Posters

Poster Titles and Abstracts (downloadable as pdf file here)

Joonwoo Bae : Operational Characterisation of Divisibility of Dynamical Maps

We show the operational characterisation to divisibility of dynamical maps in terms of distinguishability of quantum channels. It is proven that distinguishability of any pair of quantum channels does not increase under divisible maps, in which the full hierarchy of divisibility is isomorphic to the structure of entanglement between system and environment. This shows that i) channel distinguishability is the operational quantity signifying (detecting) divisibility (indivisibility) of dynamical maps and ii) the decision problem for divisibility of maps is as hard as the separability problem in entanglement theory. We also provide the information- theoretic characterisation to divisibility of maps with conditional min-entropy.

Joint work with D. Chruscinski.

Minglai Cai : Classical-Quantum Arbitrarily Varying Wiretap Channel: Common Randomness Assisted Code and Continuity

We determine the secrecy capacities under common randomness assisted coding of arbitrarily varying classical-quantum wiretap channels. Furthermore, we determine the secrecy capacity of a mixed channel model which is compound from the sender to the legal receiver and varies arbitrarily from the sender to the eavesdropper. As an application we examine when the secrecy capacity is a continuous function of the system parameters and show that resources, i.e., having access to a perfect copy of the outcome of a random experiment, are helpful for channel stability.

Joint work with Holger Boche, Christian Deppe, and Janis Nötzel.

Roberto Ferrara : Private bits, quantum data hiding and the swapping of perfect secrecy

Quantum data hiding served as intuition behind the construction of bound entangled state from which secret bits can be extracted. Here, we make this intuition formal, resulting in a new bound for the distillable entanglement of private states. As an application, we consider the problem of extending the distance of quantum key distribution with help of an intermediate station. In analogy to the quantum repeater, this paradigm has been called the quantum key repeater. We show that when extending private states, the resulting rate is bounded by the distillable entanglement and thus in this case entanglement distillation and entanglement swapping are essentially optimal.

Li Gao : Private capacity estimates via interpolation

We use operator algebra method and interpolation technique to estimate (potential) private capacity and strong converse rate for some nice classes of channels. In particular, we observe that the comparison inequality in our previous work of Schatten p-norms (Renyi p-entropy) can be improved to vector-valued L_p norms (sandwiched Renyi p-conditionl entropy). On the other hand, the conditions of our "nice" channels are simplified and expressed by the range of Stinespring isometry. The examples include depolarizing channels, Pauli channels and random unitary group channels.

Gisbert Janßen: Forward secret-key distillation from compound memoryless classical-quantum-quantum sources

We consider secret-key distillation from tripartite compound classical-quantum-quantum (cqq) sources with free forward public communication under strong security criterion. We design protocols which are universally reliable and secure in this scenario. These are shown to achieve asymptotically optimal rates as long as a certain regularity condition is fulfilled by the the set of its generating density matrices. We derive a multi-letter capacity formula for all compound cqq sources being regular in this sense. We also determine the forward secret-key distillation capacity for situations, where the legitimate sending party has perfect knowledge of his/her marginal state deriving from the source statistics. In this case regularity conditions can be dropped. Our results show, that the capacities with and without the mentioned kind of state knowledge are equal as long as the source is generated by a regular set of density matrices. We demonstrate, that regularity of cqq sources is not only a technical but also an operational issue. For this reason, we give an example of a source which has zero secret-key distillation capacity without sender knowledge, while achieving positive rates is possible if sender marginal knowledge is provided.

Joint work with Holger Boche http://arxiv.org/abs/1604.05530.

Anna Jencova : Characterization of sufficient channels by a Renyi relative entropy

At the previous Beyond IID workshop, the question was asked whether equality in the data processing inequality for the sandwiched Renyi relative entropy implies sufficiency of the channel. This question is answered in the affirmative. For the proof, an interpolating family of L_p norms with respect to a state is used. It is shown that this family can also be applied to extend the sandwiched Renyi relative entropy to infinite dimensional settings.

