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arXiv:2202.11056 · Journal of Physics A Mathematical and Theoretical

Quantum regression in dephasing phenomena

Davide Lonigro, Dariusz Chruściński

Published Apr 25, 2022• 20 citations
arXivHTMLPDF

Abstract

Abstract We investigate the validity of quantum regression for a family of quantum Hamiltonians on a multipartite system leading to phase-damping reduced dynamics. After finding necessary and sufficient conditions for the CP-divisibility of the corresponding channel, we evaluate a hierarchy of equations equivalent to the validity of quantum regression under arbitrary interventions; in particular, we find necessary conditions for a nontrivial dephasing to be compatible with quantum regression. In this framework, we study a class of dephasing-type generalized spin–boson models, investigating the existence of qubit-environment coupling functions that ensure the exact validity of quantum regression.

DOI
10.1088/1751-8121/ac6a2d

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Abstract We investigate the validity of quantum regression for a family of quantum Hamiltonians on a multipartite system leading to phase-damping reduced dynamics. After finding necessary and sufficient conditions for the CP-divisibility of the corresponding channel, we evaluate a hierarchy of equations equivalent to the validity of quantum regression under arbitrary interventions; in particular, we find necessary conditions for a nontrivial dephasing to be compatible with quantum regression. In this framework, we study a class of dephasing-type generalized spin–boson models, investigating the existence of qubit-environment coupling functions that ensure the exact validity of quantum regression.
The quantum regression formula for an open quantum system consists in an infinite hierarchy of conditions for its multitime correlation functions, thus requiring full access to the total ``system+environment\mathrm{system}+\mathrm{environment}system+environment'' evolution, and providing a stronger requirement than completely positive (CP) divisibility. Here, we analyze CP divisibility and check the validity of quantum regression beyond the Born-Markov approximation (e.g., the weak-coupling limit) for a class of generalized spin-boson models giving rise to a multilevel amplitude-damping evolution; in all cases, it is possible to engineer the system-bath coupling in such a way that quantum regression is exactly satisfied.
Noise and decoherence are ubiquitous in the dynamics of quantum systems coupled to an external environment. In the regime where environmental correlations decay rapidly, the evolution of a subsytem is well described by a Lindblad quantum master equation. In this work, we introduce a quantum algorithm for simulating unital Lindbladian dynamics by sampling unitary quantum channels without extra ancillas. Using ancillary qubits we show that this algorithm allows approximating general Lindbladians as well. For interacting dephasing Lindbladians coupling two subsystems, we develop a decoupling scheme that reduces the circuit complexity of the simulation. This is achieved by sampling from a time-correlated probability distribution - determined by the evolution of one subsystem, which specifies the stochastic circuit implemented on the complementary subsystem. We demonstrate our approach by studying a model of bosons coupled to fermions via dephasing, which naturally arises from anharmonic effects in an electron-phonon system coupled to a bath. Our method enables tracing out the bosonic degrees of freedom, reducing part of the dynamics to sampling an IQP circuit. The sampled bitstrings then define a corresponding fermionic problem, which in the non-interacting case can be solved efficiently classically. We comment on the computational complexity of this class of dissipative problems, using the known fact that sampling from IQP circuits is believed to be difficult classically.
We explore the connection between two recently introduced notions of non-Markovian quantum dynamics and the validity of the so-called quantum regression theorem. While non-Markovianity of a quantum dynamics has been defined looking at the behavior in time of the statistical operator, which determines the evolution of mean values, the quantum regression theorem makes statements about the behavior of system correlation functions of order two and higher. The comparison relies on an estimate of the validity of the quantum regression hypothesis, which can be obtained exactly evaluating two-point correlation functions. To this aim we consider a qubit undergoing dephasing due to interaction with a bosonic bath, comparing the exact evaluation of the non-Markovianity measures with the violation of the quantum regression theorem for a class of spectral densities. We further study a photonic dephasing model, recently exploited for the experimental measurement of non-Markovianity. It appears that while a non-Markovian dynamics according to either definition brings with itself violation of the regression hypothesis, even Markovian dynamics can lead to a failure of the regression relation.
One of the main frameworks to analyze the effects of the environment in a quantum computer is that of pure dephasing, where the dynamics of qubits can be characterised in terms of a well-known dynamical map. In this work we present a non-peturbative extension of such map beyond this simple pure-dephasing case, i.e. that is valid for a general spin coupled to a bosonic environment in a thermal state. To this aim, we use a Trotter decomposition and a Magnus expansion to simplify the unitary evolution operator in interaction picture. The proposed derivation can be extended to other finite-level open quantum systems including many body, initial system-environment correlated states, multiple-time correlation functions or quantum information protocols.
Non-unital noise is a fundamental feature of open quantum systems, governing energy exchange and inducing state translations on the Bloch sphere. While such translations can be useful for tasks such as state preparation and channel capacity, unital dynamics where no translation occurs are often preferred in quantum information processing, particularly for error correction. Phase-covariant dynamics provide a general framework encompassing dissipation, excitation, and dephasing processes in qubit systems; however, commonly used models such as the generalized amplitude damping (GAD) channel offer only limited control over these features. In this work, we present a constructive framework for generating a broader class of phase-covariant dynamics by mixing amplitude-damping and anti-damping channels with unequal decay parameters and time-dependent mixing probabilities. This approach enables independent control over contraction and translation, allows continuous tuning between non-unital and unital regimes, and yields an effective dephasing contribution absent in GAD. To characterize these dynamics, we employ the general theorem for P-divisibility and show that it can be used constructively by evaluating it in suitably chosen Hilbert space bases, leading to simplified conditions for both unital and non-unital cases. The framework captures a wide range of behaviors, including CP-divisible, P-divisible but not CP-divisible, and non-P-divisible dynamics. We further show that appropriate mixing can reduce the deviation from the ideal noiseless evolution and that such improvement persists even after tuning the dynamics to the unital regime. Our results provide a flexible approach to engineering open system dynamics beyond standard thermal models, with potential applications in noise control and quantum information processing.
The loss of coherence of quantum oscillations is of fundamental interest as well as of practical importance in quantum computing. In solid-state experiments the oscillations show, next to the familiar exponential decay on time scales T 1/2 , an overall loss of amplitude. We solve the spin-boson model for a large class of initial conditions without the Markov approximation at the pure dephasing point. It is shown that a loss of visibility occurs in the form of a fast initial drop for factorized initial conditions and an overall reduction for entangled initial conditions. This loss of amplitude is distinct from T 2 -decoherence with the difference being most drastic for environments with real or pseudo-gaps. This result is explained by bandwidth effects in quantum noise as well as in terms of higher order phase-breaking processes. For several experiments, such gapped environments are identified. We confirm that this physics is valid beyond the pure dephasing point.