Relating an entanglement measure with statistical correlators for two-qudit mixed states using only a pair of complementary observables

Abstract

We focus on characterising entanglement of high-dimensional bipartite states using various statistical correlators for two-qudit mixed states. The salient results obtained are as follows: (a) A scheme for determining the entanglement measure given by negativity is explored by analytically relating it to the widely used statistical correlators, viz. mutual predictability, mutual information and Pearson correlation coefficient, for different types of bipartite arbitrary-dimensional mixed states. Importantly, this is demonstrated using only a pair of complementary observables pertaining to the mutually unbiased bases. (b) The relations thus derived provide the separability bounds for detecting entanglement obtained for a fixed choice of the complementary observables, while the bounds per se are state-dependent. Such bounds are compared with the earlier suggested separability bounds. (c) We also show how these statistical correlators can enable distinguishing between the separable, distillable and bound entanglement domains of the one-parameter Horodecki two-qutrit states. Further, the relations linking negativity with the statistical correlators have been derived for such Horodecki states in the domain of distillable entanglement. Thus, this entanglement characterisation scheme based on statistical correlators and harnessing complementarity of the observables opens up a potentially rich direction of study which is applicable for both distillable and bound entangled states.

Appendices

Joint probabilities of measurement outcomes for the chosen states

Statistical correlators like MP, MI and PCC are functions of the joint probabilities of outcomes of the measurements performed on the bipartite state considered. Here we explicitly derive the relevant expressions for the classes of states that we have considered.

1.1 Noisy Bell state

The density operators for the three types of noises are given in Eqs. (9), (10) and (11). The joint probabilities of measurement outcomes for all the three types with respect to the Z basis are

$$\begin{aligned} p_{\textrm{iso},Z} =&\langle i,j|\frac{\mathbbm {1}}{d^2}|i,j\rangle = \frac{1}{d^2} \end{aligned}$$

(36)

$$\begin{aligned} p_{\textrm{cna},Z} =&\frac{1}{d}\sum \limits _{a=0}^{d-1} \langle i,j |a,a\rangle \langle a,a |i,j\rangle = \frac{1}{d}\delta _{ij} \end{aligned}$$

(37)

$$\begin{aligned} p_{\textrm{cnb},Z} =&\frac{1}{d(d-1)} \sum \limits _{\begin{array}{c} a,b=0\\ a\ne b \end{array}} \langle i,j|a,b\rangle \langle a,b|i,j\rangle = \frac{1}{d(d-1)} (1 - \delta _{ij}) \end{aligned}$$

(38)

and therefore, the mutual predictabilities of all these three states are given by

$$\begin{aligned} P_{\textrm{iso},Z} =&\frac{1}{d} \end{aligned}$$

(39)

$$\begin{aligned} P_{\textrm{cna},Z} =&1 \end{aligned}$$

(40)

$$\begin{aligned} P_{\textrm{cnb},Z} =&0 \end{aligned}$$

(41)

Similarly, with respect to the X basis, one obtains

$$\begin{aligned} p_{\textrm{iso},X} =&\langle i,j|\frac{\mathbbm {1}}{d^2}|i,j\rangle = \frac{1}{d^2} \end{aligned}$$

(42)

$$\begin{aligned} p_{\textrm{cna},X} =&\frac{1}{d}\sum \limits _{a=0}^{d-1} \langle i,j |a,a\rangle \langle a,a |i,j\rangle = \frac{1}{d^2} \end{aligned}$$

(43)

$$\begin{aligned} p_{\textrm{cnb},X} =&\frac{1}{d(d-1)} \sum \limits _{\begin{array}{c} a,b=0\\ a\ne b \end{array}} \langle i,j|a,b\rangle \langle a,b|i,j\rangle = \frac{1}{d^2} \end{aligned}$$

(44)

and, therefore, the mutual predictability of all these noises separately is 1/d.

