Session U38: Landauer and Bennett Award Session: Quantum Resource Theories and Thermodynamics

Rolf Landauer and Charles H. Bennett Award in Quantum Computing talk

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Resource theory of asymmetric distinguishability

We systematically develop the resource-theoretic perspective on distinguishability. The theory is a resource theory of asymmetric distinguishability, given that approximation is allowed for the first quantum state in general transformation tasks. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. We formally define the distillation and dilution tasks, and we find that the exact one-shot distillable distinguishability is equal to the min-relative entropy, the exact one-shot distinguishability cost is equal to the max-relative entropy, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. We also develop the resource theory of asymmetric distinguishability for quantum channels. For this setting, we prove that the exact distinguishability cost is equal to channel max-relative entropy and the distillable distinguishability is equal to the amortized channel relative entropy.   

Quantifying non-Markovianity: a quantum resource-theoretic approach

We study quantum non-Markovianity as a resource theory and introduce the robustness of non-Markovianity: an operationally-motivated, optimization-free measure that quantifies the minimum amount of Markovian noise that can be mixed with a non-Markovian evolution before it becomes Markovian. We show that this quantity is a bonafide non-Markovianity measure since it is faithful, convex, and monotonic under composition with Markovian maps. A two-fold operational interpretation of this measure is provided, with the robustness measure quantifying an advantage in both state and channel discrimination tasks. Moreover, we connect the robustness measure to single-shot information theory by using it to upper bound the min-accessible information of a non-Markovian map. Furthermore, we provide a closed-form analytical expression for this measure and show that, quite remarkably, the robustness measure is exactly equal to half the Rivas-Huelga-Plenio (RHP) measure [Phys. Rev. Lett. 105, 050403 (2010)]. As a result, we provide a direct operational meaning to the RHP measure while endowing the robustness measure with the physical characterizations of the RHP measure.   

log singularities in studying quantum capacities

Understanding the quantum capacity of a quantum channel is a long standing issue which lies at the center of quantum information theory. Determining the quantum capacity of a general channel or checking that it is positive is difficult. A major source of this difficulty arises due to non-additivity of the one-shot quantum capacity. In this work, we present a new method for checking positivity and non-additivity of the one-shot quantum capacity. This method is based on logarithmic singularities of the von-Neumann entropy. Using this log singularity method, we find a new type of non-additivity where using two very simple channels in parallel, one that doesn't have quantum capacity and a second that has positive one-shot quantum capacity, produces a channel whose one-shot quantum capacity exceeds that of the second channel.   

Quuantum Simplicity: Complexity Science in a Quantum World

Complexity and quantum science appear at first to be two fields that bear little relation. One deals with the science of the very large – seeking the understand how unexpected phenomena can emerge in vast systems consisting of many interacting components. Quantum theory, on the other hand, deals with particles at the microscopic level and is usually considered limited to the domain of individual photons and atoms. Yet, different as they appear, there is growing evidence that in interfacing ideas from quantum and complexity science, we may unveil new perspective in either both fields.

Here, I introduce computational mechanics, a branch of complexity science captures structure by building the simplest causal models of natural observations. I illustrate how many processes that require complex classical models may be simulated by remarkably simple quantum devices, and describe recent experiments to test this laboratory conditions [1-3]. I survey the potential consequences these developments, highlighting how the indicate that fundamental notions of structure, complexity and causality [4]

References:
[1] Nature Communications 3, 762
[2] Phys. Rev. Lett. 120, 24050
[3] Nature communications 10, 1630
[4] Phys. Rev. X 8, 031013   

Coherence cost for measurement and computation under conservation laws

Nature imposes many restrictions on the operations that we perform. Among such restrictiones, the restriction imposed by conservation laws on two types of the basic operations of quantum information processing, i.e. measurements and unitary operations, has been studied for a particularly long time.
One of the open problems in this field is to clarify the amount of required resource for implementing unitary operations and measurements under conservation laws.
Here, we provide a solution to this open problem.
We derive two asymptotically exact equalities that clarify the necessary and sufficient amount of quantum coherence as a resource to implement an arbitrary unitary operation and a measurement for arbitrary physical quantities within a desired error, respectively.
This work provides an optimal improvement of the Wigner-Araki-Yanase-Ozawa theorem and a proof of the long standing conjecture that WAY-type tradeoff relation holds for the implementation of an arbitrary unitary channel under conservation laws.
It also clarifies the key question of the resource theory of the quantum channels in the region of resource theory of
asymmetry, for the case of unitary channels.
The technical details of this work are in [PRL \textbf{121}, 110403], [arXiv:1906.04076] and [arXiv:1909.02904].   

