One-Way Deficit of Two Qubit States
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.
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 ()
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 ()
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
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
In the computational basis , its density matrix is
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
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 ()
After some algebra, we obtain
Next, we will evaluate the eigenvalues of by
The eigenvalues of are the same with the states , and
The eigenvalues of the equation (9) are
It can be directly verified that
Set , and
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
It converts the problem about to the problem about the function of one variable for minimum. That is
For an example, we set , and use the minimun command
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
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
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
We will only consider the following further simplified family of the states in Eq. (5), where
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.
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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