Computable measure of quantum correlation

Abstract

A general state of an $m\otimes n$ system is a classical-quantum state if and only if its associated $A$-correlation matrix (a matrix constructed from the coherence vector of the party $A$, the correlation matrix of the state, and a function of the local coherence vector of the subsystem $B$), has rank no larger than $m-1$. Using the general Schatten $p$-norms, we quantify quantum correlation by measuring any violation of this condition. The required minimization can be carried out for the general $p$-norms and any function of the local coherence vector of the unmeasured subsystem, leading to a class of computable quantities which can be used to capture the quantumness of correlations due to the subsystem $A$. We introduce two special members of these quantifiers: The first one coincides with the tight lower bound on the geometric measure of discord, so that such lower bound fully captures the quantum correlation of a bipartite system. Accordingly, a vanishing tight lower bound on the geometric discord is a necessary and sufficient condition for a state to be zero-discord. The second quantifier has the property that it is invariant under a local and reversible operation performed on the unmeasured subsystem, so that it can be regarded as a computable well-defined measure of the quantum correlations. The approach presented in this paper provides a way to circumvent the problem with the geometric discord. We provide some examples to exemplify this measure.

Appendices

Appendix A: Geometric discord and its tight lower bound

In this appendix, we provide a proof to show that Eq. (9) can be regarded as an alternative form for the geometric discord (1). To this aim, we first introduce another equivalent form for the geometric discord given by Luo and Fu [21] as

$$\begin{aligned} D_G(\rho )={\mathrm {Tr}}{(CC^{{\mathrm t}})}-\max _{A}{\mathrm {Tr}}{(ACC^{{\mathrm t}}A^{{\mathrm t}})}, \end{aligned}$$

(48)

where $C=(c_{ij})$ is an $m^2\times n^2$-dimensional matrix defined by

$$\begin{aligned} C=\frac{1}{\sqrt{mn}} \left( \begin{array}{c@{\quad }c} 1 &{} \sqrt{\frac{2}{n}}\vec {y}^{{\mathrm t}} \\ \sqrt{\frac{2}{m}}\vec {x} &{} \frac{2}{\sqrt{mn}}T \end{array} \right) . \end{aligned}$$

(49)

More precisely, $c_{ij}$ can be regarded as the expansion coefficients of $\rho$ in terms of the orthonormal basis $X_i^A\otimes X_j^B$ (for $i=0,\ldots ,m^2-1$ and $j=0,\ldots ,n^2-1$) with $X_0^A=\frac{1}{\sqrt{m}}{\mathbb I}^A$, $X_0^B=\frac{1}{\sqrt{n}}{\mathbb I}^B$, and $X_{i\ne 0}^s=\frac{1}{\sqrt{2}}\hat{\lambda }_i^s$ for $s=A,B$. Furthermore, in Eq. (48) the maximum is taken over all $m\times m^2$-dimensional matrices $A=(a_{ki})$ such that

$$\begin{aligned} a_{ki}={\mathrm {Tr}}{(|k\rangle \langle k|X_i^A)}=\langle k|X_i^A|k\rangle , \end{aligned}$$

(50)

where $\{|k\rangle \}_{k=1}^{m}$ is any orthonormal base for $\mathcal {H}^{A}$.

Alternative form for geometric discord—Following [21] we represent the projection operators corresponding to this base as

$$\begin{aligned} \Pi _{k}^{A}=|k\rangle \langle k|=\sum _{i=0}^{m^2-1}a_{ki}X_{i}^A, \end{aligned}$$

(51)

where $a_{ki}$ are defined in Eq. (50) and $k=1,\ldots ,m$. It is easy to see that we can write matrix $A=(a_{ki})$ as below

$$\begin{aligned} A=\left( \begin{array}{c@{\quad }c}\frac{1}{\sqrt{m}} &{} \vec {\mu }_1^{{\mathrm t}} \\ \vdots &{} \vdots \\ \frac{1}{\sqrt{m}} &{} \vec {\mu }_{m}^{{\mathrm t}}\end{array}\right) , \end{aligned}$$

