Quantum Information Theory

Quantum Markov chains are multipartite quantum states that can be generated from consecutive local interactions. They are characterized by the property that the conditional mutual information (CMI), an entropic quantity, vanishes. We study states that have a CMI that is non-zero but small, the so-called approximate quantum Markov chains [1]. These states still obey the property that they can be approximately generated from consecutive local interactions. This feature enables us to connect the concepts of Markov chains with the theory of quantum error correction. More precisely, if we lose part of a multipartite state that has the approximate Markov structure, we can reconstruct the lost system with a recovery map that acts only locally on a small system [2].

One major difficulty in proving the security of cryptographic protocols is the fact that an adversary could utilize complicated strategies to attack the scheme. The so-called i.i.d. (independent and identically distributed) strategies, where it is assumed that the attacking strategy for the eavesdropper is the same for each round of the protocol, are usually relatively simple to analyze. However, this does obviously not capture the full power of all possible attacking strategies. A technical tool from information theory we developed is Entropy Accumulation Theorem (EAT) [1], which allows us to reduce arbitrary strategies to i.i.d. strategies. In turn, this tool enables simple device-independent security proofs [2].

The EAT ensures that the operationally relevant quantities of a multiparty system (the smooth min-and max-entropies) can be bounded by the sum of the von Neumann entropies of its individual parts viewed in a worst-case scenario. Current research efforts aim to generalize this result and applying it to areas beyond quantum cryptography.

Quantum state tomography is the task of estimating an unknown quantum state from some measurement data, which is an essential element of current research in quantum information processing. Theoretically, it amounts to constructing an estimator using only the data from measurements on a finite number of copies of a state. Many estimators have been proposed, but the error analysis is often ignored or not  treated rigorously. We have developed a reliable and practical scheme for quantum state tomography [1], based on a post-selection technique from studying the quantum de Finetti property of a permutation-invariant sample of states. A ready-to-use software package has been developed to implement the scheme [2]. Alternatively, by extending classical statistical methods to the quantum setting, we proposed a different scheme that significantly improves the error estimates of the previous method [3].

Quantum information processing is constrained by the uncertainty principle. For instance, arbitrary information cannot be copied. Otherwise, it would be possible to accurately measure non-commuting observables such as position and momentum. This has implications for quantum error-correction and quantum cryptography, among many others. We study the possibilities of and limits on information processing protocols and how these are connected to uncertainty relations.  

Our research also involves developing new ways to bound the performance of information processing protocols. For example, we have formulated a bound on the performance of quantum error-correcting codes in terms of a semidefinite program [1], and more recently extended this technique to find tight bounds on the performance of quantum data compression [2]. We have also recently discovered that the entropic uncertainty relation implies a duality between pairs of quantum channels, so that the performance of one channel (for instance, its ability to transmit information) very tightly constrains the performance of the other [3]. 

This partly entails developing new uncertainty relations. An important contribution in this area is the realization that entanglement enables to circumventing the usual formulation of the uncertainty principle. This subtle effect is revealed by uncertainty relations formulated in terms of entropy [4, 5]. This entropic approach has subsequently led to much simpler security proofs of quantum cryptographic protocols than those known previously. More recently we have formulated error–disturbance tradeoffs for quantities that are more directly operationally relevant, which for instance implies that wave–duality relations can be understood as error-disturbance uncertainty relations [6]. 

Finally, we construct new information-processing protocols with improved performance. Here the connection to the uncertainty principle enables us to convert high-performance classical protocols to high-performance quantum protocols. The best example is our construction of quantum polar codes from classical polar codes [7]. The latter are the first discovered class of error-correcting codes that can achieve the capacity of any classical channel using efficient encoding and decoding operations. Their quantum counterpart remains the only explicitly known code family that can achieve the coherent information of arbitrary channels, and brought efficient encoding and decoding operations for high-rate codes into the quantum realm for the first time. The connection to classical codes via the uncertainty principle also leads to insight into the structure of classical polar codes [8]. Beyond polar codes, we have recently shown how to extend the standard classical 'belief propagation' decoding algorithm to the quantum domain in specific instances [9], and how to employ efficient input 'constellations' for classical Gaussian channels to construct similarly efficient quantum codes for quantum Gaussian channels [10].