One-way quantum deficit and quantum coherence in the anisotropic $XY$ chain

TL;DR

This paper explores how one-way quantum deficit and quantum coherence reveal quantum phase transitions in the anisotropic XY spin chain at both zero and finite temperatures, providing analytical and numerical insights.

Contribution

It introduces the use of one-way quantum deficit as a tool to characterize quantum phase transitions and coherence in the XY model, including analytical expressions at zero temperature.

Findings

Quantum deficit signals phase transition at λ=1

Quantum coherence aligns with quantum deficit at zero temperature

Finite temperature effects are also analyzed

Abstract

In this study, we investigate pairwise non-classical correlations measured using a one-way quantum deficit as well as quantum coherence in the $XY$ spin-1/2 chain in a transverse magnetic field for both zero and finite temperatures. The analytical and numerical results of our investigations are presented. In the case when the temperature is zero, it is shown that the one-way quantum deficit can characterize quantum phase transitions as well as quantum coherence. We find that these measures have a clear critical point at $\lambda=1$. When $\lambda\le1$, the one-way quantum deficit has an analytical expression that coincides with the relative entropy of coherence. We also study an $XX$ model and an Ising chain at the finite temperatures.

Equations

$$\Delta =\underset{{\Pi }_A^i}{\min }S(\sum _i{\Pi }_A^i({\varrho }_{AB}))-S({\varrho }_{AB}),$$

$$\varrho =\sum _{i,j}{\varrho }_{i,j}|i⟩⟨j|,$$

$${\mathcal{C}}_{l_1}(\varrho )=\sum _{i\ne j}|{\varrho }_{i,j}|,$$

$${\mathcal{C}}_{rel}(\varrho )=S({\varrho }_{\mathrm{diag}})-S(\varrho ),$$

$$H=-\sum _{j=0}^{N-1}\left\{\frac{\lambda }{2}[(1+\gamma ){\sigma }_j^x{\sigma }_{j+1}^x+(1-\gamma ){\sigma }_j^y{\sigma }_{j+1}^y]+{\sigma }_j^z\right\},$$

$${\rho }_{0n}=\frac{1}{4}\left\{I_{0n}+⟨{\sigma }^z⟩({\sigma }_0^z+{\sigma }_n^z)+\sum _{k=x,y,z}⟨{\sigma }_0^k{\sigma }_n^k⟩{\sigma }_0^k{\sigma }_n^k\right\},$$

$$\displaystyle\varrho_{0n}=\frac{1}{4}\left(\begin{array}[]{cccc}1+2\langle\sigma^{z}\rangle+\langle\sigma_{0}^{z}\sigma_{n}^{z}\rangle&0&0&\langle\sigma_{0}^{x}\sigma_{n}^{x}\rangle-\langle\sigma_{0}^{y}\sigma_{n}^{y}\rangle\cr 0&1-\langle\sigma_{0}^{z}\sigma_{n}^{z}\rangle&\langle\sigma_{0}^{x}\sigma_{n}^{x}\rangle+\langle\sigma_{0}^{y}\sigma_{n}^{y}\rangle&0\cr 0&\langle\sigma_{0}^{x}\sigma_{n}^{x}\rangle+\langle\sigma_{0}^{y}\sigma_{n}^{y}\rangle&1-\langle\sigma_{0}^{z}\sigma_{n}^{z}\rangle&0\cr\langle\sigma_{0}^{x}\sigma_{n}^{x}\rangle-\langle\sigma_{0}^{y}\sigma_{n}^{y}\rangle&0&0&1-2\langle\sigma^{z}\rangle+\langle\sigma_{0}^{z}\sigma_{n}^{z}\rangle\end{array}\right).$$

$$⟨{\sigma }^z⟩=-{\int }_0^{\pi }\frac{(1+\lambda \cos \varphi )\tanh (\beta {\omega }_{\varphi })}{2\pi {\omega }_{\varphi }}d\varphi ,$$

