Informational correlation between two parties of a quantum system: spin-1/2 chains

Abstract

We introduce the informational correlation $E^{AB}$ between two interacting quantum subsystems $A$ and $B$ of a quantum system as the number of arbitrary parameters $\varphi _i$ of a unitary transformation $U^A$ (locally performed on the subsystem $A$) which may be detected in the subsystem $B$ by the local measurements. This quantity indicates whether the state of the subsystem $B$ may be effected by means of the unitary transformation applied to the subsystem $A$. Emphasize that $E^{AB}\ne E^{BA}$ in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem $B$. However, there is some non-reducible part of $E^{AB}(t)$ which may not be decreased by any unitary transformation of the subsystem $B$ at a fixed time instant $t$. Two examples of the informational correlations between two parties of the four-node spin-1/2 chain with mixed initial states are studied. The long chains with a single initially excited spin (the pure initial state) are considered as well.

Appendices

Appendix 1: Explicit form of the matrices $\gamma _i$

We give the list of matrices $\gamma _i$ representing the basis of the Lie algebra of $SU(4)$ [35]:

$$\begin{aligned}&\gamma _1=\left[ \begin{array}{cccc} 0&{} 1&{} 0&{} 0\\ 1&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _2=\left[ \begin{array}{cccc} 0&{}-i&{} 0&{} 0\\ i&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _3=\left[ \begin{array}{cccc} 1&{} 0&{} 0&{} 0\\ 0&{}-1&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\\&\gamma _4=\left[ \begin{array}{cccc} 0&{} 0&{} 1&{} 0\\ 0&{} 0&{} 0&{} 0\\ 1&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _5=\left[ \begin{array}{cccc} 0&{} 0&{}-i&{} 0\\ 0&{} 0&{} 0&{} 0\\ i&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _6=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 1&{} 0\\ 0&{} 1&{} 0&{} 0\\ 0&{} \;\;0&{} 0&{} 0 \end{array}\right] ,\nonumber \\&\gamma _7=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{}-i&{} 0\\ 0&{} i&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _8=\frac{1}{\sqrt{3}}\left[ \begin{array}{cccc} 1&{} 0&{} 0&{} 0\\ 0&{} 1&{} 0&{} 0\\ 0&{} 0&{}-2&{} 0\\ 0&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _9=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 1\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ 1&{} 0&{} 0&{} 0 \end{array}\right] ,\nonumber \\&\gamma _{10}=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{}-i\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ i&{} 0&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _{11}=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 1\\ 0&{} 0&{} 0&{} 0\\ 0&{} 1&{} 0&{} 0 \end{array}\right] ,\;\;\gamma _{12}=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{}-i\\ 0&{} 0&{} 0&{} 0\\ 0&{} i&{} 0&{} 0 \end{array}\right] ,\nonumber \\&\gamma _{13}=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 1\\ 0&{} 0&{} 1&{} 0 \end{array}\right] ,\;\;\gamma _{14}=\left[ \begin{array}{cccc} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{}-i\\ 0&{} 0&{} i&{} 0 \end{array}\right] ,\;\;\gamma _{15}=\frac{1}{\sqrt{6}}\left[ \begin{array}{cccc} 1&{} 0&{} 0&{} 0\\ 0&{} 1&{} 0&{} 0\\ 0&{} 0&{} 1&{} 0\\ 0&{} 0&{} 0&{}-3 \end{array}\right] .\nonumber \end{aligned}$$

(148)

Appendix 2: Proof of Eq. (60) for multiple and zero eigenvalues $\lambda _i$ of the matrix $\rho ^B(\varphi ^A,t)$

Suppose that there are $(P+1)<(N^B-1)$ nonzero different eigenvalues of the matrix $\rho ^B(\varphi ,t)$. Then, in order to define the non-reducible informational correlation $E^{AB;\mathrm{min}}$, we may replace the characteristic equation (53) with the following polynomial one:

$$\begin{aligned} \prod _{i=1}^P (\lambda -\lambda _i(\varphi ^A,t)) = \lambda ^P +\sum _{j=0}^{P-1} \tilde{a}_i(\varphi ^A,t) \lambda ^i=0, \end{aligned}$$

