Showing papers by "Andreas Winter published in 2021"

First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy

Abstract: A consistent and versatile framework to understand and apply the laws of thermodynamics for quantum systems far from equilibrium is presented.

Resource theory of unextendibility and nonasymptotic quantum capacity

Abstract: In this paper, we introduce the resource theory of unextendibility as a relaxation of the resource theory of entanglement. The free states in this resource theory are the $k$-extendible states, associated with the inability to extend quantum entanglement in a given quantum state to multiple parties. The free channels are $k$-extendible channels, which preserve the class of $k$-extendible states. We define several quantifiers of unextendibility by means of generalized divergences and establish their properties. By utilizing this resource theory, we obtain nonasymptotic upper bounds on the rate at which quantum communication or entanglement preservation is possible over a finite number of uses of an arbitrary quantum channel assisted by $k$-extendible channels at no cost. These bounds are significantly tighter than previously known bounds for both the depolarizing and erasure channels. Finally, we revisit the pretty strong converse for the quantum capacity of antidegradable channels and establish an upper bound on the nonasymptotic quantum capacity of these channels.

Experimental quantum phase discrimination enhanced by controllable indistinguishability-based coherence

Abstract: Quantum coherence, a basic feature of quantum mechanics residing in superpositions of quantum states, is a resource for quantum information processing Coherence emerges in a fundamentally different way for nonidentical and identical particles, in that for the latter a unique contribution exists linked to indistinguishability which cannot occur for nonidentical particles We experimentally demonstrate by an optical setup this additional contribution to quantum coherence, showing that its amount directly depends on the degree of indistinguishability and exploiting it to run a quantum phase discrimination protocol Furthermore, the designed setup allows for simulating Fermionic particles with photons, thus assessing the role of particle statistics (Bosons or Fermions) in coherence generation and utilization Our experiment proves that independent indistinguishable particles can supply a controllable resource of coherence for quantum metrology

Thermality Versus Objectivity: Can They Peacefully Coexist?

Abstract: Under the influence of external environments, quantum systems can undergo various different processes, including decoherence and equilibration. We observe that macroscopic objects are both objective and thermal, thus leading to the expectation that both objectivity and thermalisation can peacefully coexist on the quantum regime too. Crucially, however, objectivity relies on distributed classical information that could conflict with thermalisation. Here, we examine the overlap between thermal and objective states. We find that in general, one cannot exist when the other is present. However, there are certain regimes where thermality and objectivity are more likely to coexist: in the high temperature limit, at the non-degenerate low temperature limit, and when the environment is large. This is consistent with our experiences that everyday-sized objects can be both thermal and objective.

Programmability of covariant quantum channels

Abstract: A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.

Bosonic data hiding: power of linear vs non-linear optics.

Abstract: We show that the positivity of the Wigner function of Gaussian states and measurements provides an elegant way to bound the discriminating power of "linear optics", which we formalise as Gaussian measurement operations augmented by classical (feed-forward) communication (GOCC). This allows us to reproduce and generalise the result of Takeoka and Sasaki [PRA 78:022320, 2008], which tightly characterises the GOCC norm distance of coherent states, separating it from the optimal distinguishability according to Helstrom's theorem. Furthermore, invoking ideas from classical and quantum Shannon theory we show that there are states, each a probabilistic mixture of multi-mode coherent states, which are exponentially reliably discriminated in principle, but appear exponentially close judging from the output of GOCC measurements. In analogy to LOCC data hiding, which shows an irreversibility in the preparation and discrimination of states by the restricted class of local operations and classical communication (LOCC), we call the present effect GOCC data hiding. We also present general bounds in the opposite direction, guaranteeing a minimum of distinguishability under measurements with positive Wigner function, for any bounded-energy states that are Helstrom distinguishable. We conjecture that a similar bound holds for GOCC measurements.

Infinite-Dimensional Programmable Quantum Processors

Abstract: Programmable quantum processors for continuous-variable systems are defined, and investigated in view of their memory requirements, linking universal processors to their discrete counterparts, and deriving upper and lower bounds on the programmability of Gaussian transformations of Bosonic modes.

Discrimination of quantum states under locality constraints in the many-copy setting

Abstract: We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability, thus establishing the associated Chernoff exponent. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite asymptotic separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB.

