David Sutter

Universal recovery map for approximate Markov chains

A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I(A:C|B) of a tripartite quantum state ρABC can be bounded from below by its distance to the closest recovered state RB→BC(ρAB), where the C-part is reconstructed from the B-part only and the recovery map RB→BC merely depends on ρBC. One particular application of this result implies the equivalence between two different approaches to define topological order in quantum systems.

Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map

The quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states. In this paper, we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case, we reproduce recent lower bounds on the conditional mutual information, such as the one proved by Fawzi and Renner. Our proof only relies on the elementary properties of pinching maps and the operator logarithm.

Efficient One-Way Secret-Key Agreement and Private Channel Coding via Polarization

We introduce explicit schemes based on the polarization phenomenon for the task of secret-key agreement from common information and one-way public communication as well as for the task of private channel coding. Our protocols are distinct from previously known schemes in that they combine two practically relevant properties: they achieve the ultimate rate--defined with respect to a strong secrecy condition--and their complexity is essentially linear in the blocklength. However, we are not able to give an efficient algorithm for code construction.

Achieving the capacity of any DMC using only polar codes

We construct a channel coding scheme to achieve the capacity of any discrete memoryless channel based solely on the techniques of polar coding. In particular, we show how source polarization and randomness extraction via polarization can be employed to “shape” uniformly-distributed i.i.d. random variables into approximate i.i.d. random variables distributed according to the capacity-achieving distribution. We then combine this shaper with a variant of polar channel coding, constructed by the duality with source coding, to achieve the channel capacity. Our scheme inherits the low complexity encoder and decoder of polar coding. It differs conceptually from Gallagers method for achieving capacity, and we discuss the advantages and disadvantages of the two schemes. An application to the AWGN channel is discussed.

Multivariate Trace Inequalities

We prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.

Universal polar codes for more capable and less noisy channels and sources

We prove two results on the universality of polar codes for source coding and channel communication. First, we show that for any polar code built for a source PX,Z there exists a slightly modified polar code-having the same rate, the same encoding and decoding complexity and the same error rate-that is universal for every source PX,Y when using successive cancellation decoding, at least when the channel PY|X is more capable than PZ|X and PX is such that it maximizes I(X; Y )-I(X;Z) for the given channels PY|X and PZ|X. This result extends to channel coding for discrete memoryless channels. Second, we prove that polar codes using successive cancellation decoding are universal for less noisy discrete memoryless channels.

Approximate degradable quantum channels

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that the complementary channel can be obtained from the channel by applying a degrading map. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading map which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

Efficient Quantum Polar Codes Requiring No Preshared Entanglement

We construct an explicit quantum coding scheme which achieves a communication rate not less than the coherent information when used to transmit the quantum information over a noisy quantum channel. For Pauli and erasure channels, we also present efficient encoding and decoding algorithms for this communication scheme based on polar codes (essentially linear in the blocklength), but which do not require the sender and receiver to share any entanglement before the protocol begins. Due to the existence of degeneracies in the involved error-correcting codes, it is indeed possible that the rate of the scheme exceeds the coherent information. We provide a simple criterion which indicates such performance. Finally, we discuss how the scheme can be used for secret key distillation as well as private channel coding.

Efficient Approximation of Quantum Channel Capacities

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M3 log(N)1/2ε-1) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.

Alignment of Polarized Sets

Arıkans polar coding technique is based on the idea of synthesizing n channels from the n instances of the physical channel by a simple linear encoding transformation. Each synthesized channel corresponds to a particular input to the encoder. For large n, the synthesized channels become either essentially noiseless or almost perfectly noisy, but in total carry as much information as the original n channels. Capacity can therefore be achieved by transmitting messages over the essentially noiseless synthesized channels. Unfortunately, the set of inputs corresponding to reliable synthesized channels is poorly understood, in particular, how the set depends on the underlying physical channel. In this work, we present two analytic conditions sufficient to determine if the reliable inputs corresponding to different discrete memoryless channels are aligned or not, i.e., if one set is contained in the other. Understanding the alignment of the polarized sets is important as it is directly related to universality properties of the induced polar codes, which are essential in particular for network coding problems. We demonstrate the performance of our conditions on a few examples for wiretap and broadcast channels. Finally, we show that these conditions imply that the simple quantum polar coding scheme of Renes et al. [Phys. Rev. Lett., 109, 050504, 2012] requires entanglement assistance for general channels, but also show such assistance to be unnecessary in many cases of interest.