(149)

where $\tilde{a}_i$ are expressed in terms of $\lambda _i$. In this equation, we take into account only $P$ different nonzero eigenvalues because of the identity ${\text{ Tr }}\rho ^B=\sum \nolimits _{i=1}^{P+1} q_i \lambda _i=1$, where $q_i$ is the multiplicity of the root $\lambda _i$. Then, the non-reducible informational correlation is defined by the rank of the matrix

$$\begin{aligned} J^B_\Lambda (\varphi ^A,t)=\frac{\partial (\lambda _1(\varphi ^A,t),\ldots , \lambda _P(\varphi ^A,t))}{\partial (\varphi _1,\ldots ,\varphi _{\tilde{D}^A})}, \end{aligned}$$

(150)

so that

$$\begin{aligned} E^{AB;\mathrm{min}}(\varphi ^A,t) = {\text{ ran }}\, J^B_\Lambda (\varphi ^A,t). \end{aligned}$$

(151)

Differentiating Eq. (149) with respect to the parameters $\varphi _k$, $k=1,\ldots ,\tilde{D}^A$, and solving the resulting equations for $\displaystyle \frac{\partial \lambda }{\partial \varphi _k}$ we obtain:

$$\begin{aligned} \frac{\partial \lambda }{\partial \varphi _k} = -\frac{\sum \nolimits _{i=0}^{P-1} \frac{\partial \tilde{a}_i(\varphi ^A,t)}{\partial \varphi _k} \lambda ^i}{\displaystyle P\lambda ^{P-1} + \sum \nolimits _{i=1}^{P-1} i \tilde{a}_i(\varphi ^A,t) \lambda ^{i-1}}. \end{aligned}$$

(152)

Therefore, for the matrix $J^B_\Lambda (\varphi ^A,t)$, one has

$$\begin{aligned} J^B_\Lambda (\varphi ^A,t)&= \frac{1}{\tilde{J}_0(\varphi ^A,t)}\left( \begin{array}{ccc}\displaystyle \sum _{i=0}^{P-1} \frac{\partial \tilde{a}_i(\varphi ^A,t)}{\partial \varphi _1} \lambda ^i_1(\varphi ^A,t)&{}\cdots &{}\displaystyle \sum _{i=0}^{P-1} \frac{\partial \tilde{a}_i(\varphi ^A,t)}{\partial \varphi _{\tilde{D}^A}} \lambda ^i_{1}(\varphi ^A,t)\\ \cdots &{}\cdots &{}\cdots \\ \displaystyle \sum _{i=0}^{P-1} \frac{\partial \tilde{a}_i(\varphi ^A,t)}{\partial \varphi _{1}} \lambda ^i_P(\varphi ^A,t)&{}\cdots &{}\displaystyle \sum _{i=0}^{P-1} \frac{\partial \tilde{a}_i(\varphi ^A,t)}{\partial \varphi _{\tilde{D}^A}} \lambda ^i_{P}(\varphi ^A,t) \end{array}\right) \nonumber \\&=\tilde{\Lambda }^B(\varphi ^A,t) \tilde{H}(\varphi ^A,t) , \end{aligned}$$

(153)

where

$$\begin{aligned} \tilde{J}_0(\varphi ^A,t)=(-1)^P \prod _{j=1}^P \Big (P\lambda ^{P-1}_j(\varphi ^A,t) + \sum _{i=1}^{P-1} i \tilde{a}_i(\varphi ^A,t) \lambda ^{i-1}_j(\varphi ^A,t)\Big ) \ne 0,\nonumber \\ \end{aligned}$$

(154)

while $\tilde{\Lambda }^B$ and $\tilde{H}$ are the $P\times P$ and $P\times \tilde{D}^A$ matrices respectively:

$$\begin{aligned}&\tilde{\Lambda }^B=\left( \begin{array}{cccc}\displaystyle 1 &{}\lambda _1(\varphi ^A,t)&{} \cdots &{}\lambda _1^{P-1}(\varphi ^A,t)\\ \cdots &{}\cdots &{}\cdots &{}\cdots \\ 1 &{}\lambda _P(\varphi ^A,t)&{} \cdots &{}\lambda _P^{P-1}(\varphi ^A,t) \end{array} \right) ,\end{aligned}$$

