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Complexity for Open Quantum System

Arpan Bhattacharyya, Tanvir Hanif, S. Shajidul Haque, Md. Khaledur Rahman

TL;DR

This work investigates circuit complexity in an open quantum system formed by a harmonic oscillator coupled to a one-dimensional bosonic bath. It analyzes two complexity diagnostics—the complexity of purification (COP) and the operator-state mapping—applied to the reduced density matrix obtained after tracing out the bath, for both regular and inverted oscillators. The COP saturates for the regular oscillator, with a surprising kink in the large-time saturation value as a function of damping, while the inverted oscillator exhibits linear growth with time whose slope is reduced by stronger bath coupling, indicating bath-regulated instability; by contrast, the operator-state mapping shows no kink or linear growth and is less sensitive due to fixing a purification. Together these results highlight nuanced bath-induced control of complexity in open quantum systems and suggest directions for exploring boundary conditions and experimental relevance in dissipative settings.

Abstract

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 oscillator and computed the complexity of purification (COP) and complexity by using the operator-state mapping. We found 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 behaviour for the saturation value of COP with varying damping coefficient. For the inverted oscillator, we found a linear growth of COP with time for all values of 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.

Complexity for Open Quantum System

TL;DR

This work investigates circuit complexity in an open quantum system formed by a harmonic oscillator coupled to a one-dimensional bosonic bath. It analyzes two complexity diagnostics—the complexity of purification (COP) and the operator-state mapping—applied to the reduced density matrix obtained after tracing out the bath, for both regular and inverted oscillators. The COP saturates for the regular oscillator, with a surprising kink in the large-time saturation value as a function of damping, while the inverted oscillator exhibits linear growth with time whose slope is reduced by stronger bath coupling, indicating bath-regulated instability; by contrast, the operator-state mapping shows no kink or linear growth and is less sensitive due to fixing a purification. Together these results highlight nuanced bath-induced control of complexity in open quantum systems and suggest directions for exploring boundary conditions and experimental relevance in dissipative settings.

Abstract

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 oscillator and computed the complexity of purification (COP) and complexity by using the operator-state mapping. We found 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 behaviour for the saturation value of COP with varying damping coefficient. For the inverted oscillator, we found a linear growth of COP with time for all values of 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.
Paper Structure (9 sections, 77 equations, 4 figures)

This paper contains 9 sections, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Complexity for a Density Matrix
  3. Our Model
  4. Complexity of Purification
  5. Complexity by Operator-State Mapping
  6. Discussion
  7. Periodic Boundary Conditions
  8. Characteristic Function
  9. Correlation Functions

Figures (4)

  • Figure 1: COP vs time. For regular oscillator with: $\omega_0=1$ and various values of the damping $\Gamma = 0.3 \ (\rm red \ dotted), 0.5\ (\rm blue. \ dotted ), 0.7\ (\rm black \ dotted), 0.9\ (\rm green \ dotted), 1.1 \ (\rm cyan), 1.5 \ (\rm red),\\ 2 \ (\rm blue)$.
  • Figure 2: COP vs time. Regular oscillator: the green and magenta line is for $\omega_0=2$ and the red and orange line is for $\omega_0=1$. The critical value of $\Gamma$ appears for $\Gamma_k=0.5$ and $0.65$ respectively.
  • Figure 3: Left: COP vs time for the inverted harmonic oscillator with $\omega_0=i$ and various values of damping $\Gamma = 0.2 {\rm \ (red \ dotted)}, 0.4 {\rm \ (blue \ dotted)},\ 0.6 \ {\rm (black \ dotted)}, \ 0.8 \ {\rm (yellow\ dotted)}, \ 1.2 \\ {\rm (green \ dotted)}, 2.0 \ {\rm (cyan \ dotted)\, respectively;}$Right: Rate of change of slope of COP with $\Gamma$.
  • Figure 4: Complexity vs time by operator-state mapping. Left: Regular oscillator with $\omega_0=1$ and damping: $\Gamma = 0.3 \ (\rm green), 1.2 \ (\rm red)$; Right: Inverted oscillator with $\omega_0=i$ and damping: $\Gamma = 0.3 \ (\rm green), 1.2 \ (\rm red)$.