One-Way Deficit of Two Qubit X States

One-Way Deficit of Two Qubit States

Yao-Kun Wang Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China College of Mathematics, Tonghua Normal University, Tonghua, Jilin 134001, China    Naihuan Jing School of Mathematical Sciences, South China University of Technology, Guangzhou, Guangdong 510640, China Department of Mathematics, North Carolina State University, Raleigh, NC27695, USA    Shao-Ming Fei School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany    Zhi-Xi Wang School of Mathematical Sciences, Capital Normal University, Beijing 100048, China    Jun-Peng Cao Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing 100190, China    Heng Fan Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
Abstract

Quantum deficit originates in questions regarding work extraction from quantum systems coupled to a heat bath [Phys. Rev. Lett. 89, 180402 (2002)]. It links quantum correlations with quantum thermodynamics and provides a new standpoint for understanding quantum non-locality. In this paper, we propose a new method to evaluate the one-way deficit for a class of two-qubit states. The dynamic behavior of the one-way deficit under decoherence channel is investigated and it is shown that the one-way deficit of the states with five parameters is more robust against the decoherence than the entanglement.

I Introduction

The quantum entanglement is a resource in quantum information processing such as teleportationCH2 (), super-dense codingCH (), quantum cryptographyAK (), remote-state preparationAK2 () and so on. However, there are quantum correlations other than entanglement which are also useful and has attracted much attention recently zurek1 (); 1modi (); Giorgi (); 1Streltsov (); modi2 (). One remarkable and widely accepted quantity of quantum correlation is quantum discord. Quantum discord is a measure of the difference between the mutual information and maximum classical mutual information, which is generally difficult to calculate even for two qubit quantum system Ali (); Li (); chen (); shi (); Vinjanampathy ().

Other nonclassical correlations besides entanglement and quantum discord have arisen recently. For example, the quantum deficit oppenheim (); horodecki (), measurement-induced disturbance luo (), geometric discord luoandfu (); dakic (), and continuous-variable discord adesso (); giorda (), see a review modi2 (). Quantum deficit originates in question how to use nonlocal operation to extract work from a correlated system coupled to a heat bath oppenheim (). It is also closely related with other forms of quantum correlations. Oppenheim et al. define the work deficit oppenheim ()

(1)

where is the information of the whole system and is the localizable informationhorodecki2 (). As with quantum discord, quantum deficit is also equal to the difference of the mutual information and classical deficit oppenheim2 (). Recently, Streltsov et al. Streltsov0 (); chuan () give the definition of the one-way information deficit (one-way deficit) by the relative entropy, which reveals an important role of quantum deficit as a resource for the distribution of entanglement. One-way deficit by von Neumann measurement on one side is given bystreltsov ()

(2)

From the definition we can find that the one-way deficit and quantum discord are exactly different kinds of quantum correlation. One may wonder whether the analytical formula or the calculation method for a class of two-qubit states like quantum discord can be obtained. In this paper, we will endeavor to calculate the one-way deficit for quantum states with five parameters.

Ii One-Way Deficit for States with Five Parameters

We first introduce the form of two qubit states. By using proper local unitary transformations, we can write as

(3)

where r and s are Bloch vectors and are the standard Pauli matrices. When r=s=0, reduces to the two-qubit Bell-diagonal states. Then, we assume that the Bloch vectors are in direction, that is, , . The state in Eq. (3) turns into the following form

(4)

In the computational basis , its density matrix is

(5)

From Eq. (4) in chen (), after some algebraic calculations, we can obtain that parameters in chen () can be substituted for of the states in Eq. (5) successively and

(6)

One can also change them to be or direction via an appropriate local unitary transformation without losing its diagonal property of the correlation terms kim ().

The eigenvalues of the states in Eq. (5) are given by

The entropy is given by

(7)

Next, we evaluate the one-way deficit of the states in Eq. (5). Let

be the local measurement for the particle along the computational base ; then any von Neumann measurement for the particle can be written as

for some unitary . For any unitary , we have

with , , and After the measurement , the state will be changed to the ensemble with

To evaluate and , we write

By the relations luo ()

and

After some algebra, we obtain

where

Next, we will evaluate the eigenvalues of by

(8)

and

The eigenvalues of are the same with the states , and

(9)

The eigenvalues of the equation (9) are

(10)

It can be directly verified that

Set , and

(11)

Let us put then , , then , and the equality can be readily attained by appropriate choice of luo (). Therefore, we see that the range of values allowed for is .

The entropy of is

From Eq.(6), (11), we can obtain and

(13)

It converts the problem about to the problem about the function of one variable for minimum. That is

By Eqs. (2), (II), (II), the one-way deficit of the states in Eq. (5) is given by

where

For an example, we set , and use the minimun command

(16)

in “Wolfram Mathematics8.0” software, and obtain the value of the one-way deficit .

When , reduces to the two-qubit Bell-diagonal states. One-way deficit of Bell-diagonal states is

(17)

which is in consistent with the result using the simultaneous diagonalization theorem obtained in wang ().

It is worth mentioning that we have obtained a formula for solving one-way deficit. It is more simpler than the method using the joint entropy theoremshao ().

Iii Dynamics of one-way deficit under local nondissipative channels

The concurrence of the states in Eq. (5) can be calculated in terms of the eigenvalues of , where . The eigenvalues of are

The concurrence of the states in Eqs. (5) is given by

(18)

In the following we consider that the states in Eq. (5) undergoes the phase flip channel Maziero (), with the Kraus operators diag, diag, diag, diag, where , is the phase damping rate Maziero (); yu (). Let represent the operator of decoherence. Then under the phase flip channel we have

(19)

We will only consider the following further simplified family of the states in Eq. (5), where

(20)

As satisfies conditions in Eqs. (5), (20) and the one-way deficit of the under the phase flip channel is given by

As an example, for , , the dynamic behavior of correlation of the state under the phase flip channel is depicted in Fig.1. Here one sees that the concurrence become zero after the transition. We find that sudden death of entanglement appears at . Therefore for these states the entanglement is weaker against the decoherence than the one-way deficit.

Figure 1: (Color online) Concurrence(blue dashed line) and one-way deficit(red solid line) under phase flip channel for , , and .

Iv summary

We have given a new method to evaluate the one-way deficit for states with five parameters. By this way, we can evaluate one-way deficit of the wide range states than the method using the simultaneous diagonalization theorem. Meanwhile, this way is more simpler than the method using the joint entropy theorem. The dynamic behavior of the one-way deficit under decoherence channel is investigated. It is shown that one-way deficit of the states is more robust against the decoherence than quantum entanglement.


Acknowledgments This work was supported by NSFC (11175248).

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