1.2 Werner State

$$\begin{aligned} \rho _\textrm{w} =&a \frac{2}{d(d+1)} P_\textrm{sym}+ (1-a) \frac{2}{d(d-1)} P_\textrm{as} \end{aligned}$$

(45)

where

$$\begin{aligned} P_\textrm{sym} =&\frac{1}{2} \left( \mathbbm {1} + P \right) \end{aligned}$$

(46)

$$\begin{aligned} P_\textrm{as} =&\frac{1}{2} \left( \mathbbm {1} - P \right) \end{aligned}$$

(47)

$$\begin{aligned} P =&\sum \limits _{i,j} |i\rangle \langle j| \otimes |j\rangle \langle i| \end{aligned}$$

(48)

The joint probability of outcomes in the Z basis for the operator P is

$$\begin{aligned} p_{P,Z} =&\sum \limits _{a,b} \langle i,j|a,b\rangle \langle b,a|i,j\rangle = \delta _{ij} \end{aligned}$$

(49)

Note that due to the $U \otimes U$ invariance of Werner states, the probability of measurement outcomes with respect to the X basis is the same as that with respect to the Z basis.

State-dependent separability bounds

As discussed in §4, with the fixed choice of measurement bases, which in this work are the X and Z bases, our separability bounds are state-dependent. This is because the relations between the negativity and the statistical correlators are state-dependent. On the other hand, with the chosen bases in this paper, the separability criteria of [15, 16] are not suitable to detect all the entangled states (among the classes of states considered here). We show this by using the sums of statistical correlators with respect to the Z and X bases in the figures described below. For comparison, we also indicate in these figures the respective bounds of [15, 16].

In Figs. 13, 14, 15, we have plotted the negativity of the Noisy Bell state given by Eq. (14) with respect to the sums of MPs, MIs and PCCs for the X and Z bases, respectively, in the case of $d=3$. Note that, for all the three statistical correlators, the corresponding state-dependent separability bounds depend upon the mixedness parameter a as indicated by Eqs. (15) and (16). In Figs. 16, 17 and 18, we have plotted the negativity of Werner state with respect to the sums of MPs, MIs and PCCs for the X and Z bases, respectively, in the case of $d=3$. For these observables, there exists entangled states such that the sums of the statistical correlators are less than the state-independent bounds [15, 16]. In other words, the use of the observables X and Z is not suitable to detect certain entangled states using the state-independent bounds but which can be detected using the state-dependent bounds.

Fig. 13

Negativity $(\mathcal {N}$) of the noisy Bell state versus the sum of MPs $(P_{Z}+P_{X})$ with respect to the X and Z bases for $d=3$. The vertical dotted line corresponds to the state-independent but observable dependent separability bound $1 + 1/d$. The line for each value of the state parameter a intersects the horizontal line corresponding to $\mathcal {N}=0$, with the point of intersection indicating the state-dependent separability bound for a given value of a

Fig. 14

Negativity $(\mathcal {N}$) of the noisy Bell state versus the sum of MIs $(I_{Z}+I_{X})$ with respect to the X and Z bases for $d=3$. The vertical dotted line corresponds to the state-independent but observable dependent separability bound $\log _2 d$. The line for each value of the state parameter a intersects the horizontal line corresponding to $\mathcal {N}=0$, with the point of intersection indicating the state-dependent separability bound for a given value of a

Fig. 15

Negativity $(\mathcal {N})$ of the noisy Bell state versus the sum of PCCs $(PCC_{X}+PCC_{Z})$ with respect to the X and Z bases for $d=3$. The vertical dotted line corresponds to the state-independent but observable dependent separability bound 1. The line for each value of the state parameter a intersects the horizontal line corresponding to $\mathcal {N}=0$, with the point of intersection indicating the state-dependent separability bound for a given value of a

Fig. 16

Negativity $(\mathcal {N})$ of the Werner state versus the sum of MPs $(P_{Z}+P_{X})$ with respect to the X and Z bases for $d=3$. The vertical dotted line corresponds to the state-independent but observable dependent separability bound $1 + 1/d$

Fig. 17

Negativity $(\mathcal {N})$ of the Werner state versus the sum of MIs $(I_{Z}+I_{X})$ with respect to the X and Z bases for $d=3$. The vertical dotted line corresponds to the state-independent but observable dependent separability bound $\log _2 d$