(155)

$$\begin{aligned}&\tilde{H}(\varphi ^A,t)=\left( \begin{array}{ccc}\displaystyle \frac{\partial \tilde{a}_0(\varphi ^A,t)}{\partial \varphi _1} &{}\cdots &{} \displaystyle \frac{\partial \tilde{a}_0(\varphi ^A,t)}{\partial \varphi _{\tilde{D}^A}} \\ \cdots &{}\cdots &{}\cdots \\ \displaystyle \displaystyle \frac{\partial \tilde{a}_{P-1}(\varphi ^A,t)}{\partial \varphi _1} &{}\cdots &{}\displaystyle \frac{\partial \tilde{a}_{P-1}(\varphi ^A,t)}{\partial \varphi _{\tilde{D}^A}} \end{array} \right) . \end{aligned}$$

(156)

Then, Eq. (151) yields

$$\begin{aligned} E^{AB;\mathrm{min}}(\varphi ^A,t) ={\text{ ran }} \, J^B_\Lambda (\varphi ^A,t)={\text{ ran }} \,\tilde{H}(\varphi ^A,t). \end{aligned}$$

(157)

Now notice that $P$ coefficients $\tilde{a}_i$ in Eq. (149) are defined by $P$ different nonzero eigenvalues $\lambda _i$, $i=1,\ldots ,P$. From another hand, the coefficients $a_i$ in Eq. (53) are defined by the same $P$ independent eigenvalues $\lambda _i$, $i=1,\ldots ,P$, and consequently by $P$ coefficients $\tilde{a}_i$, $i=1,\ldots ,P$. The last statement is provided by the relation between sets $\tilde{a}_i$ and $\lambda _i$. This relation follows from Eq. (149), where all $\lambda _i$, $i=1,\ldots ,\lambda _P$, are different by our requirement. Consequently,

$$\begin{aligned} \left| \frac{\partial (\tilde{a}_0,\ldots ,\tilde{a}_{P-1})}{\partial (\lambda _1,\ldots ,\lambda _P)} \right| \ne 0. \end{aligned}$$

(158)

Thus, for the matrix $H$ represented by Eq. (59), we may write

$$\begin{aligned} H(\varphi ^A,t) = F(\varphi ^A,t) \tilde{H}(\varphi ^A,t), \end{aligned}$$

(159)

where $F$ is $(N^B-1) \times P$ matrix,

$$\begin{aligned} F(\varphi ^A,t)=\left( \begin{array}{ccc}\displaystyle \frac{\partial a_0(\varphi ^A,t)}{\partial \tilde{a}_0} &{}\cdots &{}\displaystyle \frac{\partial a_0(\varphi ^A,t)}{\partial \tilde{a}_{P-1}} \\ \cdots &{}\cdots &{}\cdots \\ \displaystyle \frac{\partial a_{N^B-2}(\varphi ^A,t)}{\partial \tilde{a}_{0}} &{}\cdots &{}\displaystyle \frac{\partial a_{N^B-2}(\varphi ^A,t)}{\partial \tilde{a}_{P-1}} \end{array}\right) . \end{aligned}$$

(160)

It may be readily shown that the rank of the matrix $F$ takes its maximal possible value, ${\text{ ran }}\, F = P$, $P\le (N^B-1)$. In fact, since $a_i$, $i=0,1,\ldots , N^B-2$, are expressed in terms of $\lambda _i$ [see Eqs. (52, 53)] and there are only $P$ independent eigenvalues $\lambda _i$, $i=1,\ldots ,P$, then ${\text{ ran }}\displaystyle \frac{\partial (a_0,a_1,\ldots ,a_{N^B-2})}{\partial (\lambda _1,\ldots ,\lambda _P)}=P$. But $\displaystyle \frac{\partial (a_0,a_1,\ldots ,a_{N^B-2})}{\partial (\lambda _1,\ldots ,\lambda _P)} = F \frac{\partial (\tilde{a}_0,\ldots ,\tilde{a}_{P-1})}{\partial (\lambda _1,\ldots ,\lambda _P)}$. Consequently, in virtue of condition (158), we conclude that $\displaystyle {\text{ ran }}\, F ={\text{ ran }}\displaystyle \frac{\partial (a_0,a_1,\ldots ,a_{N^B-2})}{\partial (\lambda _1,\ldots ,\lambda _P)}=P$. Thus, the rank of the product $F \tilde{H}$ equals to the rank of $\tilde{H}$ in Eq. (159), which yields

