Decoherence Entanglement Negativity and Circuit Complexity for Open Quantum System Arpan Bhattacharyyaa1Tanvir Hanifb2 S. Shajidul Haquecd3

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Decoherence, Entanglement Negativity and Circuit

Complexity for Open Quantum System

Arpan Bhattacharyyaa, 1,Tanvir Hanif b, 2, S. Shajidul Haquec,d, 3,

Arpon Paul e, 4

aIndian Institute of Technology, Gandhinagar, Gujarat-382355, India

bDepartment of Theoretical Physics, University of Dhaka, Dhaka-1000, Bangladesh

cHigh Energy Physics, Cosmology & Astrophysics Theory Group

and

The Laboratory for Quantum Gravity & Strings

Department of Mathematics and Applied Mathematics,

University of Cape Town, Cape Town-7700, South Africa

dNational Institute for Theoretical and Computational Sciences (NITheCS)

South Africa

eUniversity of Minnesota Twin Cities, Minneapolis, Minnesota 55455, USA

Abstract

In this paper, we compare the saturation time scales 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 time scale for both types of

complexity is smaller than the saturation time scale for linear entropy. On top of this, we

found that the saturation time scale for linear entropy and entanglement negativity is of

the same order for the Caldeira Leggett model.

1abhattacharyya@iitgn.ac.in

2thanif@du.ac.bd

3shajid.haque@uct.ac.za

4paul1228@umn.edu

arXiv:2210.09268v1 [hep-th] 17 Oct 2022

Contents

1 Introduction 2

2 Linear Entropy and Entanglement Negativity 4

3 Circuit Complexity for Density Matrix 6

4 Open Quantum Systems 8

4.1 Two-oscillatorModel................................. 8

4.1.1 Time Evolution of Linear Entropy, Negativity and Complexity . . . . . . 14

4.2 CaldeiraLeggettModel................................ 17

4.2.1 Time Evolution of Linear Entropy, Negativity and Complexity . . . . . . 22

5 Discussion 24

A Circuit Complexity 26

B Mode expansion for Caldeira Leggett Model and Periodic Boundary Con-

ditions 27

C Characteristic Function 28

D Correlation Functions of Caldeira Leggett Model 30

1 Introduction

Understanding the transition of quantum to classical behaviour is an important problem in

many branches of physics, such as in low-temperature physics [1, 2], early universe cosmology

[3], quantum computation [4, 5] etc. The ﬁrst step towards this transition is believed to be

decoherence [6], which means loss of quantum coherence that originates from the interaction

of a quantum system with its surrounding environment. For a comprehensive review of this

subject, interested readers are referred to [7,8]. The decoherence or the amount of mixedness for

an open quantum system 5can be quantiﬁed by a quantity known as the linear entropy [8, 10].

Apart from the amount of mixedness, quantum entanglement is also a crucial feature of

a quantum state. For a closed system, von Neumann entropy [11], is a useful measure. But

von Neumann entropy is not always a useful measure to quantify quantum correlation for an

open system since it also captures the classical correlations. For an open quantum system, the

entanglement negativity is a useful quantity that measures the quantum correlation between the

system and its environment. In [12–16] the negativity was proposed as a measure of quantum

entanglement for mixed states. Entanglement negativity, which stems from the criteria for

5For a comprehensive review of recent developments towards understanding the physics of open quantum

systems, interested readers are referred to [9].

separability of mixed state, is computed by taking the trace norm of the partial transpose of

the density matrix. In recent times, the interplay between mixedness and quantum correlation

between has been subject to lots of study [17–25] 6.

Another quantity that got much attention in recent years to characterize various aspects of

quantum systems is the circuit complexity. Although the original interest came from AdS/CFT

in some black hole settings [26–30] as an extension to the entanglement entropy, it was already

quite well known in quantum information theory and is now widely used for probing various

quantum systems [31–72] 7. The idea of circuit complexity comes from the theory of quantum

computation. It is based on quantifying the minimal number of operations or gates required to

build a circuit that will take one from a given reference state (|ψRi) to the desired target state

(|ψTi). In this paper, we will follow the approach pioneered by Nielsen [75–77] to compute the

circuit complexity.

In [78, 79], it has been shown that the complexity can be a useful probe of open quantum

systems. In this paper, we further explore whether the circuit complexity can be a useful indi-

cator to estimate if the system is in a mixed or pure state. This is an important question from

the perspective of the open quantum system and complements the analysis done in [78,79]. We

investigate the time evolution of linear entropy and negativity along with complexity for several

representative systems. We ﬁnd some general features of the behaviour of the linear entropy, neg-

ativity and complexity in diﬀerent systems. We have considered two particular solvable models

of a system coupled to a bath with several parameters. First, we consider two harmonic oscilla-

tors, where we treat one of the oscillators as the system and the other as the environment/bath.

Secondly, we study a harmonic oscillator coupled to a one-dimensional bosonic (free) ﬁeld theory.

The target states are generated by quench in both models. Speciﬁcally, we start with the system

and bath decoupled and we suddenly turn on the coupling between the system and the bath

and also change the parameters in the Hamiltonian abruptly. We take the resulting state, trace

out the bath, and scrutinize the reduced density matrix of the remaining oscillator — we show

results (as a function of time) for the complexity and the linear entropy, which is a measure of the

purity of the system. Furthermore, we investigate the quantum correlation between the system

and bath using negativity by considering the total density matrix of the combined system and

bath. We found that entanglement negativity negativity saturates only for the Caldeira Leggett

model and the saturation time scale of linear entropy and negativity coincides. Moreover, the

circuit complexity always saturates when the system becomes completely mixed.