Cecilia Lancien : Parallel repetition and concentration for (sub-)no-signalling games via a flexible constrained de Finetti reduction

We use a recently discovered constrained de Finetti reduction (aka post-selection lemma) to study the parallel repetition of multi-player non-local games under no-signalling strategies. Since the technique allows us to reduce general strategies to independent plays, we obtain parallel repetition (corresponding to winning all rounds) in the same way as exponential concentration of the probability to win a fraction larger than the value of the game. Our proof technique leads us naturally to a relaxation of no-signalling (NS) strategies, which we dub sub-no-signalling (SNOS). While for two players the two concepts coincide, they differ for three or more players. Our results are most complete and satisfying for arbitrary number of sub-no-signalling players, where we get universal parallel repetition and concentration for any game, while the no-signalling case is obtained as a corollary, but only for games with full support.

Based on arXiv[quant-ph]:1506.07002, joint work with A. Winter.

Felix Leditzky : Degradable states : New bounds on one-way distillable entanglement and quantum capacity

Smith et al. [IEEE Trans. on Inf. Th. 54.9 (2008), pp. 4208–4217.] defined degradable states in analogy to degradable quantum channels. A degradable state is a bipartite state shared between Alice and Bob and purified by an environment system (Eve), for which there exists a quantum operation that degrades Bob's share of the system to the environment. We also define the notion of conjugate degradable states, for which the above holds up to entry-wise complex conjugation, and antidegradable states, for which there exists a quantum operation that degrades Eve's share to Bob's. Our goal is to use the concept of degradable states to find upper bounds on the one-way distillable entanglement (oDE) as well as the quantum capacity, both of which are known to be generally given by a regularized formula. To this end, we first show that the oDE of (conjugate) degradable states is equal to the coherent information from Alice to Bob, and therefore given by a single-letter formula. Moreover, for tensor products of (conjugate) degradable states and antidegradable states, the oDE can be bounded from above by that of the (conjugate) degradable part alone. These two observations yield single-letter upper bounds on the oDE of a bipartite state by decomposing it into a convex combination of degradable and antidegradable states. Applying this technique to the qubit depolarizing channel with parameter p yields an upper bound on its quantum capacity that is stronger than known upper bounds for p≳0.055.

Joint work with Nilanjana Datta and Graeme Smith.

Iman Marvian : Universal Quantum Emulator

We propose a quantum algorithm that emulates the action of an unknown unitary transformation on a given input state, using multiple copies of some unknown sample input states of the unitary and their corresponding output states. The algorithm does not assume any prior information about the unitary to be emulated, or the sample input states. To emulate the action of the unknown unitary, the new input state is coupled to the given sample input-output pairs in a coherent fashion. Remarkably, the runtime of the algorithm is logarithmic in D, the dimension of the Hilbert space, and increases polynomially with d, the dimension of the subspace spanned by the sample input states. Furthermore, the sample complexity of the algorithm, i.e. the total number of copies of the sample input-output pairs needed to run the algorithm, is independent of D, and polynomial in d. In contrast, the runtime and the sample complexity of the incoherent methods, i.e. methods that use tomography, are both linear in D. The algorithm is blind, in the sense that at the end it does not learn anything about the given samples, or the emulated unitary. This algorithm can be used as a subroutine in other algorithms, such as quantum phase estimation.

William Matthews : On the tightness of hypothesis testing bounds for coding over quantum channels

In coding over classical channels, the hypothesis testing converse given by Polyanskiy, Poor and Verdú (2010) has been shown to be a tight bound on the success probability in that it is equal to the success probability for a given code when the input distribution is spread uniformly over the codewords [1]. The maxmimum over *all* input distributions is an upper-bound for all codes with the advantage of being easier to compute. Extremely recently, Vazquez-Vilar demonstrated that the analagous hypothesis testing bound for coding over classical-quantum channels is tight in a similar sense [2]. Here we establish a result, similar in spirit, which shows how a fully quantum hypothesis testing converse of [3] can be seen as a relaxation of an exact expression for the channel fidelity of unassisted quantum codes, and relate this result and [2] to earlier work on the operational meaning of the conditional min-entropy [4].

