Quantum Discords of Tripartite Quantum Systems

The quantum discord usually involves with quantum entanglement and umentangled quantum correlations in quantum systems. It measures the total non-classical correlation in a quantum system, and has attracted widespread attention since its appearance. Applications of the non-entanglement quantum correlations in quantum information processings have been extensively studied, including the quantum computing scheme of DQC1 [1] and Grover search algorithm [2] etc. This partly explains why quantum schemes surpass classical schemes. Meanwhile, the quantum discord as a non-classical correlation is one of the important quantum resources and is ubiquitous in many areas of modern physics ranging from condensed matter physics, quantum optics, high-energy physics to quantum chemistry, thus can be regarded as one of the fundamental non-classical correlations besides entanglement and EPR-steerable states [3, 4].

The quantum discord is defined as the maximal difference between the quantum mutual information without and with a von Neumann projective measurement applying to one part of the bipartite system. For tripartite and lager systems, some generalizations of the discord have been proposed [5, 6, 7, 8, 9, 10], and have been used in quantum information processings. It is well-known that quantum discord is extremely difficult to evaluate and most exact solutions are only for the X-type quantum states (cf. [11, 12, 13, 14]). This paper is devoted to quantification of the quantum correlation in tripartite and larger systems to derive some exact solutions for non-X-type states, and we hope it can contribute to better understanding and more effective use of quantum states in realizing quantum information processing schemes.

The paper is organized as follows. We first introduce the generalized discord for tripartite systems [10] based on that of bipartite systems [3]. We derive analytic solutions for tripartite states with fourteen parameters. Furthermore, the quantum discord of some well-known states (such as GHZ states) are computed.

For a bipartite state $\phi^{bc}$ on system $H_{B}\otimes H_{C}$, the quantum mutual information is $I(\phi^{bc}):=S_{B}(\phi^{b})+S_{C}(\phi^{c})-S_{BC}(\phi^{bc})$, where $S(\phi^{X})=\mathrm{Tr}\phi^{X}\log_{2}(\phi^{X})$ is the von Neumann entropy of the quantum state on system X. Set $\{\Pi^{B}_{k}\}$ to be an one-dimensional von Neumann projection operator on subsystem $B$ which satisfies $\sum_{k}\Pi_{k}^{B}=I,(\Pi_{k}^{B})^{2}=\Pi_{k}^{B},\Pi_{k}^{B}\Pi_{k^{\prime}}^{B}=\delta_{kk^{{}^{\prime}}}$. Then the state $\phi^{bc}$ under the measurement $\{\Pi^{B}_{k}\}$ is changed into

with the probability $p_{k}=\mathrm{Tr}(I\otimes\Pi^{B}_{k})\phi^{bc}(I\otimes\Pi^{B}_{k})$. For simplicity, we denote by $\Pi^{X}$ the measurement $\{\Pi_{k}^{X}\}$ on system $X$. The quantum conditional entropy is simply given by $S_{C|\Pi^{B}}(\phi^{bc})=\sum_{k}p_{k}S(\phi^{c}_{k})$. Then the measurement-induced quantum mutual information is given by

By Olliver and Zurek [3], the original definition of the quantum discord $Q(\rho)$ is the difference of the quantum mutual information $I(\phi^{bc})$ and the measurement-induced quantum mutual information $C(\phi^{bc})$, i.e.

where $S_{C|B}(\phi^{bc})=S_{BC}(\phi^{bc})-S_{B}(\phi^{b})$ is the unmeasured conditional state on subsystem $C$.

For the tripartite system $H_{A}\otimes H_{B}\otimes H_{C}$, we consider the $BC$ composite system as the first subsystem and $A$-system as the second subsystem. The state $\rho^{abc}$ of system $H_{A}\otimes H_{B}\otimes H_{C}$ gives arise to a state on $BC$-subsystem after the von Neumann measurement $\{\Pi_{j}^{A}\}$ on $A$ subsystem. Namely, it takes the following form:

with probability $p^{bc}_{j}=\mathrm{Tr}(\Pi_{j}^{A}\otimes I)\rho^{abc}(\Pi_{j}^{A}\otimes I)$.The measured quantum mutual information of $\rho^{abc}$ is naturally given by

The quantity of classical correlation of the tripartite state $\rho^{abc}$ is

We know that the quantum mutual information $I(\rho^{abc})=S_{A}(\rho^{a})+S_{BC}(\rho^{bc})-S_{ABC}(\rho^{abc})$. Similar to Eq.(2.1), the generalized quantum discord of the tripartite state $\rho^{abc}$ can be defined as

where $S_{BC|A}(\rho^{abc})=S_{ABC}(\rho^{abc})-S_{A}(\rho^{a})$ is the unmeasured conditional entropy on $BC$-bipartite subsystem.

In order to evaluate the quantity $\min_{\Pi^{A}}S_{BC|\Pi^{A}}(\rho^{abc})$, the multipartite measurement based on conditional operators can be constructed as follows: [15]

with the measurement ordering from $A$ to $B$. The projector $\Pi_{k|j}^{B}$ on subsystem $B$ is conditional measurement outcome of $A$. These projectors satisfy $\sum_{k}\Pi^{B}_{k|j}=I^{B},\sum_{j}\Pi_{j}^{A}=I^{A}$. Then after the measurement $\Pi_{jk}^{AB}$, the state $\rho^{abc}$ is collapsed to a state on subsystem $C$, i.e.

with the probability $p^{c}_{jk}=\mathrm{Tr}(\Pi_{jk}^{AB}\otimes I)\rho^{abc}(\Pi_{jk}^{AB}\otimes I)$. The conditional entropy after the $AB$-bipartite measurement is

where $\lambda_{l}^{(jk)}$ are eigenvalues of state $\rho^{c}_{jk}$.

