Complexity and entanglement for thermofield double states (1810.05151v4)

Abstract: Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t > 0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.

• The paper demonstrates that circuit complexity at t=0 is proportional to thermodynamic entropy, aligning with holographic complexity ideas.
• It reveals that for t>0, complexity evolves over time and saturates around the inverse temperature, challenging existing holographic predictions.
• The study introduces a novel application of covariance matrices and symplectic transformations to analytically treat Gaussian states in free QFTs.

Complexity and Entanglement for Thermofield Double States

The paper "Complexity and Entanglement for Thermofield Double States" explores the implementation of circuit complexity as a tool for analyzing thermofield double (TFD) states within the framework of free scalar quantum field theories (QFTs), employing the Nielsen approach. This research is motivated by the potential of holographic complexity proposals to provide new insights into black hole spacetimes.

The authors begin by investigating TFD states at $t=0$. They establish that the complexity of forming these states is proportional to the thermodynamic entropy, aligning qualitatively with holographic complexity paradigms. However, a divergence arises for TFD states at times $t > 0$. Here, the complexity exhibits a temporal evolution, saturating in a timeframe approximately equivalent to the inverse of the temperature. This finding contradicts predictions from holographic models, and it is attributed to the Gaussian characteristics intrinsic to the TFD state within free bosonic QFTs.

A unique computational advancement presented in this paper is the employment of covariance matrices and symplectic transformations. This mathematical framework grants a natural formalism for addressing Gaussian states, proving invaluable for conducting complexity calculations.

In the case of free QFTs in 1+1 dimensions, the research further dissects the dynamics of circuit complexity in relation to the entanglement entropy over simple bipartitions of TFDs. The authors connect their entanglement entropy results to previous studies examining non-equilibrium entanglement progression post-quench events. Notably, this paper introduces a new analytic derivation elucidating the logarithmic contributions from the zero momentum mode, applicable in situations where the mass approaches zero. They argue convincingly why similar logarithmic behavior should manifest across larger subsystems.

The paper's implications are broad, offering novel theoretical insights with potential applications in further exploring quantum information complexities facilitated by field theories. These findings provide a robust platform for speculating on the future trajectory of AI development within theoretical physics, particularly where quantum mechanics and complex systems intersect.

The convergence of complexity and entanglement analysis favored here could also inspire subsequent studies attempting to bridge the gap between two seemingly disparate yet profoundly interconnected quantum phenomena. By leveraging a symmetry-founded, Gaussian-centric approach, the authors lay the groundwork for future explorations aimed at enhancing our understanding of quantum field complexities and the ways these complexities encapsulate critical facets of spacetime and particle interactions.