Fig. 18

Negativity $(\mathcal {N})$ of the Werner state versus the sum of PCCs $(PCC_{X}+PCC_{Z})$ with respect to the X and Z bases. The vertical dotted line corresponds to the state-independent but observable dependent separability bound given by 1

Proof of Maccone et al. Conjecture for pure and coloured noise A states

Here we deal with the problem of finding pairs of complementary observables $\lbrace \hat{A},\hat{C} \rbrace$ and $\lbrace \hat{B},\hat{D} \rbrace$ such that the sum of PCCs $|PCC_{AB}|+|PCC_{CD}|>1$ for a bipartite arbitrary-dimensional entangled state. For two given observables, say, X and Y, $PCC_{XY}$ is given by

$$\begin{aligned} PCC_{XY} = \frac{\langle X\otimes Y\rangle -\langle X\otimes \mathbbm {1}\rangle \langle \mathbbm {1}\otimes Y\rangle }{\sqrt{\langle X^{2}\otimes \mathbbm {1}\rangle -\langle X\otimes \mathbbm {1}\rangle ^{2}}\sqrt{\langle \mathbbm {1}\otimes Y^{2}\rangle -\langle \mathbbm {1}\otimes Y^2\rangle }} \end{aligned}$$

(50)

Taking the negativity ($\mathcal {N}$) as the measure of entanglement, we show that, using suitable pairs of complementary observables $\{A,C\}$ and $\{B,D\}$, $|PCC_{AB}|+|PCC_{CD}|-1\propto \mathcal {N}$ for certain types of states, viz. pure and coloured noise A state.

We proceed as follows. First, we construct the relevant observables and demonstrate their mutual unbiasedness. Next, we obtain the values of the relevant Pearson correlators for a d-dimensional bipartite state. Finally, we derive the desired relationship between the sum of Pearson correlators and the negativity for pure and coloured noise A states.

1.1 Construction of the required observables and their properties

The complementary observables for each subsystem are the generalised Z observable and a modification of the generalised X observable which is given by

$$\begin{aligned} W = \sum _{i,j:\langle i|j\rangle = 0}|j\rangle \langle i| \end{aligned}$$

(51)

In Eq. (51), the summation is over both i and j. For a given i, we sum over only those values of j which are orthogonal to i, i.e. $\langle i|j\rangle =0$. Another way to express the above observable is to consider generalised Pauli basis $\hat{\sigma }_{j,k}=|j\rangle \langle k|+|k\rangle \langle j|$ with $j>k$. We can then write $W=\underset{j,k}{\sum }\hat{\sigma }_{j,k}$.

The observable defined in Eq. (51) projects each computational basis vector $|j\rangle$ to $\sum _{k:\langle k|j\rangle =0}|k\rangle$, i.e. to its complete orthogonal subspace, $|i\rangle \longrightarrow \sum _{j:\langle i|j\rangle =0}|j\rangle$.

Now, we show that the observables Z and W are complementary to each other, i.e. the corresponding eigenstates are mutually unbiased.

1.1.1 Demonstration of maximum complementarity of the W and Z observables

Note that the eigenstates of Z form the computational basis $\{|j\rangle \}$. Therefore, to show the maximum complementarity of Z and W we only need to show that the eigenvectors of W are mutually unbiased to the states of the computational basis. For this purpose, we define a suitable basis, mutually unbiased with respect to the computational basis $\{|j\rangle \}$ given by $\lbrace |k\rangle =\frac{1}{\sqrt{d}}\sum _{j}\textrm{e}^{\frac{2\textrm{i}\pi jk}{d}}|j\rangle \rbrace$.

