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arXiv:2210.09268 · Physical review. D/Physical review. D.

Decoherence, entanglement negativity, and circuit complexity for an open quantum system

Arpan Bhattacharyya, T. Hanif, S. Shajidul Haque, Arpon Paul

Published May 4, 2023• 19 citations
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Abstract

In this paper, we compare the saturation timescales for complexity, linear entropy, and entanglement negativity for two open quantum systems. Our first model is a coupled harmonic oscillator, where we treat one of the oscillators as the bath. The second one is a type of Caldeira-Leggett model, where we consider a one-dimensional free scalar field as the bath. Using these open quantum systems, we discovered that both the complexity of purification and the complexity from operator-state mapping is always saturated for a completely mixed state. More explicitly, the saturation timescale for both types of complexity is smaller than the saturation timescale for linear entropy. On top of this, we found that the saturation timescale for linear entropy and entanglement negativity is of the same order for the Caldeira-Leggett model.

DOI
10.1103/physrevd.107.106007

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In this paper, we compare the saturation timescales for complexity, linear entropy, and entanglement negativity for two open quantum systems. Our first model is a coupled harmonic oscillator, where we treat one of the oscillators as the bath. The second one is a type of Caldeira-Leggett model, where we consider a one-dimensional free scalar field as the bath. Using these open quantum systems, we discovered that both the complexity of purification and the complexity from operator-state mapping is always saturated for a completely mixed state. More explicitly, the saturation timescale for both types of complexity is smaller than the saturation timescale for linear entropy. On top of this, we found that the saturation timescale for linear entropy and entanglement negativity is of the same order for the Caldeira-Leggett model.
We study the complexity for an open quantum system. Our system is a harmonic oscillator coupled to a one-dimensional massless scalar field, which acts as the bath. Specifically, we consider the reduced density matrix by tracing out the bath degrees of freedom for both regular and inverted oscillators and compute the complexity of purification (COP) and complexity by using the operator-state mapping. We find that when the oscillator is regular, the COP saturates quickly for both underdamped and overdamped oscillators. Interestingly, when the oscillator is underdamped, we discover a kink-like behavior for the saturation value of the COP with a varying damping coefficient. For the inverted oscillator, we find a linear growth of the COP with time for all values of the bath-system interaction. However, when the interaction is increased the slope of the linear growth decreases, implying that the unstable nature of the system can be regulated by the bath.
A bstract We investigate Krylov complexity in open quantum systems using Lindblad master equations for bosonic bath models, with particular emphasis on the Caldeira-Leggett model. Krylov complexity is computed from the moments of the two-point function within the standard master equation framework. For the damped harmonic oscillator, the results reveal clear dissipative features in Krylov complexity. In the Caldeira-Leggett model, in the high-temperature limit, we find that Krylov complexity saturates in the full system and reproduces the expected dissipative behavior when the decoherence term is suppressed in the master equation. Conversely, when the dissipative term is suppressed, the contribution from decoherence exhibits the familiar oscillatory dynamics of the coherent system, along with additional novel features. However, Krylov complexity appears insensitive to the onset of decoherence, as no clear distinctive signature is observed. We attribute this to the fact that Krylov complexity is defined in the Krylov basis, which does not coincide with the conventional basis typically used to study decoherence.
We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models.The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators.The entropy production for this model in the regimes of "complete" and "incomplete" decoherence is discussed.As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a two-level system into the field vacuum.It is shown that, for all strengths of coupling, the open system passes through the mixed state with the maximum information entropy.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
We study the influence of entanglement on the relation between the statistical entropy of an open quantum system and the heat exchanged with a low temperature environment. A model of quantum Brownian motion of the Caldeira–Leggett type—for which a violation of the Clausius inequality has been stated by Th M Nieuwenhuizen and A E Allahverdyan (2002 Phys. Rev. E 66 036102)—is re-examined and the results of the cited work are put into perspective. In order to address the problem from an information theoretical viewpoint a model of two coupled Brownian oscillators is formulated that can also be viewed as a continuum version of a two-qubit system. The influence of an additional internal coupling parameter on heat and entropy changes is described and the findings are compared to the case of a single Brownian particle.
A bstract We purify the thermal density matrix of a free harmonic oscillator as a two-mode squeezed state, characterized by a squeezing parameter and squeezing angle. While the squeezing parameter is fixed by the temperature and frequency of the oscillator, the squeezing angle is otherwise undetermined, so that the complexity of purification is obtained by minimizing the complexity of the squeezed state over the squeezing angle. The resulting complexity of purification of the thermal state is minimized at non-zero values of the squeezing angle and saturates to an order one number at low frequencies, indicating that there is no additional operator cost required to build thermal mixed states when the oscillator probes length scales that are large compared to the thermal length scale. We also review applications in which thermal density matrices arise for quantum fields on curved spacetimes, including Hawking radiation and a simple model of decoherence of cosmological density perturbations in the early Universe. The complexity of purification for these mixed states also saturates as a function of the effective temperature, which may have interesting consequences for the quantum information stored in these systems.