Let $\rho_{\Pi^{X}}=\sum_{\Pi^{X}}\Pi^{X}\rho\Pi^{X}$ be the state after measurement $\Pi^{X}$. Then for a bipartite state $\rho^{ab}$, the conditional entropy on subsystem $B$ after the measurement on subsystem $A$ is

By [10, Eq.(6)], the entropy of the measured system can always be decomposed as

For the tripartite system, using the measurement $\Pi^{AB}$, we have

when the measurement on $A$ system is $\Pi^{A}$, then we have

By Eq.(2.9), Eq.(2.10), Eq.(2.11), we have that

Meanwhile, $S_{A}(\rho^{abc}_{\Pi^{AB}})=S_{A}(\rho^{abc}_{\Pi^{A}})$, so the generalization discord of a tripartite state can be written as [10]

For the product states in the tripartite system, the discord has the special property that it reduces to the standard bipartite discord when only bipartite quantum correlations are present. This means $\mathcal{Q}_{ABC}(\rho^{x}\otimes\rho^{y})=\mathcal{Q}_{X}(\rho^{x})$ for $X=AB,BC$ and $AC$ subsystem. We consider the following tripartite states

where $I_{d}$ represents the unit matrix of order $d$, and $\sigma_{i}(i=1,2,3)$ are Pauli matrices. The parameters $a_{3},b_{3},c_{i},r_{i},s_{i},T_{i}\in\mathbb{R}$ and they are confined within the internal $[-1,1]$. Its matrix has the following form:

Let $\{|j\rangle\langle j|,j=0,1\}$ be the computational base, then any von Neumann measurement on system $X$ can be written as $\{\Pi_{j}^{X}=V|j\rangle\langle j|V^{\dagger}|,j=0,1\}$ for some unitary matrix $V\in\mathrm{SU}(2)$. Any unitary matrix can be written as $V=tI+\sqrt{-1}\sum_{k}y_{k}\sigma_{k}$ with $t,y_{k}\in\mathbb{R}$. When the measurement $\{\Pi^{X}_{j}\}$ is performed locally on one part of the composite system $Y\otimes X$, the ensemble $\{\rho_{j}^{Y},p_{j}^{Y}\}$ is given by $\rho_{j}^{Y}=\frac{1}{p_{j}^{Y}}\mathrm{Tr}_{X}(I\otimes\Pi^{X}_{j})\rho^{YX}(I\otimes\Pi^{X}_{j})$ with the probability $p_{j}^{Y}=\mathrm{Tr}[\rho^{YX}(I\otimes\Pi_{j}^{X})]$.

It follows from symmetry that

Introduce new variables $z_{1}^{X}=2(-ty_{2}+y_{1}y_{3}),z_{2}^{X}=2(ty_{1}+y_{2}y_{3}),z_{3}^{X}=(t^{2}+y_{3}^{2}-y_{1}^{2}-y_{2}^{2})$, then $(z_{1}^{X})^{2}+(z_{2}^{X})^{2}+(z_{3}^{X})^{2}=1$. Therefore $\Pi_{j}^{X}\sigma_{k}\Pi_{j}^{X}=(-1)^{j}z_{k}^{X}\Pi_{j}^{X}$ for $j=0,1$ and $k=1,2,3$.

For the tripartite state $\rho^{abc}$, the conditional state on $BC$ subsystem after measurement $\{\Pi_{j}^{A}(j=0,1)\}$ on subsystem $A$ is

where the probabilities are

and $\sum_{i}^{3}(z^{A}_{i})^{2}=1$. Therefore the reduced state of $\rho_{j}^{bc}$ is

with the probability $p_{j}^{b}=p_{j}^{bc}=\frac{1}{2}[1+(-1)^{j}a_{3}z_{3}^{A}]$. The eigenvalues of $\rho_{j}^{b}$ are

We define the following entropy function

Then measured conditional entropy of $B$ subsystem can be obtained as [3, 4, 14, 16, 17, 18]

where $A_{\pm}=\sqrt{(b_{3}\pm r_{3}z_{3}^{A})^{2}+\sum_{i}^{2}(r_{i}z_{i}^{A})^{2}}$.

After measurement $\Pi_{k|j}^{B}$ on $BC$ system, the state $\rho_{j}^{bc}$ is changed to

with the probability ($k=0,1$)

where $\alpha_{k}=a_{3}z_{3}^{A}+(-1)^{k}(b_{3}z_{3}^{B}+\sum_{i}^{3}r_{i}z_{i}^{A}z_{i}^{B}),\beta_{k}=-a_{3}z_{3}^{A}+(-1)^{k}(b_{3}z_{3}^{B}-\sum_{i}^{3}r_{i}z_{i}^{A}z_{i}^{B})$. The non-zero eigenvalues of $\rho^{c}_{jk}$ are given by

where

According to the fact that the eigenvalues in Eq.(3.9) are nonnegative, we have $\sqrt{\sum_{i}^{3}a_{i}^{2}}+\sqrt{\sum_{i}^{3}b_{i}^{2}}+\sqrt{\sum_{i}^{3}r_{i}^{2}}\leq 1$.

The entropy of $\rho^{abc}$ under the measurement $\Pi^{AB}$ is given by

In particularly, $a_{3}z_{3}^{A}=\frac{\alpha_{0}+\alpha_{1}}{2}$.

Let $G(z_{1}^{A},z_{2}^{A},z_{3}^{A})=1-S_{B|\Pi^{A}}(\rho)$ and $F(z_{1}^{A},z_{2}^{A},z_{3}^{A},z^{B}_{1},z^{B}_{2},z^{B}_{3})=2-S_{C|\Pi^{AB}}(\rho)$, then we have the following result.