$$\displaystyle\langle\sigma_{0}^{x}\sigma_{n}^{x}\rangle=\left|\begin{array}[]{cccc}F_{-1}&F_{-2}&\cdots&F_{-n}\cr F_{0}&F_{-1}&\cdots&F_{-n+1}\cr\vdots&\vdots&\ddots&\vdots\cr F_{n-2}&F_{n-3}&\cdots&F_{-1}\cr\end{array}\right|,$$

$$\displaystyle\langle\sigma_{0}^{y}\sigma_{n}^{y}\rangle=\left|\begin{array}[]{cccc}F_{1}&F_{0}&\cdots&F_{-n+2}\cr F_{2}&F_{1}&\cdots&F_{-n+3}\cr\vdots&\vdots&\ddots&\vdots\cr F_{n}&F_{n-1}&\cdots&F_{1}\cr\end{array}\right|,$$

$$⟨{\sigma }_0^z{\sigma }_n^z⟩=⟨{\sigma }^z{⟩}^2-F_n{F}_{-n},$$

$$F_n=$$

$$\displaystyle\varrho_{0}=\varrho_{i}=\frac{1}{2}\left(\begin{array}[]{cc}1+\langle\sigma^{z}\rangle&0\cr 0&1-\langle\sigma^{z}\rangle\end{array}\right),$$

$$S({\varrho }_{0n})=-\sum _{i=0}^1({\eta }_i\log {\eta }_i+{\xi }_i\log {\xi }_i),$$

$${\eta }_i=[1+⟨{\sigma }_0^z{\sigma }_n^z⟩+(-1{)}^i\sqrt{4⟨{\sigma }^z{⟩}^2+(⟨{\sigma }_0^x{\sigma }_n^x⟩-⟨{\sigma }_0^y{\sigma }_n^y⟩{)}^2}]/4,$$

$${\xi }_i=[1-⟨{\sigma }_0^z{\sigma }_n^z⟩+(-1{)}^i(⟨{\sigma }_0^x{\sigma }_n^x⟩+⟨{\sigma }_0^y{\sigma }_n^y⟩)]/4.$$

$$\displaystyle V=\left(\begin{array}[]{cc}\cos(\theta/2)&e^{i\varphi}\sin(\theta/2)\cr-e^{-i\varphi}\sin(\theta/2)&\cos(\theta/2)\end{array}\right).$$

$$S(\sum _i{\Pi }_n^i({\varrho }_{0n}))=-\sum _{i,j=0}^1{\xi }_{ij}\log {\xi }_{ij},$$

$${\xi }_{ij}=[1+(-1{)}^i⟨{\sigma }^z⟩\cos \theta +(-1{)}^j\sqrt{[⟨{\sigma }_0^x{\sigma }_n^x{⟩}^2{\cos }^2\phi +⟨{\sigma }_0^y{\sigma }_n^y{⟩}^2{\sin }^2\phi ]{\sin }^2\theta +[⟨{\sigma }^z⟩+(-1{)}^i⟨{\sigma }_0^z{\sigma }_n^z⟩\cos \theta {]}^2}]/4.$$

$$\Delta =\underset{\theta ,\phi }{\min }[-\sum _{i,j=0}^1{\xi }_{ij}\log {\xi }_{ij}]+\sum _{i=0}^1({\eta }_i\log {\eta }_i+{\xi }_i\log {\xi }_i).$$

$${\mathcal{C}}_{l_1}({\varrho }_{0n})=⟨{\sigma }_0^x{\sigma }_n^x⟩.$$

$${\mathcal{C}}_{rel}({\varrho }_{0n})=\sum _{i=0}^1(-{\zeta }_i\log {\zeta }_i+{\eta }_i\log {\eta }_i+{\xi }_i\log {\xi }_i)-2\epsilon \log \epsilon ,$$

$${\zeta }_i=[1+⟨{\sigma }_0^z{\sigma }_n^z⟩+(-1{)}^{i}2⟨{\sigma }^z⟩]/4$$

Full text

Keywords: One-way quantum deficit, quantum coherence, quantum phase transitions, $XY$ chain