Capacities of Gaussian Quantum Channels with Passive Environment Assistance

Abstract: Passive environment assisted communication takes place via a quantum channel modeled as a unitary interaction between the information carrying system and an environment, where the latter is controlled by a passive helper, who can set its initial state such as to assist sender and receiver, but not help actively by adjusting her behaviour depending on the message. Here we investigate the information transmission capabilities in this framework by considering Gaussian unitaries acting on Bosonic systems. We consider both quantum communication and classical communication with helper, as well as classical communication with free classical coordination between sender and helper (conferencing encoders). Concerning quantum communication, we prove general coding theorems with and without energy constraints, yielding multi-letter (regularized) expressions. In the search for cases where the capacity formula is computable, we look for Gaussian unitaries that are universally degradable or anti-degradable. However, we show that no Gaussian unitary yields either a degradable or anti-degradable channel for all environment states. On the other hand, restricting to Gaussian environment states, results in universally degradable unitaries, for which we thus can give single-letter quantum capacity formulas. Concerning classical communication, we prove a general coding theorem for the classical capacity under and energy constraint, given by a multi-letter expression. Furthermore, we derive an uncertainty-type relation between the classical capacities of the sender and the helper, helped respectively by the other party, showing a lower bound on the sum of the two capacities. Then, this is used to lower bound the classical information transmission rate in the scenario of classical communication between sender and helper.

LOCC protocols with bounded width per round optimize convex functions

Abstract: We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two...

Products of synchronous games

Abstract: We show that the *-algebra of the product of two synchronous games is the tensor product of the corresponding *-algebras. We prove that the product game has a perfect C*-strategy if and only if each of the individual games does, and that in this case the C*-algebra of the product game is *-isomorphic to the maximal C*-tensor product of the individual C*-algebras. We provide examples of synchronous games whose synchronous values are strictly supermultiplicative.

Information theoretic parameters of non-commutative graphs and convex corners.

Abstract: We establish a second anti-blocker theorem for non-commutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimisation parameters for a given convex corner that generalise well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterise its maximum value in terms of a generalised fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behaviour of the introduced parameters with respect to tensoring. Specialising to non-commutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of non-commutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a non-commutative graph and compute the values of our parameters in some specific cases.

Thermality versus objectivity: can they peacefully coexist?.

Abstract: Under the influence of external environments, quantum systems can undergo various different processes, including decoherence and equilibration. We observe that macroscopic objects are both objective and thermal in the long time limit, thus leading to the expectation that both objectivity and thermalisation can peacefully coexist on the quantum regime too. Crucially, however, objectivity relies on distributed classical information that could conflict with thermalisation. Here, we examine the overlap between thermal and objective states. We find that in general, one cannot exist when the other is present. However, there are certain regimes where thermality and objectivity are more likely to coexist: in the high temperature limit, at the non-degenerate low temperature limit, and when the environment is large. This is consistent with our experiences that everyday-sized objects can be both thermal and objective.

A genuinely natural information measure.

Abstract: The theoretical measuring of information was famously initiated by Shannon in his mathematical theory of communication, in which he proposed a now widely used quantity, the entropy, measured in bits. Yet, in the same paper, Shannon also chose to measure the information in continuous systems in nats, which differ from bits by the use of the natural rather than the binary logarithm. We point out that there is nothing natural about the choice of logarithm basis, rather it is arbitrary. We remedy this problematic state of affairs by proposing a genuinely natural measure of information, which we dub gnats. We show that gnats have many advantages in information theory, and propose to adopt the underlying methodology throughout science, arts and everyday life.

Geometry of the $p$-adic special orthogonal group $SO(3)_p$

Abstract: We derive explicitly the structural properties of the $p$-adic special orthogonal groups in dimension three, for all primes $p$, and, along the way, the two-dimensional case. In particular, starting from the unique definite quadratic form in three dimensions (up to linear equivalence and rescaling), we show that every element of $SO(3)_p$ is a rotation around an axis. An important part of the analyis is the classification of all definite forms in two dimensions, yielding a description of the rotation subgroups around any fixed axis, which all turn out to be abelian and parametrised naturally by the projective line. Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$ admits a representation in terms of Cardano angles of rotations around the reference axes, in close analogy to the real orthogonal case. However, this works only for certain orderings of the product of rotations around the coordinate axes, depending on the prime; furthermore, there is no general Euler angle decomposition. For $p=2$, no Euler or Cardano decomposition exists.

Asymptotic Separation Between Adaptive and Non-adaptive Strategies in Quantum Channel Discrimination

Abstract: We present a broad investigation of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel multiple times and observing its output at the end. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. Our main result is the first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. Full details in [1].