The paper is organized as follows. In Section 2 and 3 we provide a brief review of linear

entropy, entanglement negativity and the complexity of puriﬁcation, complexity by operator-

state mapping, respectively. Section 4 provides the details of the two open quantum systems

that we have considered. It also includes the main results and the ﬁndings. Finally, we conclude

with a discussion and future direction in Section 5. Some useful details regarding computation

6These references are not exhaustive by no means. Interested readers are encouraged to consult the references

and citations of these references.

7This list is by no means exhaustive. Interested readers are referred to these reviews [73,74].

of circuit complexity and correlation function for the Caldeira Leggett model are given in the

Appendix A,B, C and D.

2 Linear Entropy and Entanglement Negativity

Consider the system as a combination of two subsystems Aand B(for example, the harmonic

oscillator, and the bath); one considers the reduced density matrix of subsystem-A, upon tracing

out subsystem-B

ˆρA= TrB[ˆρ],(1)

where ˆρis the density matrix of the entire system. A central measure of entanglement is the

αth-Renyi entropy, Sα, (α > 1) is deﬁned by

Sα=−1

α−1ln Tr[ˆρα

A] ; (2a)

taking α→1+gives the entanglement entropy,S,

S=−Tr[ˆρAln ˆρA].(2b)

Entanglement Negativity:

However, when the system is described by a mixed state then the entanglement entropy is not

a good measure of quantum correlation (e.g SA6=SBin general for mixed states). There are

several other measures [11, 80], one of them is Entanglement Negativity 8. This is deﬁned by

taking the trace norm of the partial transpose of the density matrix. Consider a bipartite state

ρAB, consists of sub-systems A and B. The partial transposition with respect to sub-system B of

ρAB that is expanded in a given local orthonormal basis as ρ=Pρij,kl |iihj|⊗|kihl|is deﬁned

ρTB:= X

i,j,k,l

ρij,kl |iihj|⊗|lihk|(3)

Note that the spectrum of the partial transposition of the density matrix is independent of the

choice of the basis and is also independent of whether the partial transposition is taken over

sub-system A or B. The positivity of the partial transpose of a state is a necessary condition for

separability of the density matrix [81]. This criterion of separability motivates one to consider

the entanglement negativity, which is related to the absolute value of the sum of the negative

8There are other measures of quantum entanglement in mixed states e.g. Mutual Information, Entanglement

of Puriﬁcation. Interested readers are referred to [80] for more details of various correlation measures.

eigenvalues of ρTB[82–84]. Explicitly, this is deﬁned as

N(ρ) := 1

2kρTBk − 1,(4)

where kAk ≡ Tr √A†A, is the trace norm. It can be shown that N(ρ) vanishes for unentangled

states, i.e, for separable density matrices ρ. Another relevant quantity named as the Logarithmic

Negativity,ENis deﬁned as below:

EN(ρ) := log2

ρTB

(5)

= log2(2N(ρ)−1).(6)

These quantities have been explored in recent times in the context of ﬁeld theory and quantum

many-body systems [85–103].

Entanglement Negativity for Gaussian State: In this paper, we are interested in the entanglement

negativity for a bipartite mixed Gaussian state. A Gaussian state can be characterized by the

ﬁrst moments of the phase-space variables and the covariance matrix σdeﬁned as below:

σij =1

2hXiXj+XjXii−hXiihXji,(7)

where X0

is denote the phase-space variables that satisfy the canonical commutation relations,

[Xi, Xj] = 2 iΩij . Here Ωij is the symplectic form. In case of bipartite system, the covariance

matrix can be constructed out of three 2 ×2 matrices, α, β, γ, in the following way:

σ= α γ

γTβ!,(8)

where αand βcorresponds to the correlators among the phase-space variables of subsystem A

and B respectively, whereas γrefers to the cross-correlators between subsystem A and B. Now,

the partial transposition of the bipartite Gaussian density matrix ρchanges the covariance matrix

σinto a new matrix ˜σwhere the det of the cross-correlator γﬂips the sign, i.e, det γ→ −det γ

[104, 105]. The negativity can be deﬁned in terms of the symplectic eigenvalues ˜ν±of the new

covariance matrix ˜σin the following way:

N(ρ) = max 0,1−˜ν−

2˜ν−.(9)

Here, symplectic eigenvalues of ˜σcan be evaluated as

˜ν±=r1

2∆(˜σ)∓p∆(˜σ)2−4 det ˜σ(10)

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Decoherence,EntanglementNegativityandCircuitComplexityforOpenQuantumSystemArpanBhattacharyyaa;1,TanvirHanifb;2,S.ShajidulHaquec;d;3,ArponPaule;4aIndianInstituteofTechnology,Gandhinagar,Gujarat-382355,IndiabDepartmentofTheoreticalPhysics,UniversityofDhaka,Dhaka-1000,BangladeshcHighEnergyPhysics,Cosmo...

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Decoherence Entanglement Negativity and Circuit Complexity for Open Quantum System Arpan Bhattacharyyaa1Tanvir Hanifb2 S. Shajidul Haquecd3

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