I Introduction

Quantum entanglement is considered to be a resource in many quantum information processing tasks Amico2008 ; Horodecki2009 , and entangled states have been experimentally created using 14 trapped ions Monz2011 , five superconducting qubits Barends2014 , and optical systems Lin2015a ; Heilmann2015 ; Wang2016 . Quantum entangled states can been used as resources in quantum cryptography Ekert1991 , quantum dense coding Bennett1992 , quantum communications Zou2014 , and quantum key distribution Cao2015 . Nonetheless, in the past few decades, it is realized that quantum correlations beyond entanglement also play essential roles in quantum information processing Modi2012 . Many measures have been proposed to quantify quantum correlations in physical systems, e.g., quantum discord Ollivier2001 ; Henderson2001 , one-way quantum deficit Oppenheim2002 , measurement induced disturbance Luo2008 , geometric discord Dakic2010 , quantum dissonance Modi2010 , and measurement induced non-locality Luo2011 .

In recent years, quantum coherence has also attracted considerable attention Streltsov2016 ; consequently, reasonable measures of quantum coherence have been discussed extensively Baumgratz2014 . As an analogy of quantum entanglement, quantum coherence may be also considered as a resource to characterize the classical-quantum boundary Adesso2016 .

The study of various quantum correlation measures in the ground states of spin models has been an active area of research. Entanglement in the finite size $XY$ chain has been investigated Osborne2002 . Multi-particle entanglement in an anisotropic $XY$ model in a transverse field has been explored by using different criteria for detecting the entanglement Giampaolo2013 ; Hofmann2014 , which shows that it obeys a scaling behavior near the critical point of the quantum phase transition in the model. Interest in the subject has increased since the introduction of quantum discord Ollivier2001 , which is one of the most important quantum correlations that characterizes the quantum phase transition Maziero2010 . Following quantum discord, many other measures have been introduced to explore the correlations in the context of the $XY$ model. The quantum phase transition is studied using local quantum uncertainty and Wigner-Yanase skew information Cakmak2015 . It has been shown that single-spin coherence reliably identifies the quantum phase transition in the thermal ground state of the anisotropic spin-1/2 $XY$ chain in the transverse magnetic field. Geometric discord is used to characterize the quantum phase transition for the $XY$ model Cheng2012 . There are many followed results dedicated to the quantum phase transition in other spin chain models Altintas2012 , such as $XXZ$, $XYT$, Lipkin-Meshkov-Glick (LMG), $XY$ with the Dzyaloshinskii-Moriya (DM) interaction Liu2011 , both in the thermodynamic limit and in few body cases.

In this article, we consider more general quantum correlations in the $XY$ model. The one-way quantum deficit is one of the popular measures that can characterize and quantify the quantum correlations Oppenheim2002 . Nevertheless, the one-way quantum deficit has not been studied with regards to characterizing the $XY$ spin-1/2 chain. Quantum coherence based on $l_{1}$ norm and relative entropy measures is also a basic and important method for characterizing quantum systems Baumgratz2014 . We capture the quantum phase transition using these measures to study quantum phase transitions for the $XY$ chain in a transverse field.

The article is organized as follows. In Section II, we recall the basic notation and concepts of the one-way quantum deficit and quantum coherence. The spin-1/2 anisotropic $XY$ chain is introduced in Section III. The numerical results regarding the quantum phase transition are presented in Section IV. Finally, we present our conclusions in Section V.

II One-way quantum deficit and

quantum coherence

Let us first review the basic definitions of the one-way quantum deficit and quantum coherence.

