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 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.
Highlights
We consider the circuit complexity of an open quantum system, i.e., a system not in isolation but coupled to a “bath.” The motivation for such a system stems from the fact that most experimentally accessible systems are open quantum systems [1,2]
Our system is a harmonic oscillator coupled to a onedimensional massless scalar field, which acts as the bath
In this work we explored the complexity of an open quantum system
Summary
We consider the circuit complexity of an open quantum system, i.e., a system not in isolation but coupled to a “bath.” The motivation for such a system stems from the fact that most experimentally accessible systems are open quantum systems [1,2]. We consider complexity by using the state operator mapping, which essentially picks out a particular purification and lacks the scanning for the minimum path. One might be tempted to compute the operator complexity [40] for the reduced density operator Computing the COP and complexity by operator-state mapping is the most natural approach for realizing complexity for the reduced density matrix. We would like to explore how complexity behaves when the instability of the inverted harmonic oscillator is regulated by the bath, for which we will consider a string. We compute the complexity of purification of the reduced density matrix by tracing out the string. VI we conclude with a discussion of our results and future directions
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