multiplicativity of output 2-norm similar to that exhibited in [Montanaro 2013] for output œ-norm of random quantum channels.
We obtain a lower bound on the maximum number of qubits, Qn, ε(N), which can be transmitted over n uses of a quantum channel N, for a given non-zero error threshold ε. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the second order. In our method to prove this achievability bound, the decoding map, used by the receiver on the output of the channel, is chosen to be the Petz recovery map (also known as the transpose channel). Our result, in particular, shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission. Applying our achievability bound to the 50-50 erasure channel (which has zero quantum capacity), we find that there is a sharp error threshold above which Qn, ε(N) scales as n.
We find an upper bound for the minimum output entropy of a unital quantum channel, and obtain an exact formula for general qubit channels. Our techniques incorporate the Rényi entropies, particularly, with Rényi parameter α = 2. Moreover, since our upper bound is additive under tensor product, we get as a corollary an upper bound for the classical capacity of unital quantum channels. Interestingly, our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless Hermitian operators, and is tight in the sense that it gives the precise quantity of classical capacity of the Werner-Holevo channel. As an example, we study quantum channels with operator sum representation that is made of the discrete Weyl operators (generalized Pauli operators), and explain how our formula works in this case. Finally, we find new examples for which the minimum output Rényi 2-entropy is additive.
We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters fff when f>0.80944f>0.80944f>0.80944. Random coding has nonzero capacity only for f>0.81071f>0.81071f>0.81071; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.
We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.
Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity of such channels can be accounted via Holevo upper bound, which is evaluated here. Upper estimation is the difference between maximal and minimal output entropy, and lower estimation is given by an ensemble of states with minimal output entropy, such that the output entropy for medium state is as big as possible. For these computations, we use complementary channels because they are much easier to deal with. Both estimations coincide in case of dimension 2. Moreover, in this case the complementary channel is unital cubit, so that the Holevo upper bound equals to the capacity. In large dimensions, the asymptotical estimation is an interval of length log⁡(e2)\log(\frac{e}{2})log(2e​) .
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 -- ddd-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.