$$\begin{aligned} {\text{ ran }}\, H(\varphi ^A,t) = {\text{ ran }}\, \tilde{H}(\varphi ^A,t). \end{aligned}$$

(161)

In turn, Eq. (161) means that Eq. (60) holds for the multiple and/or zero eigenvalues as well.

Appendix 3: Informational correlation in systems with arbitrary initial state

The results obtained in Sects. 2 and 3 are based on the tensor product initial state (13). If the initial state is more general, then Eq. (15) is not valid as well as Eq. (28). In other words, the $\varphi ^A$-dependence may not be collected in the density matrix $\rho ^A(\varphi ^A,0)$. In this case, we also may introduce informational correlation $E^{AB}$ by Eqs. (36, 37). In turn, the number $E^{AA}$ of parameters encoded into the subsystem $A$ may be introduced by Eqs. (38, 39). Therewith the vector $\hat{X}$ is defined by Eq. (25) together with Eqs. (22). Again, the number of parameters encoded into $\rho ^A(\varphi ^A,0)$ is defined by the multiplicity of the eigenvalues of $\rho ^A(0)$. However, the representation (23) for $X(\rho ^B(\varphi ^A,t))$, $Y(\rho ^B(\varphi ^A,t))$ and $Z(\rho ^B(\varphi ^A,t))$ is not valid any more. Inequality (41) between $E^{AB}$ and $E^{AA}$ has no place as well. At first glance, this inequality must be replaced by the more formal one:

$$\begin{aligned} E^{AB}(t)\le E^{AA}. \end{aligned}$$

(162)

However, inequality (162) is not evident and might be wrong in general. In fact, applying the local transformation to the subsystem $A$ we influence on the whole density matrix $\rho (0)$ yielding the density matrix $\rho (\varphi ^A,0)$ . However, only certain combinations of the elements of $\rho (\varphi ^A,0)$ appear in $\rho ^A(\varphi ^A,0)$. Thus, some of the parameters $\varphi _i$ might be missed from the local density matrix $\rho ^A(\varphi ^A,0)$, but might be detected in the whole density matrix $\rho (\varphi ^A,0)$. This forces us to denote the number of all parameters encoded into the initial density matrix $\rho (\varphi ^A, 0)$ by $E^A$, $E^A\ge E^{AA}$. This quantity is defined by the equation (similar to Eqs. (37) and (39))

$$\begin{aligned} E^A(\varphi ^A,0)= {\text{ ran }} J(\rho (\varphi ^A,0)) , \end{aligned}$$

(163)

where

$$\begin{aligned} J(\rho (\varphi ^A,0))=\frac{\partial (\hat{X}_1(\rho (\varphi ^A,0)),\ldots , \hat{X}_{N^2-1}\rho ^A(\varphi ^A,0) }{\partial (\varphi _1,\ldots ,\varphi _{\tilde{D}^A})}. \end{aligned}$$

(164)

Thus, there might be such parameters $\varphi _i$ that are not encoded into the initial reduced density matrix $\rho ^A(\varphi ^A,0)$, but might appear in the reduced density matrix $\rho ^B(\varphi ^A,t)$ in the course of evolution. The number of these parameters may not exceed the value $\delta E^{A}$,

$$\begin{aligned} \delta E^{A}(\varphi ^A,0) = E^A(\varphi ^A,0) - E^{AA}. \end{aligned}$$

(165)

Consequently, instead of (162), the following inequality holds:

$$\begin{aligned} E^{AB}(\varphi ^A,t) \le E^{A}(\varphi ^A,0). \end{aligned}$$

(166)

Emphasize that $E^{AB}$ depends on $\varphi ^A$ in the case of arbitrary initial state $\rho (0)$. It is obvious that the normalized informational correlation $E^{AB}_\mathrm{norm}$ defined by formula (43) might be bigger then one in this case.