For $k=0$, we can show that

$$\begin{aligned} W|k=0\rangle&= \frac{1}{\sqrt{d}}\sum _{j}W|j\rangle \end{aligned}$$

(52)

$$\begin{aligned}&=\frac{1}{\sqrt{d}}\sum _{j,k:\langle k|j\rangle =0}|k\rangle \end{aligned}$$

(53)

$$\begin{aligned}&=\frac{(d-1)}{\sqrt{d}}\sum _{k}|k\rangle \end{aligned}$$

(54)

$$\begin{aligned}&=(d-1)|k=0\rangle \end{aligned}$$

(55)

The above calculation shows us that $|k=0\rangle$ is one of the eigenstates of W with eigenvalue $(d-1)$. For $1 \le k \le (d-1)$, taking $\omega =\textrm{e}^{\frac{2\textrm{i}\pi }{d}}$ we obtain

$$\begin{aligned} W|k\rangle&= \frac{1}{\sqrt{d}}\sum _{j}\omega ^{jk}W|j\rangle \nonumber \\&=\frac{1}{\sqrt{d}}\sum _{j,l:\langle l|j\rangle =0}\omega ^{jk}|l\rangle \end{aligned}$$

(56)

$$\begin{aligned}&=\frac{1}{\sqrt{d}}\sum _{l}|l\rangle \sum _{j:\langle l|j\rangle =0}\omega ^{jk} \end{aligned}$$

(57)

Note that for any k which is not a multiple of d we have

$$\begin{aligned} \sum _{j=0}^{d-1}\omega ^{jk}=\frac{1-\textrm{e}^{2\textrm{i}\pi k}}{1-\textrm{e}^{\frac{2\textrm{i}\pi k}{d}}}=0. \end{aligned}$$

(58)

Using Eq. (58), we can rewrite Eq. (57) as

$$\begin{aligned} W|k\rangle&=-\frac{1}{\sqrt{d}}\sum _{l}|l\rangle \omega ^{lk}\\&=-|k\rangle \end{aligned}$$

In other words, the states $|k\rangle =\frac{1}{\sqrt{d}}\sum _{j}\omega ^{jk}|j\rangle$ with $1\le k\le (d-1)$ are the eigenstates of W with eigenvalues $-1$.

Next, we outline the steps used in proving a relation which will be crucial for calculating the denominator of the expression of PCC given by Eq. (50).

• The following relation holds for the observable given by Eq. (51)

  $$\begin{aligned} W^{2}=\left[ (d-1)\mathbbm {1}+(d-2)W\right] \end{aligned}$$

  (59)

  Note that from Eq. (51) it is evident that we can write the matrix form of the observable W as follows:

  $$\begin{aligned} W=\begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} \dots &{} 1 \\ \vdots &{} \ddots &{} \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 0 \end{bmatrix} \end{aligned}$$

  (60)

  Consequently, for $W^{2}$ we have the following expression

  $$\begin{aligned} W^{2}&=\begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} \dots &{} 1 \\ \vdots &{} \ddots &{} \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 0 \end{bmatrix} \begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} \dots &{} 1 \\ \vdots &{} \ddots &{} \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 0 \end{bmatrix}\nonumber \\&= \begin{bmatrix} (d-1) &{} (d-2) &{} (d-2) &{} (d-2) &{} \dots &{} (d-2) \\ (d-2) &{} (d-1) &{} (d-2) &{} (d-2) &{} \dots &{} (d-2) \\ (d-2) &{} (d-2) &{} (d-1) &{} (d-2) &{} \dots &{} (d-2) \\ (d-2) &{} (d-2) &{} (d-2) &{} (d-1) &{} \dots &{} (d-2) \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ (d-2) &{} (d-2) &{} (d-2) &{} (d-2) &{} \dots &{} (d-1) \\ \end{bmatrix}\nonumber \\&=(d-1)\mathbbm {1}+(d-2)W \end{aligned}$$

  (61)

1.2 Pearson correlators for a d-dimensional bipartite state

Here we find the values of $PCC_{Z}$ and $PCC_{W}$ for a general bipartite qudit state.

