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arXiv:2412.11595

Monotones in Resource Theories for Dynamical Decoupling

Graeme D. Berk, Simon Milz, Kavan Kishore Modi

Published Dec 16, 2024• 0 citations
arXivHTMLPDF

Abstract

In arXiv:2110.02613, we presented a generalised dynamical resource theory framework that enabled noise reduction techniques including dynamical decoupling (DD) to be studied. While this fundamental contribution remains correct, it has been found that the main resource quantifiers we employed to study these resource theories -- based on the relative entropies between Choi states of multitime processes -- are not monotonic under the allowed transformations. In this letter we detail modified relative entropy-based resource quantifiers, prove that they are indeed monotonic in our resource theories. We re-interpret our numerical results in terms of these new relative entropy monotones, arriving at the same empirical conclusions: DD can be understood as temporal resource distillation, and improvements to noise reduction via our multitimescale optimal dynamical decoupling (MODD) method coincide with a decrease in the corresponding non-Markovianity monotone.

DOI
10.48550/arxiv.2412.11595

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In arXiv:2110.02613, we presented a generalised dynamical resource theory framework that enabled noise reduction techniques including dynamical decoupling (DD) to be studied. While this fundamental contribution remains correct, it has been found that the main resource quantifiers we employed to study these resource theories -- based on the relative entropies between Choi states of multitime processes -- are not monotonic under the allowed transformations. In this letter we detail modified relative entropy-based resource quantifiers, prove that they are indeed monotonic in our resource theories. We re-interpret our numerical results in terms of these new relative entropy monotones, arriving at the same empirical conclusions: DD can be understood as temporal resource distillation, and improvements to noise reduction via our multitimescale optimal dynamical decoupling (MODD) method coincide with a decrease in the corresponding non-Markovianity monotone.
Abstract A great many efforts are dedicated to developing noise reduction and mitigation methods. One remarkable achievement in this direction is dynamical decoupling (DD), although its applicability remains limited because fast control is required. Using resource theoretic tools, we show that non-Markovianity is a resource for noise reduction, raising the possibility that it can be leveraged for noise reduction where traditional DD methods fail. We propose a non-Markovian optimisation technique for finding DD pulses. Using a prototypical noise model, we numerically demonstrate that our optimisation-based methods are capable of drastically improving the exploitation of temporal correlations, extending the timescales at which noise suppression is viable by at least two orders of magnitude, compared to traditional DD which does not use any knowledge of the non-Markovian environment. Importantly, the corresponding tools are built on operational grounds and can be easily implemented to reduce noise in the current generation of quantum devices.
The correlation structure of multitime quantum processes - succinctly described by quantum combs - is an important resource for many quantum information protocols and control tasks. Inspired by approaches for quantum states, we introduce quantifiers of the practical utility of quantum processes that satisfy monotonicity properties, thus overcoming shortcomings in previous state-motivated approaches. Applying these quantifiers to the problem of noise reduction of a quantum process under open-loop control, they are shown to represent the largest amount of temporal mutual information that a process can possibly exhibit. In addition, we study their resource composition behaviour and connect them to the recently introduced notion of generalised comb divergences. Finally, in light of these new quantifiers, we re-interpret the numerical findings of npj Quantum Information 9, 104 (2023) on the relationship of dynamical decoupling and non-Markovian memory, which were based on insufficient resource quantifiers, and show that its main conclusion - the interpretation of dynamical decoupling as a resource distillation - still holds.
The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over deterministic ones, significantly strengthening previous findings and extending recent no-go theorems. We apply our results to obtain a substantial improvement in bounds for the errors and overheads of probabilistic distillation protocols, directly applicable to tasks such as entanglement or magic state distillation, and computable through convex optimization. In broad classes of resources, we strengthen our results to show that the monotone completely governs probabilistic transformations-it serves as a necessary and sufficient condition for state convertibility. This endows the monotone with a direct operational interpretation, as it can exactly quantify the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol.
We establish the connection between non-Markovianity and negative entropy production rate for various classes of quantum operations. Generalizing the definition of entropy production rate for non-equilibrium case we connect it with the rate of change of free energy of the system and establish complementary relations between non-Markovianity and maximum loss of free energy. We naturally conclude that non-Markovianity, in terms of divisibility breaking is a necessary resource for backflow of other resources like purity or free energy, under corresponding allowed operations. Based on this we propose a resource theory of non-Markovianity by constructing the free operations, free states and a generalized measure of non-Markovianity. The framework satisfies the basic properties of a consistent resource theory. The proposed resource quantifier is lower bounded by the optimization free Rivas-Huelga-Plenio (RHP) measure of non-Markovianity.
Many quantum information processing tasks require the consumption of various types of informational resource. Entanglement is a well-studied example of this type of resource; frameness is another. Both types of resource may be quantified through the use of realvalued functions known as monotones, a collection of which can fully characterize the resourcefulness of a state. A complete characterization of the entanglement or frameness of even a finite-dimensional quantum state might require an infinite collection of monotones, despite finiteness of the descriptions of the states in question. We propose a new framework of relative monotones. A relative monotone quantifies the resourcefulness of a quantum state relative to another quantum state. We give an example of the efficacy of relative monotones by characterizing a simple type of frameness with a single relative monotone. This type of frameness is unlikely to yield to analysis with 'absolute' monotones.
The Rains relative entropy of a bipartite quantum state is the tightest known upper bound on its distillable entanglement – which has a crisp physical interpretation of entanglement as a resource – and it is efficiently computable by convex programming. It has not been known to be a selective entanglement monotone in its own right. In this work, we strengthen the interpretation of the Rains relative entropy by showing that it is monotone under the action of selective operations that completely preserve the positivity of the partial transpose, reasonably quantifying entanglement. That is, we prove that Rains relative entropy of an ensemble generated by such an operation does not exceed the Rains relative entropy of the initial state in expectation, giving rise to the smallest, most conservative known computable selective entanglement monotone. Additionally, we show that this is true not only for the original Rains relative entropy, but also for Rains relative entropies derived from various Rényi relative entropies. As an application of these findings, we prove, in both the non-asymptotic and asymptotic settings, that the probabilistic approximate distillable entanglement of a state is bounded from above by various Rains relative entropies.Full version available at https://arxiv.org/abs/2201.00835