(52)

where $\vec {\mu }_k=\frac{\sqrt{2}}{m}\vec {\alpha }_k$ with $\vec {\alpha }_k$ as defined in Eqs. (16) and (17). Therefore, vectors $\{\vec {\mu }_k\}_{k=1}^{m}$ make the $(m-1)$-dimensional simplex $\Delta ^{m-1}_{\{\vec {\mu }_k\}\in \mathbb {R}^{m^2-1}}$. Using Eq. (49) we get

$$\begin{aligned} {\mathrm {Tr}}{(CC^T)}=\frac{1}{mn}\left[ \left( 1+\frac{2}{n}\vec {y}^{{\mathrm t}}\vec {y}\right) +\frac{2}{m}{\mathrm {Tr}}{G}\right] , \end{aligned}$$

(53)

and

$$\begin{aligned}{}[ACC^TA^T]_{kk^{\prime }}&= \frac{1}{m^2n}\left[ \left( 1+\frac{2}{n}\vec {y}^{{\mathrm t}}\vec {y}\right) +2\vec {\mu }_k^{{\mathrm t}} G\vec {\mu }_{k^{\prime }} \right] \nonumber \\&\quad +\frac{\sqrt{2}}{m^2n}\left[ \vec {\mu }_k^{{\mathrm t}} \left( \vec {x}+\frac{2}{n}T\vec {y}\right) +\left( \vec {x}^{{\mathrm t}} +\frac{2}{n}\vec {y}^{{\mathrm t}} T^{{\mathrm t}}\right) \vec {\mu }_{k^{\prime }}\right] , \end{aligned}$$

(54)

where $G$ is defined by Eq. (8). We find, therefore,

$$\begin{aligned} {\mathrm {Tr}}{[ACC^TA^T]}=\frac{1}{mn}\left[ \left( 1+\frac{2}{n}\vec {y}^{{\mathrm t}}\vec {y}\right) +\frac{2}{m}\sum _{k=1}^{m}\vec {\mu }_k^{{\mathrm t}}G\vec {\mu }_k \right] , \end{aligned}$$

(55)

where we have used the fact that $\sum _{k=1}^{m}\vec {\mu }_k=\vec {0}$. Substituting Eqs. (53) and (55) into Eq. (48), we arrive at the following form for the geometric discord

$$\begin{aligned} D_{G}(\rho )=\frac{2}{m^2 n}\left[ {\mathrm {Tr}}{G}-\max _{\{\vec {\mu }_k\}} \sum _{k=1}^{m}\vec {\mu }_k^{{\mathrm t}} G \vec {\mu }_k\right] . \end{aligned}$$

(56)

Here maximum is taken over all simplexes $\Delta ^{m-1}_{\{\vec {\mu }_k\}\in \mathbb {R}^{m^2-1}}$, i.e., over all vectors $\{\vec {\mu }_k\}_{k=1}^{m}\in \mathbb {R}^{m^2-1}$ fulfilling conditions $\vec {\mu }_k\cdot \vec {\mu }_{k^\prime }=\left( \delta _{kk^\prime }-\frac{1}{m}\right)$ and $\sum _{k=1}^{m}\vec {\mu }_k=\vec {0}$. To gain further insight into the meaning of the above equation, it is worth to compare it with Eq. (25) for $p=2$, $f_1(y)=f_2(y)=1$. It turns out that the calculation of $[D_{{\mathcal T},f=1}^{(2)}(\rho )]^2$ needs to perform optimization over $(m-1)$-dimensional projection operators $P$, which can be solved exactly, but in calculation of $D_G(\rho )$ we have to make optimization over $(m-1)$-dimensional simplexes $\Delta ^{m-1}_{\{\vec {\mu }_k\}\in {\mathbb R}^{m^2-1}}$, where does not have an exact solution in general. Two definitions become identical when $m=2$, namely for $2\otimes n$ systems. This happens because in case $m=2$, calculation of the geometric discord leads to the problem of optimization over one-dimensional simplexes $\Delta ^{1}_{\{\vec {\mu }_1,\vec {\mu }_2\}\in {\mathbb R}^{3}}$ with $\vec {\mu }_1=-\vec {\mu }_2=\frac{1}{\sqrt{2}}\vec {\alpha }_1$ and $|\vec {\alpha }_1|=1$, which is the same as the problem of optimization over one-dimensional projection operators $P$, and get