To set the stage, we will define a few mathematical notations. The vector $|\bar{i}\rangle$ denotes an arbitrary vector from the computational basis spanning the orthogonal subspace of $|i\rangle$. For example, for $d=3$, and the computational basis $\{|0\rangle ,|1\rangle ,|2\rangle \}$, the vector $|\bar{0}\rangle$ would denote an arbitrary vector from the set $\{|1\rangle ,|2\rangle \}$. We use this notation in Eq. (51). For a fixed i, the summation $\sum _{j:\langle i|j\rangle =0}$ is equivalent to $\sum _{\bar{i}}$. In this notation, we can rewrite Eq. (51) as follows:

$$\begin{aligned} W =\sum _{i,\bar{i}}|\bar{i}\rangle \langle i| \end{aligned}$$

(62)

Given a $d_{a}\times d_{b}$ bipartite state

$$\begin{aligned} \rho _{ab}=\underset{i,j;k,l}{\sum } \epsilon _{i,j;k,l}|i\rangle \langle j|\otimes |k\rangle \langle l|, \end{aligned}$$

(63)

for $W_{a}=\sum _{i,\bar{i}}|\bar{i}\rangle \langle i|$ and $W_{b}=\sum _{k,\bar{k}}|\bar{k}\rangle \langle k|$, the following relationships hold

$$\begin{aligned} \langle W_{a}\otimes W_{b}\rangle&= \sum _{i,\bar{i};k,\bar{k}}\epsilon _{i,\bar{i};k,\bar{k}} \end{aligned}$$

(64)

$$\begin{aligned} \langle W_{a}\otimes \mathbbm {1}_{b}\rangle&= \sum _{i,\bar{i};k,k}\epsilon _{i,\bar{i};k,k} \end{aligned}$$

(65)

$$\begin{aligned} \langle \mathbbm {1}_{a}\otimes W_{b}\rangle&= \sum _{i,i;k,\bar{k}}\epsilon _{i,i;k,\bar{k}} \end{aligned}$$

(66)

1.2.1 Proofs of Eqs. (64)–(66)

For

$$\begin{aligned} \rho =\underset{i,j;k,l}{\sum }\ \epsilon _{i,j;k,l}|i\rangle \langle j|\otimes |k\rangle \langle l| \end{aligned}$$

given by Eq. (63), we have

$$\begin{aligned} W_{a}\otimes W_{b}\rho =\underset{\bar{i},i,j;\bar{k},k,l}{\sum } \epsilon _{i,j;k,l}|\bar{i}\rangle \langle j|\otimes |\bar{k}\rangle \langle l|.\end{aligned}$$

Note that $\bar{i}\ne i$ and $\bar{k} \ne k$ by definition. Consequently, for the expectation value, we have

$$\begin{aligned} \langle W_{a}\otimes W_{b}\rho \rangle =\underset{i,\bar{i};k,\bar{k}}{\sum } \epsilon _{i,\bar{i};k,\bar{k}} \end{aligned}$$

thus proving Eq. (64).

Similarly, we can prove Eqs. (65) and (66). Then, combining Eqs. (64)–(66), we can obtain the following expression for the numerator of the expression for $PCC_{W}$ as follows:

$$\begin{aligned} \sum _{i,\bar{i};k,\bar{k}}\epsilon _{i,\bar{i};k,\bar{k}}-\left( \sum _{i,\bar{i};k}\epsilon _{i,\bar{i};k,k}\right) \left( \sum _{i;k,\bar{k}}\epsilon _{i,i;k,\bar{k}}\right) \end{aligned}$$

(67)

Similarly, from Eq. (63) using $Z=\sum _{k}e_{k}|k\rangle \langle k|$, we can obtain the expression of the numerator of $PCC_{Z}$ as follows

$$\begin{aligned} \sum _{j,k}e_{j}e_{k}\epsilon _{j,j;k,k} -\left( \sum _{j,k}e_{j}\epsilon _{j,j;k,k}\right) \left( \sum _{j,k}e_{k}\epsilon _{j,j;k,k}\right) \end{aligned}$$

(68)

Note that $Tr(W)=0$ as well as $Tr(Z)=0$ in the computational basis. Using Eqs. (68) and (67), one can then seek to obtain the sum of PCCs for the various types of states and relate them to the negativity for the respective states.