One-way quantum deficit  The one-way quantum deficit is defined as the difference in the von Neumann entropy of a bipartite state, $\varrho_{AB}$, before and after a measurement is performed, without a loss of generality, on particle $A$ Streltsov2011 ,

[TABLE]

where $\Pi_{A}$ is the measurement on subsystem $A$ and $S(\varrho_{AB})=-{\rm Tr}\varrho_{AB}\log\varrho_{AB}$ is the von Neumann entropy. Throughout the article, $\log$ is in base 2. The minimum is taken over all local measurements $\Pi_{A}.$

Quantum coherence  We consider the $l_{1}$ norm and relative entropy of coherence measures in this article. For a fixed basis set $\{|i\rangle\}$, the set of incoherent states $\mathcal{I}$ is the set of quantum states with diagonal density matrices, with respect to this basis. For an arbitrary quantum state

[TABLE]

the $l_{1}$ norm coherence, $\mathcal{C}_{l_{1}}(\varrho)$, of the state $\varrho$ is defined by

[TABLE]

which is the sum of the absolute values of all the non-diagonal entries for $\varrho$.

The relative entropy of the coherence is defined as

[TABLE]

where $S(\cdot)$ is the von Neumann entropy and $\varrho_{\rm diag}$ denotes the state obtained from $\varrho$ by deleting all the off-diagonal elements.

III Anisotropic $XY$ chain

The Hamiltonian, $H$, of the one-dimensional anisotropic spin-$\frac{1}{2}$ $XY$ chain in a transverse magnetic field is given by

[TABLE]

where $\sigma_{j}^{k}(k=x,y,z)$ is the $k$th component of the spin-1/2 Pauli operator acting on the $j$th spin, $\gamma$ is the degree of anisotropy (for simplicity we take this to be $0\leq\gamma\leq 1$), and $\lambda$ is the strength of the inverse of the external transverse magnetic field. In this study, we focus on the infinite chain case, $N\rightarrow\infty.$ When $\gamma=0$, the Hamiltonian reduces to the $XX$ chain, and the Ising model in transverse field when $\gamma=1$.

The diagonalization procedure for the $XY$ model includes the well-established techniques of Jordan-Wigner and the Bogoliubov transformation Barouch1970 ; Barouch1971 . By considering the thermal ground state, the reduced density operator for sites 0 and $n$ can be described by

[TABLE]

where $I_{0n}$ is the identity matrix acting on the state space of sites [math] and $n$. Here, $n$ indicates for the distance between two spins. The two-spin reduced density matrix is only dependent on the distance between the spins $n=|j-i|$, with $j,i$ denoting two different spins. The Hamiltonian exhibits global $Z_{2}$ symmetry. The density matrix is of the alphabet $X$ form,

[TABLE]

Owing to the fact that the system is invariant under translations, the entries of the two-site reduced-density depend only on the distance, $n$. The transverse magnetization is given by

[TABLE]

where $\omega_{\phi}=\sqrt{(\gamma\lambda\sin\phi)^{2}+(1+\lambda\cos\phi)^{2}}/2$ and $\beta=1/(kT)$ with $k$ being the Boltzmann’s constant and $T$ the absolute temperature. The two-point correlation function is given by

[TABLE]

[TABLE]

and

[TABLE]

with

[TABLE]

Tracing out one of the two spins, we have the reduced density matrix of a single-spin,

[TABLE]

where $\langle\sigma^{z}\rangle$ is the transverse magnetization in Eq.(12). All of the single-spin reduced density matrices are of the same form (27).

IV Behaviors of correlations

From the one-way quantum deficit defined in Eq.(1), we have

[TABLE]

where

[TABLE]

and

[TABLE]

We perform the complete set of orthonormal projective measurements, $\Pi_{n}$, on the $n$th nearest spins, with $\Pi_{n}^{i}=V|i\rangle\langle i|V^{\dagger}$, where $i\in\{0,1\}$ and

[TABLE]

Thus, we obtain the first term of Eq.(1) as follows

[TABLE]

where

[TABLE]

Therefore, the one-way quantum deficit is given by $\{\theta,\varphi\}$

[TABLE]

The $l_{1}$ norm of the coherence can be directly shown to be

[TABLE]

The relative entropy of the coherence is given by

[TABLE]

with

[TABLE]

and $\varepsilon=[1-\langle\sigma_{0}^{z}\sigma_{n}^{z}\rangle]/4.$

Remarkably, the numerical analysis implies that the extremization is achieved when $\lambda\leq 1$. The one-way quantum deficit can be represented by an analytical expression by choosing $\theta=\varphi=0$, which is in coincident with the relative entropy of the coherence $\Delta=\mathcal{C}_{rel}(\varrho_{0n})$. In the region where $\lambda>1$, it is only possible to obtain the numerical solution for the one-way quantum deficit.

