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arXiv:2107.08969 · Journal of High Energy Physics

Saturation of thermal complexity of purification

S. Shajidul Haque, Chandan Jana, Bret Underwood

Published Jan 1, 2022• 18 citations

Abstract

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.

DOI: 10.1007/jhep01(2022)159

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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.

Mode entanglement in a two-mode squeezed state is studied. The entanglement spectrum is derived and interpreted as Boltzmann coefficients of harmonic oscillator eigenstates at thermal equilibrium. The effective temperature is expressed in terms of squeezing parameters and is equal to zero when the state is unsqueezed and increases without bound as the state is infinitely squeezed. It is demonstrated that the reduced density operator can be identified as a canonical ensemble of single harmonic oscillator at some effective temperature Te. The Renyi entanglement entropy is calculated and found to increase without bound as the squeezing parameter increases. The single-copy entanglement, which is a limiting value of the Renyi entropy, is interpreted as a partition function.

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.

Abstract We study the complexity of Gaussian mixed states in a free scalar field theory using the ‘purification complexity’. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of ‘mode-by- mode purifications’ where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. We explore the purification complexity for thermal states of a free scalar QFT in any number of dimensions, and for subregions of the vacuum state in two dimensions. We compare our results to those found using the various holographic proposals for the complexity of subregions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the ‘mutual complexity’ in the various cases studied in this paper.

It is shown that n>0 simple-harmonic-oscillator squeezed states have the same time-evolution behavior as the fundamental (n=0) squeezed state. That is, the probability density \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ retains its original shape as time progresses, except that the scales of the ordinate and abscissa oscillate in a reciprocal manner with frequency 2\ensuremath{\omega}. This allows the time-evolved squeezing operator S${\mathrm{^}}_{\mathit{H}}$(\ensuremath{\lambda},t) to be written as a product of simple factors. Moreover, if a thermal distribution with density matrix \ensuremath{\rho}^ is squeezed, then the mixed squeezed state with minimum free energy has a density matrix given by a unitary transformation \ensuremath{\rho}${\mathrm{^}}_{\mathit{s}}$=S^(\ensuremath{\lambda})\ensuremath{\rho}^S^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(\ensuremath{\lambda}), where \ensuremath{\lambda} is a suitable squeezing parameter.

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.