Now, let us calculate $E^{AB}$ using Eqs. (36, 37). In general, the rank of the matrix $J(\rho ^B)$ must be calculated numerically. For this purpose, we fix the time interval $T_1<t<T_2$ taken for the parameter detection in the subsystem $B$ and introduce the set of auxiliary functions

$$\begin{aligned}&\mathcal{{M}}_n(t)\!=\! \frac{\hat{M}_n(t)}{\hat{M}_{n;\mathrm{max}}},\, \hat{M}_n(t) \!=\! \int \limits _{ G^A} \sum _i |M_{ni}(\varphi ^A,t)| d\Omega (\varphi ^A),\, \hat{M}_{n;\mathrm{max}} \!=\!\max \limits _{T_1<t<T_2} \hat{M}_n(t),\nonumber \\&n=1,\ldots , \tilde{D}^A, \end{aligned}$$

(167)

where $M_{ni}$ are the $n$th-order minors of $J(\rho ^B)$, sum is over all minors, integration is over the whole $G^A$, and $\Omega (\varphi ^A)$ is some measure. The function $\mathcal{{ M}}_n(t)$ is positive if only at least some of the $n$th-order minors $M_{ni}$ are nonzero on the non-zero volume subregion $g(t)$ of the region $G^A$ (note that $g$ may depend on $t$). Then, we define $E^{AB}(g(t),t)$ as the maximal order $n_0$ of the positive functions $\mathcal{{ M}}_n(t)$, $n=1,\ldots ,n_0$, i.e.,

$$\begin{aligned} E^{AB}(g(t),t) = \max \limits _{\begin{array}{c} \mathcal{{M}}_n(t)> 0\\ {\varphi ^A\in g(t)} \end{array}}\, n(t) = n_0(t), \end{aligned}$$

(168)

so that $n_0$ depends on $t$ in general. For the practical purpose, we might need to replace the positivity condition of $\mathcal{{M}}_n(t)$ by the following one:

$$\begin{aligned} \mathcal{{M}}_{n}(t)\ge \varepsilon , \end{aligned}$$

(169)

where $\varepsilon >0$ is some parameter predicted by the errors of calculations and/or experiment.

Let us consider the case of stationary region $\hat{g}, \hat{g}\subset G^A$. The time intervals suitable for the detection of the transfered parameters in the subsystem $B$ might be defined numerically by the algorithm similar to that used in examples of Sect. 3.2.3. First of all, we introduce the auxiliary function ${{ M}}(t)$ defined as follows:

$$\begin{aligned} M(t)=\frac{M^g_{n_0}(t)}{M^g_{max}}, \;\; M^g_{n_0}(t)=\min _{\varphi ^A\in g} \sum _i |M_{n_0i}(\varphi ^A,t)|, \;\; M^g_{max} = \max \limits _{T_1\le t \le T_2} M^g_{n_0}(t).\nonumber \\ \end{aligned}$$

(170)

Formally, any time instant corresponding to the positive $M$ is suitable for the parameter detection. However, if $M$ is too small, then there might be some obstacles for the correct detection of these parameters (for instance, fluctuations). Thus, we take only such time subintervals inside of the taken interval $T_1 < t < T_2$ that satisfy the following condition:

$$\begin{aligned} M(t) > \tilde{\varepsilon }. \end{aligned}$$

(171)

Here $\tilde{\varepsilon }$ is some positive parameter, predicted by the required accuracy. For instance, $\displaystyle \tilde{\varepsilon } =\frac{1}{2}$ in examples of Sect. 3.2.3.

In a similar way, we may study the non-reducible correlations. The formulas (167–171) hold with replacement $E^{AB} \rightarrow E^{AB;\mathrm{min}}$, therewith $M_{ni}$ must be the $n$th-order minors of the matrix $H$ defined by Eq. (59).