IV.1 Quantum phase transition and correlations at zero

temperature

The one-way quantum deficit, relative entropy of coherence and $l_{1}$ norm for the first, second and fifth nearest neighbors in the thermal ground state (6) near zero temperature are depicted in Fig.1. The figures in the first row show the quantum coherence by the $l_{1}$ norm, the relative entropy of coherence and the one-way quantum deficit for $n=1,2$ and $5$, respectively, from left to right, with $\gamma=0.5$. The figures in the second row are the first order derivations of the corresponding quantum correlations with respect to the parameter $\lambda$.

As expected, all these three types of quantum correlations decrease as the distance, $n$, increases. Nonetheless, we see a clear difference between the regions where $\lambda<1$ and $\lambda>1$. It is evident that the first order derivations of the corresponding quantum correlations with respect to the parameter, $\lambda$, are singular at the quantum phase transition point $\lambda=1$. Namely, these quantum correlations can be used to characterize the quantum phase transition in this quantum system.

Some properties are obvious: the quantum coherence via the $l_{1}$ norm is greater than both the relative entropy of coherence and the one-way quantum deficit. The one-way quantum deficit coincides with the relative entropy of coherence in the region where $\lambda\leq 1$. While in the region where $\lambda>1$, the $l_{1}$ norm is greater than the relative entropy of coherence, which is greater than one-way quantum deficit. A critical point appears at $\lambda=1$. The first order derivations of these quantum correlations show the quantum phase transition clearly: they change sharply at $\lambda=1$ for $n=1,2$ and $5$, with $\gamma=0.5$.

IV.2 Quantum phase transition

and correlations at finite temperature

We now consider the thermal state of the $XX$ ($\gamma=0$) chain at finite temperatures. In this case, we plot the three kinds of quantum correlations for the first nearest neighbors, as shown in Fig.2. Similar results can be obtained for the other nearest neighbors, $n$. It can be seen that for a given $\lambda$ $(\lambda>1)$, the quantum correlations decrease as the temperature increases. However, in a given region of $\lambda$ $(\lambda\leq 1)$, the quantum correlations indeed increase as the temperature increases when near the critical value of $\lambda=1$. The quantum phase transition phenomenon disappears as the temperature increases.

For the case of the transverse Ising model ($\gamma=1$) at finite temperatures, as can be seen in the figures in the second row of Fig.2, the one-way quantum deficit increases when $\lambda$ increases until it is nearly 1. It then decreases as $\lambda$ increases, even in the high temperature region. Meanwhile the coherence always increases when $\lambda$ increases for a given temperature.

V Conclusions

In this study, we investigated the pairwise quantum correlations in the thermodynamic limit of the anisotropic $XY$ spin-1/2 chain in the presence of an external transverse magnetic field. The cases of both zero and finite temperatures have been considered. We have shown that the one-way quantum deficit, the $l_{1}$ norm, and the relative entropy of coherence can be used to characterize the quantum phase transition for the anisotropic $XY$ chain. It has been shown that in the region where $\lambda\leq 1$, the one-way quantum deficit has an analytical expression that coincides with the relative entropy of coherence. The critical point is at $\lambda=1.$ The $XX$ chain and the Ising models have also been studied at finite temperature. These results can highlight the investigation of the relations between quantum correlations and quantum phase transition.

VI Acknowledgments

This work is supported by the NSFC under grant numbers 11275131, 11305105, 11675113 and Jiangxi Education Department Fund (KJLD14088).