$$\begin{aligned}{}[D_{{\mathcal T},f=1}^{(2)}(\rho )]^2=D_{G}(\rho )&= \frac{1}{2 n}\left[ {\mathrm {Tr}}{G}-\max _{\vec {\alpha }_1}\{\vec {\alpha }_1^{{\mathrm t}} G \vec {\alpha }_1\}\right] \nonumber \\&= \frac{1}{2 n}\left[ {\mathrm {Tr}}{G}-\eta _{1}\right] =\frac{1}{2 n}\left[ \eta _2+\eta _3\right] . \end{aligned}$$

(57)

where we have defined $\eta _{1}\ge \eta _{2}\ge \eta _{3}\ge 0$ as the eigenvalues of $G$. This agrees with the result obtained in Refs. [3, 25].

Tight lower bound on the geometric discord [23, 24]—Unfortunately, for $m>2$, the maximization involved in Eq. (56) can not be solved analytically and we need to obtain lower bound. To do so, let $\{|s\rangle \}_{s=1}^{m}$ be the standard base of the space $\mathcal {H}^{A}$, namely the one which the SU$(m)$ generators $\{\hat{\lambda }^A_i\}_{i=1}^{m^2-1}$ are expanded in terms of them. Similar to Eq. (51), we can write

$$\begin{aligned} \Pi _{s}^{A}=|s\rangle \langle s|=\sum _{i=0}^{m^2-1}b_{si}X_{i}^A, \end{aligned}$$

(58)

where

$$\begin{aligned} b_{si}={\mathrm {Tr}}{[|s\rangle \langle s|X_{i}^A]}=\langle s|X_{i}^A|s\rangle , \end{aligned}$$

(59)

for $s=1,\ldots ,m$ and $i=0,\ldots ,m^2-1$. Now if we choose the basis of the algebra in such a way that Cartan subalgebra makes the first $m-1$ generators, then we can write matrix $B=(b_{si})$ as follows

$$\begin{aligned} B=\left( \begin{array}{c@{\quad }c}\frac{1}{\sqrt{m}} &{} \vec {{\tilde{\nu }}}_1^{{\mathrm t}} \\ \vdots &{} \vdots \\ \frac{1}{\sqrt{m}} &{} \vec {{\tilde{\nu }}}_{m}^{{\mathrm t}} \end{array}\right) . \end{aligned}$$

(60)

Here $\{\vec {{\tilde{\nu }}}_s\}_{s=1}^{m}$ are vectors in ${\mathbb R}^{m^2-1}$ such that only first $m-1$ components of them are nonzero. So, we can write $\vec {{\tilde{\nu }}}_s=(\vec {\nu }_s,\vec {0})$ where $\{\vec {\nu }_s\}_{s=1}^{m}$ are vectors in ${\mathbb R}^{m-1}$, and $\vec {0}$ denotes null vectors in ${\mathbb R}^{m(m-1)}$. It is worth to mention that vectors $\{\vec {\nu }_s\}_{s=1}^{m}$ are in fact weight vectors of the SU$(m)$ Lie algebra in the defining representation [41] and satisfy the following orthonormality condition

$$\begin{aligned} \sum _{s=1}^{m}(\vec {\nu }_s)_k(\vec {\nu }_s)_l=\delta _{kl}. \end{aligned}$$

(61)

In view of this, the zero vectors $\vec {0}$ of the definition $\vec {{\tilde{\nu }}}_s=(\vec {\nu }_s,\vec {0})$ arise from the diagonal elements of the root operators of the algebra, which are all zero. Therefore, vectors $\{\vec {\tilde{\nu }}_s\}_{s=1}^{m}$ makes simplex $\Delta ^{m-1}_{\{\vec {\tilde{\nu }}_s\}\in \mathbb {R}^{m^2-1}}$, or equivalently simplex $\Delta ^{m-1}_{\{\vec {\nu }_s\}\in \mathbb {R}^{m-1}}$. Evidently, the general base $\{|k\rangle \}_{k=1}^{m}$ can be obtained from the standard one by a unitary transformation $U\in \mathrm{{SU}}(m)$ as $\{|k\rangle \}=U\{|s\rangle \}$. Corresponding to this, there exists orthogonal transformation $\tilde{R}\in \mathrm{{SO}}(m^2-1)$ such that the general simplex $\Delta ^{m-1}_{\{\vec {\mu }_k\}\in \mathbb {R}^{m^2-1}}$ can be obtained from $\Delta ^{m-1}_{\{\vec {\tilde{\nu }}_s\}\in \mathbb {R}^{m^2-1}}$, i.e.,

