# Complexity of mixed states in QFT and holography

@article{Cceres2020ComplexityOM, title={Complexity of mixed states in QFT and holography}, author={Elena C{\'a}ceres and Shira Chapman and Josiah D. Couch and Juan Hernandez and Robert C. Myers and Shan-Ming Ruan}, journal={Journal of High Energy Physics}, year={2020}, volume={2020}, pages={1-120} }

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…

## 50 Citations

Saturation of thermal complexity of purification

- PhysicsJournal of High Energy Physics
- 2022

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…

Holographic and QFT complexity with angular momentum

- PhysicsJournal of High Energy Physics
- 2021

Abstract
We study the influence of angular momentum on quantum complexity for CFT states holographically dual to rotating black holes. Using the holographic complexity=action (CA) and…

Complexity for charged thermofield double states

- PhysicsJournal of High Energy Physics
- 2020

We study Nielsen’s circuit complexity for a charged thermofield double state (cTFD) of free complex scalar quantum field theory in the presence of background electric field. We show that the ratio of…

Geometry of quantum complexity

- PhysicsPhysical Review D
- 2021

Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum…

Switchback effect of holographic complexity in multiple-horizon black holes

- Physics
- 2020

In this paper, we use the “complexity equals action” (CA) conjecture to explore the switchback effect in the strongly-coupled quantum field theories with finite N and finite coupling effects. In the…

Purification complexity without purifications

- Biology
- 2020

Due to Uhlmann's theorem, it is shown that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification.

Complexity for an open quantum system

- PhysicsPhysical Review D
- 2022

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…

Quantum computational complexity from quantum information to black holes and back

- PhysicsThe European Physical Journal C
- 2022

Quantum computational complexity estimates the difficulty of constructing quantum states from elementary operations, a problem of prime importance for quantum computation. Surprisingly, this quantity…

Information geometry in quantum field theory: lessons from simple examples

- PhysicsSciPost Physics
- 2020

Motivated by the increasing connections between information theory and high-energy physics, particularly in the context of the AdS/CFT correspondence, we explore the information geometry associated…

Subsystem complexity after a local quantum quench

- Computer Science
- 2021

Numerical results for the complexity for the entire chain and the subsystem complexity for a block of consecutive sites, obtained by exploiting the Fisher information geometry of the covariance matrices are discussed.

## References

SHOWING 1-10 OF 102 REFERENCES

Complexity and entanglement for thermofield double states

- PhysicsSciPost Physics
- 2019

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…

Subsystem complexity and holography

- PhysicsJournal of High Energy Physics
- 2019

A bstractAs a probe of circuit complexity in holographic field theories, we study sub-system analogues based on the entanglement wedge of the bulk quantities appearing in the “complexity = volume”…

Circuit complexity for coherent states

- PhysicsJournal of High Energy Physics
- 2018

A bstractWe examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The complexity of the coherent states have the same UV…

Toward a Definition of Complexity for Quantum Field Theory States.

- PhysicsPhysical review letters
- 2018

We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations…

Complexity of formation in holography

- Physics
- 2016

A bstractIt was recently conjectured that the quantum complexity of a holographic boundary state can be computed by evaluating the gravitational action on a bulk region known as the Wheeler-DeWitt…

Entanglement of purification: from spin chains to holography

- Physics
- 2017

A bstractPurification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of…

Entanglement of purification in free scalar field theories

- Physics
- 2018

A bstractWe compute the entanglement of purification (EoP) in a 2d free scalar field theory with various masses. This quantity measures correlations between two subsystems and is reduced to the…

Complexity as a Novel Probe of Quantum Quenches: Universal Scalings and Purifications.

- PhysicsPhysical review letters
- 2019

The recently developed notion of complexity for field theory is applied to a quantum quench through a critical point in 1+1 dimensions, and it is demonstrated that complexity is capable of probing features to which the entanglement entropy is insensitive.

Quantum circuits with mixed states

- Computer ScienceSTOC '98
- 1998

A solution for the subroutine problem: the general function that a quantum circuit outputs is a probabilistic function, but using pure state language, such a function can not be used as a black box in other computations.

Holographic purification complexity

- BiologyPhysical Review D
- 2019

It is found that within the complexity = volume and complexity = spacetime volume conjectures, the subregion complexity is equal to the holographic purification complexity, though it is shown where this bound is not saturated.