Fusion rules from entanglement

We derive some of the axioms of the algebraic theory of anyon [A. Kitaev, Ann. Phys., 321, 2 (2006)] from a conjectured form of entanglement area law for two-dimensional gapped systems. We derive the fusion rules of topological charges and show that the multiplicities of the fusion rules satisfy these axioms. Moreover, even though we make no assumption about the exact value of the constant sub-leading term of the entanglement entropy, this term is shown to be equal to the logarithm of the total quantum dimension of the underlying anyon theory. These derivations are rigorous and follow from the entanglement area law alone. More precisely, our framework starts from two local entropic constraints, which are implied by the area law. They allow us to prove what we refer to as the isomorphism theorem, which enables us to define superselection sectors and fusion multiplicities without a Hamiltonian. These objects and the axioms of the anyon theory are shown to emerge from the structure and the internal self-consistency relations of an object known as the information convex.   

Nonequilibrium work relations of open quantum systems in one-time measurement scheme

Because of the ambiguity in defining work and heat in open quantum systems, it is challenging to formulate non-equilibrium work relations for general quantum channels, beyond the unital case. To overcome this challenge, here we introduce well-defined notions of quantum heat and work in open quantum systems, based on the notion of guessed state defined by the one-time measurement scheme, as developed in [Phys. Rev. E 94, 010103(R)]. This allows us to derive non-equilibrium work relations for general quantum channels, and formulate a modified maximum work theorem with respect to the defined “guessed” quantum work.   

An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states

Quantum channels represent the most general physical changes of a quantum system. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. Such maps vastly generalize stochastically independent maps (e.g., random independence) or equality of the channel maps (i.e., translation invariance). The repeated composition of an ergodic sequence of maps could represent the effect of repeated application of a given quantum channel subject to arbitrary correlated noise or decoherence. Under such a hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one -- entanglement breaking-- channel. As an application, we describe the thermodynamic limit of ergodic Matrix Product States and derive a formula for the expectation value of a local observable and prove that the 2-point correlations of local observables in such states decay exponentially in the bulk with their distance.   

Quantifying Structure and Information Processing in One-Dimensional Quantum Systems

We develop a framework for studying one-dimensional quantum systems with stationary, ergodic dynamics and introduce quantum information properties related to the von Neumann entropies of blocks of qubits within a structured chain. Some of the resulting statistical features are unique to quantum systems and others can be recreated with purely classical ensembles. Applying sequential measurements on these quantum states yields classical stochastic processes whose information properties are determined by the structure of the initial quantum state and the measurement protocol. Examples states with short- and long-range entanglement, as well as specific Hamiltonian ground states are analyzed within this framework.   

Measurement-Induced Randomness and Structure in Controlled Qubit Processes

When an observer measures a time series of qubits, the outcomes generate a classical stochastic process. We present a model family of classically controlled qubit time series and show that measurement induces high complexity in these processes in two specific senses: they are inherently unpredictable (positive Shannon entropy rate) and they require an infinite number of features for optimal prediction (divergent statistical complexity). We argue that nonunifilarity is the mechanism underlying the resulting complexities and identify the different contributions to the randomness of the observed process. We examine the influence that the choice of measurement has on the randomness and structure of the measured qubit process and discuss measurement choices of potential interest in obtaining information about the underlying time series of qubits.   

A study about complete cohering/decohering power with ancillary system.

In this talk we will discuss the cohering/decohering power of a quantum operation which quantifies the maximum amount of coherence which can be generated/eliminated through that operation taken over all input states.
Complete cohering/decohering power (CCP and CDP respectively) of an operation is defined as cohering power of the product of the original operation and an identical operation on ancillary system. In our talk we will compare the properties of CCP and CDP with the general cohering/decohering power when only one system is considered.

We first observe that the CCP with L1 measure is not the same with general cohering power, or even not bounded. Taking dimension to be a large number, CCP increases unboundedly too. In contrast, for the relative entropy measure, the two cohering powers have the same value.
Then we consider the decohering power. Here we will have an unexpected result. For L1 measure, CDP does not have a boundary again. On the other hand, CDP with the relative entropy measure is larger than the general decohering power which can be seen with an entangled input state. This result strongly implies that entanglement helps CDP exceed the general cohering power. We will conclude with some open problems.