$$\begin{aligned} (\vec {\mu }_k)_i=\sum _{j=1}^{m^2-1}{\tilde{R}}_{ij}(\vec {\tilde{\nu }}_k)_j=\sum _{j=1}^{m-1}R_{ij}(\vec {\nu }_k)_j, \end{aligned}$$

(62)

for $i=1,2,\ldots , m^2-1$. In the second equality $R=(R_{ij})=(\hat{n}_j)_i$ is an $(m^2-1)\times (m-1)$ left orthogonal matrix [23, 24], i.e., $R^{{\mathrm t}}R=I_{m-1}$, and $\hat{n}_j\in \mathbb {R}^{m^2-1}$ ($j=1,\ldots ,m-1$) are orthonormal vectors, i.e., $\hat{n}_i\cdot \hat{n}_{i^\prime }=\delta _{ii^\prime }$. Using this and Eq. (61), we get

$$\begin{aligned} \max _{\{\vec {\mu }_k\}} \sum _{k=1}^{m}\vec {\mu }_k^{{\mathrm t}} G \vec {\mu }_k\le \max _{\{\hat{n}_i\}} \sum _{i=1}^{m-1}\hat{n}_i^{{\mathrm t}} G \hat{n}_i = \sum _{i=1}^{m-1}\eta _{i}^\downarrow , \end{aligned}$$

(63)

where $\{\eta _k^{\downarrow }\}_{k=1}^{m^2-1}$ are eigenvalues of $G$ in nonincreasing order. Using this in Eq. (56), we find the desired lower bound (7) for the geometric discord, which is already obtained in Refs. [23, 24]. It is worth to mention that in the particular case $m=2$, the obtained bound gives exact result for the geometric discord (see Eq. (57)). This follows from the homomorphism SU$(2)\sim \mathrm{{SO}}(3)$, happens only for $m=2$. On the other hand, for $m>2$ the set of all unitary transformations $U\in \mathrm{{SU}}(m)$ acting on the $m$-dimensional Hilbert space $\mathcal {H}^A$ will be a subset of the matrices in $\mathrm{{SO}}(m^2-1)$. This implies that there exist rotations $\tilde{R}\in \mathrm{{SO}}(m^2-1)$ that are not correspond to any $U\in \mathrm{{SU}}(m)$, leading, therefore, to the inequality (63).

Appendix B: A proof for Theorem 2

In this appendix, we provide a proof for Theorem 2. To this aim, we need the following lemma.

Lemma 7

• (i) If $\rho$ is a zero-discord state on the space $\mathcal {H}^{A}\otimes \mathcal {H}^{B}$, then its corresponding local coherence vectors $\vec {x}$, $\vec {y}$, and the correlation matrix $T$ can be represented by the following equations

  $$\begin{aligned} \vec {x}&= \sum _{k=1}^{m}p_k \vec {\alpha }_{k},\qquad \vec {y}=\sum _{k=1}^{m}p_k \vec {\xi }_{k}, \end{aligned}$$

  (64)
  $$\begin{aligned} T&= \sum _{k=1}^{m}p_k (\vec {\alpha }_{k})(\vec {\xi }_{k})^{{\mathrm t}}, \end{aligned}$$

  (65)

  where $\{\vec {\alpha }_{k}\}_{k=1}^m$ denote coherence vectors associated to the orthonormal projection operators of the subsystem $A$, hence satisfy Eqs. (16) and (17), but $\{\vec {\xi }_{k}\}_{k=1}^m$ are coherence vectors of arbitrary states of the subsystem $B$.