In what follows, we will show that for both pure and coloured noise A states, the following relation holds good

$$\begin{aligned} PCC_{Z}+PCC_{W}-1 \propto \mathcal {N} \end{aligned}$$

(69)

1.3 The sum of Pearson correlators for the pure and coloured noise A states

1.3.1 Pure state

For pure entangled states, we have the Schmidt bases $\lbrace |i\rangle \rbrace$ and $\lbrace |\alpha _{i}\rangle \rbrace$ such that

$$\begin{aligned} |\psi \rangle _{d}&= \sum _{i}\sqrt{\lambda _{i}}|i\rangle |\alpha _{i}\rangle \end{aligned}$$

(70)

$$\begin{aligned} \rho _{d}&= \sum _{i;j}\sqrt{\lambda _{i}\lambda _{j}}|i\rangle \langle \alpha _{j}|\otimes |i\rangle \langle \alpha _{j}|. \end{aligned}$$

(71)

The negativity for a pure state is given by

$$\begin{aligned} \mathcal {N}_{p}=\frac{1}{2}\sum _{i,j:i\ne j}\sqrt{\lambda _{i}\lambda _{j}}=\frac{1}{2}\sum _{i,\bar{i}}\sqrt{\lambda _{i}\lambda _{\bar{i}}}. \end{aligned}$$

(72)

Writing Eq. (71) in the form of Eq. (63), we obtain

$$\begin{aligned} \epsilon _{i,j;k,l}= {\left\{ \begin{array}{ll} \sqrt{\lambda _{i}\lambda _{j}} &{} \text {for }i=k\hbox { and }j=l, \epsilon _{i,j;i,j},\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(73)

Equation (73) implies that for $j=\bar{i}$ the nonzero terms are $\epsilon _{i,\bar{i};i,\bar{i}}=\sqrt{\lambda _{i}\lambda _{\bar{i}}}$. From Eq. (64), we obtain $\langle W_{a}\otimes W_{b}\rangle =\sum _{i,\bar{i}}\epsilon _{i,\bar{i};i,\bar{i}}=\sum _{i,\bar{i}}\sqrt{\lambda _{i}\lambda _{\bar{i}}}=2\mathcal {N}_{p}$. From Eqs. (65) and (66), we have $\langle W_{a}\otimes \mathbbm {1}\rangle =\langle \mathbbm {1}\otimes W_{b}\rangle =0$. Consequently, the following relationship is obtained

$$\begin{aligned} PCC_{W}=\frac{2\mathcal {N}_{p}}{\Delta _{a}\Delta _{b}} \end{aligned}$$

(74)

Here the variances $\Delta _{a},\Delta _{b}$ are given by

$$\begin{aligned}{} & {} \Delta _{a}=\sqrt{\langle W_{a}^{2}\otimes \mathbbm {1}\rangle -\langle W_{a}\otimes \mathbbm {1}\rangle }\\{} & {} \Delta _{b}=\sqrt{\langle \mathbbm {1}\otimes W_{b}^{2}\rangle -\langle \mathbbm {1}\otimes W_{b}\rangle }. \end{aligned}$$

Using Eq. (61) and the result that $\langle W_{a}\otimes \mathbbm {1}\rangle =\langle \mathbbm {1}\otimes W_{b}\rangle =0$, we can obtain

$$\begin{aligned} \Delta _{a}=\Delta _{b}=\sqrt{d-1} \end{aligned}$$

Thus, Eq. (74) can be rewritten as

$$\begin{aligned} PCC_{W}=\frac{2\mathcal {N}_{p}}{(d-1)} \end{aligned}$$

(75)

Now, considering $PCC_{Z}$, note that since with respect to the Schmidt decomposition, the outcomes are perfectly correlated, and it follows that

$$\begin{aligned} PCC_{Z}=1 \end{aligned}$$

(76)

Combining Eqs. (75) and (76), we then obtain

$$\begin{aligned} PCC_{Z}+PCC_{W} = 1 + \frac{2\mathcal {N}_{p}}{(d-1)} \end{aligned}$$

(77)

Note that for the pure product state, the above sum is 1.