[1] Gonzalo Vazquez-Vilar, Adrià Tauste Campo, Albert Guillén i Fàbregas, Alfonso Martinez, IEEE Trans. Inf. Theory, vol. 62, no. 5, pp. 2324-2333, May 2016. arXiv:1411.3292

[2] Gonzalo Vazquez-Vilar, ``Multiple Quantum Hypothesis Testing Expressions and Classical-Quantum Channel Converse Bounds,'' ISIT 2016.

[3] Matthews, Wehner, IEEE Trans. Inf. Theory, vol. 60, no. 11, pp. 7317-7329, Nov. 2014. arXiv:1210.4722.

[4] Robert Koenig, Renato Renner, Christian Schaffner, IEEE Trans. Inf. Th., vol. 55, no. 9 (2009). arXiv:0807.1338.

Alexander Müller-Hermes : Relative Entropy Bounds on Quantum, Private and Repeater Capacities

We find a strong-converse bound on the private capacity of a quantum channel assisted by unlimited two-way classical communication. The bound is based on the max-relative entropy of entanglement and its proof uses a new inequality for the sandwiched R\'{e}nyi divergences based on complex interpolation techniques. We provide explicit examples of quantum channels where our bound improves both the transposition bound (on the quantum capacity assisted by classical communication) and the bound based on the squashed entanglement introduced by Takeoka et al.. As an application we study a repeater version of the private capacity assisted by classical communication and provide an example of a quantum channel with negligible private repeater capacity.

George Negulescu : Limiting Characterization of Capacity for Interference Channels with Finite Memory

In this work we examine the capacity of interference channels with finite memory. Since a single-letter region is not even known for memoryless interference channels (with the exception of those satisfying the strong interference property), we derive a characterization based on limiting expressions. Limiting characterizations can be obtained for many memoryless channels for which no single-letter region is known. Even though limiting expressions have been considered unsatisfactory, we show that by using these expressions we can establish a strong relation between memoryless interference channels and interference channels with finite memory. Since this relation holds for all interference channels, we obtain a characterization of interference channels with memory in terms of limiting expressions. This work generalizes generalizes the results from [1], which obtains a limiting expression for the capacity region of the MAC channel with finite memory by relying on the use of a single-letter capacity region expression for the memoryless MAC channel. We show how this result can be extended to interference channels with finite memory, even though memoryless ICs do not have, in general, a single-letter expression for their capacity region.

[1] S. Verdu, “Multiple-access channels with memory with and without frame synchronization,” IEEE Trans. Inf. Theory, vol. 35, no. 3, pp. 605–619, May 1989.

Alex Streltsov : Entanglement and coherence in quantum state merging

Understanding the resource consumption in distributed scenarios is one of the main goals of quantum information theory. A prominent example for such a scenario is the task of quantum state merging where two parties aim to merge their parts of a tripartite quantum state. In standard quantum state merging, entanglement is considered as an expensive resource, while local quantum operations can be performed at no additional cost. However, recent developments show that some local operations could be more expensive than others: it is reasonable to distinguish between local incoherent operations and local operations which can create coherence. This idea leads us to the task of incoherent quantum state merging, where one of the parties has free access to local incoherent operations only. In this case the resources of the process are quantified by pairs of entanglement and coherence. Here, we develop tools for studying this process, and apply them to several relevant scenarios. While quantum state merging can lead to a gain of entanglement, our results imply that no merging procedure can gain entanglement and coherence at the same time. We also provide a general lower bound on the entanglement-coherence sum, and show that the bound is tight for all pure states. Our results also lead to an incoherent version of Schumacher compression: in this case the compression rate is equal to the von Neumann entropy of the diagonal elements of the corresponding quantum state.