• (ii) If $\rho$ is an arbitrary bipartite state, then its corresponding local coherence vectors $\vec {x}$ and $\vec {y}$ can be represented by Eq. (64).

Proof

• (i) Use the coherence vector representations for $\Pi _k^A$ and $\rho _k^B$ as

  $$\begin{aligned} \Pi _k^{A}=\frac{1}{m}\left( {\mathbb I}+\vec {\alpha }_{k}\cdot \hat{\lambda }^A\right) ,\quad \rho _k^{B}=\frac{1}{n}\left( {\mathbb I}+\vec {\xi }_{k}\cdot \hat{\lambda }^B\right) , \end{aligned}$$

  (66)

  and insert them in the definition of zero-discord state (19). Comparing the result with the definition of $\rho$ given in Eq. (3), one can obtain the coherence vectors $\vec {x}$, $\vec {y}$ and the correlation matrix $T$ as given by Eqs. (64) and (65).

• (ii) Let $\rho ^{A}=\sum _{k=1}^{m}p_k\Pi _k^A$, with $\{\Pi _k^A\}_{k=1}^{m}$ orthonormal projections on $\mathcal {H}^{A}$, be the eigenspectral decomposition of $\rho ^{A}$. Then denoting coherence vectors of $\{\Pi _k^A\}_{k=1}^{m}$ by $\{\vec {\alpha }_k\}_{k=1}^{m}$, we find that $\vec {x}=\sum _{k=1}^{m}p_k \vec {\alpha }_{k}$. Now having $\{p_k\}_{k=1}^{m}$, we can always find set $\{\rho _k^B\}_{k=1}^{m}$ such that ensemble $\{p_k,\rho _k^B\}_{k=1}^{m}$ realizes $\rho ^B$, i.e., $\rho ^B=\sum _{k=1}^{m}p_k\rho _k^B$. Now letting $\{\vec {\xi }_k\}_{k=1}^{m}$ be coherence vectors of $\{\rho _k^B\}_{k=1}^{m}$, we get $\vec {y}=\sum _{k=1}^mp_k\vec {\xi }_k$. Note that for a given probability set $\{p_k\}_{k=1}^{m}$, states $\{\rho _k^B\}_{k=1}^{m}$ which realize $\rho ^B$ are not unique, so associated coherence vectors $\{\vec {\xi }_k\}_{k=1}^{m}$ are not unique too.$\square$

Now we are in a position to present the proof for Theorem 2. If $\rho$ is a zero-discord state, then by Lemma 7 its corresponding local coherence vectors $\vec {x}$, $\vec {y}$ and correlation matrix $T$ can be represented by Eqs. (64) and (65), with $\{\vec {\alpha }_k\}_{k=1}^{m}$ as coherence vectors corresponding to orthonormal projections. Defining ${P}$ as (18) and using the properties $\{\vec {\alpha }_k\}_{k=1}^{m}$ given in Eq. (16), one can easily shows that conditions (20) are satisfied. Conversely, we have to proof that if Eq. (20) is satisfied, then $\rho$ is a zero-discord state, i.e., its corresponding $\vec {x}$, $\vec {y}$ and $T$ have the form given by Eqs. (64) and (65). To do this, we first note that Eq. (64) is satisfied for a general state $\rho$. But by assumption Eq. (20) is also satisfied, leading, therefore, to the following form for the correlation matrix $T$

$$\begin{aligned} T=\sum _{k=1}^{m}\sum _{l=1}^{m}p_{kl}(\vec {\alpha }_{k})(\vec {\eta }_{l})^{{\mathrm t}}. \end{aligned}$$

(67)

Since $\{\vec {\xi }_i\}_{i=1}^m$ are not unique, we can, therefore, choose them in such a way that they can be expanded in terms of $\{\vec {\eta }_l\}_{l=1}^m$ as $p_k\vec {\xi }_k=\sum _{l=1}^m p_{kl}\vec {\eta }_l$. Substituting this into Eq. (67) we get Eq. (65), therefore, $\vec {x}$, $\vec {y}$ and $T$ take the form given by Eqs. (64) and (65), hence $\rho$ is a zero-discord state.