1.3.2 Coloured noise A state

First, we consider the coloured noise A state defined as follows

$$\begin{aligned} \rho _{cna}&= \frac{p}{d}\sum _{j=0,k=0}^{d-1}|j\rangle \langle k|\otimes |j\rangle \langle k|+\frac{(1-p)}{d}\sum _{j=0}^{d-1}|j\rangle \langle j|\otimes |j\rangle \langle j|\\&= \frac{1}{d}\sum _{j}|j\rangle \langle j|\otimes |j\rangle \langle j|+\frac{p}{d}\sum _{j,\bar{j}}|j\rangle \langle \bar{j}|\otimes |j\rangle \langle \bar{j}| \nonumber \end{aligned}$$

(78)

The negativity $(\mathcal {N}_{a})$ of a colored noise A state is given by

$$\begin{aligned} \mathcal {N}_{a}=\frac{p(d-1)}{2} \end{aligned}$$

(79)

Writing Eq. (78) in the form of Eq. (63), we obtain

$$\begin{aligned} \epsilon _{i,j;k,l}= {\left\{ \begin{array}{ll} \frac{1}{d} &{} \text {if }i=j=k=l, \epsilon _{j,j;j,j} \\ \frac{p}{d} &{} \text {if }i=k\ne j=l, \epsilon _{j,\bar{j};j,\bar{j}}\\ \text {zero} &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

(80)

It is evident from Eq. (80) that the terms $\epsilon _{j,j;k,\bar{k}}=\epsilon _{j,\bar{j};k,k}=0$. Consequently, R.H.S of Eqs. (65) and (66) are zero and we have $\underset{j,\bar{j},k,\bar{k}}{\sum }\epsilon _{j,\bar{j};k,\bar{k}}=p(d-1)$.

Using Eq. (64), we then have $\langle W_{a}\otimes W_{a}\rangle =p(d-1)=2\mathcal {N}_{a}$, whence the numerator of the expression of $PCC_{W}$ becomes $2\mathcal {N}_{a}$. Using Eq. (59), the denominator of $PCC_{W}$ is obtained as $(d-1)$. Thus, for the coloured noise A we obtain

$$\begin{aligned} PCC_{W} = \frac{2\mathcal {N}_{a}}{(d-1)} \end{aligned}$$

(81)

Next, in order to obtain the numerator of the expression for $PCC_{Z}$ for the coloured noise A state, we can use Eq. (68) along with Eq. (80) which yield

$$\begin{aligned} \langle Z_{a}\otimes Z_{b}\rangle -\langle Z_{a}\otimes \mathbbm {1}_{b}\rangle \langle \mathbbm {1}_{a}\otimes Z_{b}\rangle = \sum _{j}e_{j}^{2} \end{aligned}$$

(82)

The denominator of the expression for $PCC_{Z}$ can then be obtained as follows

$$\begin{aligned} \langle Z_{a}^{2}\otimes \mathbbm {1}\rangle&= p\sum _{j}e_{j}^{2}+(1-p)\sum _{j}e_{j}^{2}\nonumber \\&= \sum _{j}e_{j}^{2} = \langle \mathbbm {1}\otimes Z_{b}^{2}\rangle \end{aligned}$$

(83)

Using $Tr(Z_{a})=Tr(Z_{b})=\sum _{j}j=0$, one can verify that

$$\begin{aligned} \langle Z_{a}\otimes \mathbbm {1}\rangle&= \langle \mathbbm {1}\otimes Z_{b}\rangle \nonumber \\&= \sum _{j}e_{j}\nonumber \\&= 0 \end{aligned}$$

(84)

Combining Eqs. (82)–(84), the value of $PCC_{Z}$ is obtained as

$$\begin{aligned} PCC_{Z}=1 \end{aligned}$$

(85)

Combining Eqs. (85) and (81), the final relation is then derived as

$$\begin{aligned} PCC_{Z}+PCC_{W}=1+\frac{2\mathcal {N}_{a}}{(d-1)} \end{aligned}$$

(86)

To sum up, the sums of the two PCCs, $PCC_{Z}$ and $PCC_{W}$, as functions of the negativity of the states considered, given by Eqs. (77) and (86) justify the conjecture made by Maccone et al. [16] for the bipartite arbitrary-dimensional pure and coloured noise A states, respectively.