Eyuri Wakakuwa : The Cost of Randomness for Converting a Tripartite Quantum State to be Approximately Recoverable

We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρ on system ABC, and is transformed by a random unitary operation on A to another state which is approximately recoverable from its reduced state on AB (Case 1) or BC (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery, mainly focusing on the case of pure states. We prove that the minimum cost in Case 1 is equal to the Markovianizing cost of the state, for which a single-letter formula is known. With an additional requirement on the convergence speed of the recovery error, we prove that the minimum cost in Case 2 is also equal to the Markovianizing cost. Our results have an application for distributed quantum computation.

Joint work with Akihito Soeda and Mio Murao available at http://arxiv.org/abs/1512.06920

Xin Wang : Asymptotic entanglement manipulation under PPT operations: new SDP bounds and irreversibility

We study various aspects of asymptotic entanglement manipulation of general bipartite states under operations that completely preserve positivity of partial transpose (PPT). Our key findings include: i) an additive semidefinite programming (SDP) entanglement measure which is an improved upper bound of the distillable entanglement than the logarithmic negativity; ii) a succinct SDP characterization of the one-copy deterministic distillation rate and an additive upper bound; iii) nonadditivity of Rains’ bound for a class of two-qubit states; and iv) two additive SDP lower bounds to the Rains’ bound and relative entropy of entanglement, respectively. These findings enable us to efficiently evaluate the asymptotic distillable entanglement and entanglement cost for several classes of mixed states. As applications, we show that for any rank-two mixed state supporting on the 3-level anti-symmetric subspace, both the Rains’ bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. That also implies the irreversibility of asymptotic entanglement manipulation under PPT operations, one of the major open problems in quantum information theory.

Joint work with Runyao Duan, available at arXiv:1601.07940, 1605.00348 and 1606.09421.

Mark M. Wilde : Converse bounds for private communication over quantum channels

We establish several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state [Horodecki et al., PRL 94, 160502 (2005)], which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a "privacy test" to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish several converse bounds on the private transmission capabilities of all phase-insensitive Bosonic channels.

Joint work with Mario Berta and Marco Tomamichel available at arXiv:1602.08898

Mischa Woods : A bound on entropy production for quantum dynamical semigroups via the Petz recovery map.

Markovian master equations (formally known as quantum dynamical semigroups), can be used to describe the evolution of a quantum state ρ when in contact with a memoryless thermal bath. This approach has had much success in describing the dynamics of real-life open quantum systems in the lab. Such dynamics increase the entropy of the state ρ until thermalizing with the bath, at which point entropy production stops. Our main result is an upper bound in terms of the relative entropy for the entropy production, when the state is evolving according to a quantum dynamical semigroup which satisfies a physically relevant property known as detailed balance. The problem of deriving bounds on entropy production has attracted much interest in the literature. Most of the proofs take advantage of the hypercontractivity of quantum dynamical semigroups, to bound the so-called log-Sobolev constant. We take a very different and novel approach motivated from results coming from Shannon information theory. Namely the Petz recovery map. Furthermore, in doing so, we extend the known classes of CPTP maps for which the main conjecture in arXiv:1410.4184 holds. The bound also has a nice physical interpretation and we conjecture that it is tight in a certain sense, motivated by numerical studies.

Joint work with Alvaro M. Alhambra.

Mischa Woods : Work and reversibility in quantum thermodynamics

It is a central question in quantum thermodynamics to determine how much work can be gained by a process that transforms an initial state ρ to a final state σ. For example, we might ask how much work can be obtained by thermalizing ρ to a thermal state σ at temperature T of an ambient heat bath. Here, we show that for large systems, or when allowing slightly inexact catalysis, the amount of work is characterized by how reversible the process is. More specifically, the amount of work to be gained depends on how well we can return the state σ to its original form ρ without investing any work. We proceed to exhibit an explicit reversal operation in terms of the Petz recovery channel coming from quantum information theory. Our result establishes a quantitative link between the reversibility of thermodynamical processes and the corresponding work gain.

Joint work with Stephanie Wehner and Mark M. Wilde.