A Note on Holevo quantity of $SU(2)$-invariant states

The Holevo bound characterizes the capacity of quantum states in classical communication holevo ; bena . As an exceedingly useful upper bound on the accessible information it plays an important role in many quantum information processing nielsen . It is a keystone in the proof of many results in quantum information theory lupo ; zhang ; lloyd ; roga ; wu ; guo . With respect to the Holevo bound, the maximal Holevo quantity related to weak measurements has been studied in wang . The Holevo quantity of an ensemble $\{p_{i};\rho_{A|i}\}$, corresponding to a bipartite quantum state $\rho_{AB}$ with the projective measurements $\{\Pi^{B}_{i}\}$ performed on the subsystem $B$, is given by wang

$\displaystyle\chi\{\rho_{AB}|\{\Pi^{B}_{i}\}\}=\chi\{p_{i};\rho_{A|i}\}\equiv S(\sum_{i}p_{i}\rho_{A|i})-\sum_{i}p_{i}S(\rho_{A|i}),$

(1)

where

$$p_{i}=\mbox{tr}_{AB}[(I_{A}\otimes\Pi^{B}_{i})\rho_{AB}({I}_{A}\otimes\Pi^{B}_{i})],~{}~{}\rho_{A|i}=\frac{1}{p_{i}}\mbox{tr}_{B}[({I}_{A}\otimes\Pi^{B}_{i})\rho_{AB}({I}_{A}\otimes\Pi^{B}_{i})].$$

(2)

It denotes the A’s accessible information about the B’s measurement outcome when B projects its system by the projection operators $\{\Pi^{B}_{i}\}$.

When measuring one particle in a composite quantum system, partial measurement in quantum mechanics has recently been generalized in long . It collapses-in or collapses-out the measured states. Measure of correlation is also very important in quantum gravity gyongyosi ; van , and characterizing coherence in quantum systems han .

The Holevo quantity (1) related to projective measurements depends on the bipartite states. In the following we study the Holevo quantity for a particular class of $SU(2)$-invariant states. The $SU(2)$-invariant states originate from thermal equilibrium states of spin systems with $SU(2)$-invariant Hamiltonian Durkin . With respect to two spins $\vec{S}_{1}$ and $\vec{S}_{2}$, a density matrices $\rho$ is said to be $SU(2)$-invariant if $U_{1}\otimes U_{2}\rho U^{\dagger}_{1}\otimes U^{\dagger}_{2}=\rho$, where $U_{a}=\exp(i\vec{\eta}\cdot\vec{S}_{a})$, $a\in\{1,2\}$, are the usual rotation operator representation of $SU(2)$ with real parameter $\vec{\eta}$ and $\hbar=1$ Schliemann1 ; Schliemann2 . Those $SU(2)$-invariant states $\rho$ commute with the total spin $\vec{J}=\vec{S}_{1}+\vec{S}_{2}$. The entanglement of SO(3)-invariant bipartite quantum states have been studied in Breuer1 ; Breuer2 . For the $SU(2)$-invariant quantum spin systems it is shown that the negativity is the necessary and sufficient condition of separability Schliemann1 ; Schliemann2 . The relative entropy of entanglement has been analytically calculated in zwang for $(2j+1)\otimes 2$ and $(2j+1)\otimes 3$ dimensional $SU(2)$-invariant quantum spin states. The entanglement of formation, I-concurrence, I-tangle and convex-roof-extended negativity of the $SU(2)$-invariant states of spin-$j$ and spin-$\frac{1}{2}$ systems have been investigated in Manne by using the approach in Vollbrecht . The quantum discord and one way deficit for the $SU(2)$-invariant states in spin-$j$ and spin-$\frac{1}{2}$ bipartite systems have been discussed respectively in cakmak ; wang1 .

As an $SU(2)$-invariant state commutes with all the components of $\vec{J}$, $\rho$ has the general form Schliemann1 ,

$$\rho=\sum_{J=|S_{1}-S_{2}|}^{S_{1}+S_{2}}\frac{A(J)}{2J+1}\sum_{J^{z}=-J}^{J}|J,J^{z}\rangle_{0}{{}_{0}\langle J,J^{z}|}\,,$$

(3)

where the constants $A(J)\geq 0$, $\sum_{J}A(J)=1$, $|J,J^{z}\rangle_{0}$ denotes the state of total spin $J$ and $z$-component $J^{z}$. We discuss the case with $\vec{S}_{1}$ of arbitrary spin $j$ and $\vec{S}_{2}$ of spin $\frac{1}{2}$. A general $SU(2)$-invariant density matrix has the form,

$\displaystyle\rho^{ab}=\frac{F}{2j}\sum_{m=-j+\frac{1}{2}}^{j-\frac{1}{2}}|j-\frac{1}{2},m\rangle\langle j-\frac{1}{2},m|+\frac{1-F}{2(j+1)}\sum_{m=-j-\frac{1}{2}}^{j+\frac{1}{2}}|j+\frac{1}{2},m\rangle\langle j+\frac{1}{2},m|,$

(4)

where $F\in[0,1]$, which is a function of temperature in thermal equilibrium. $\rho^{ab}$ is a $(2j+1)\otimes 2$ bipartite state. It has two eigenvalues $\lambda_{1}={F}/{(2j)}$ and $\lambda_{2}=({1-F})/({2j+2})$ with degeneracies $2j$ and $2j+2$, respectively cakmak . As the eigenstates of the total spin can be given by the Clebsch-Gordon coefficients shankar in coupling a spin-$j$ to spin-$\frac{1}{2}$,

$\displaystyle|j\pm\frac{1}{2},m\rangle=\pm\sqrt{\frac{j+\frac{1}{2}\pm m}{2j+1}}|j,m-\frac{1}{2}\rangle\otimes|\frac{1}{2},\frac{1}{2}\rangle+\sqrt{\frac{j+\frac{1}{2}\mp m}{2j+1}}|j,m+\frac{1}{2}\rangle\otimes|\frac{1}{2},-\frac{1}{2}\rangle,$

the density matrix (4) can be written in product basis form cakmak ,

	$\displaystyle\rho^{ab}$	$\displaystyle=$	$\displaystyle\frac{F}{2j}\sum_{m=-j+\frac{1}{2}}^{j-\frac{1}{2}}(a_{-}^{2}|m-\frac{1}{2}\rangle\langle m-\frac{1}{2}|\otimes|\frac{1}{2}\rangle\langle\frac{1}{2}|$
			$\displaystyle+a_{-}b_{-}(|m-\frac{1}{2}\rangle\langle m+\frac{1}{2}|\otimes|\frac{1}{2}\rangle\langle-\frac{1}{2}|$
			$\displaystyle+|m+\frac{1}{2}\rangle\langle m-\frac{1}{2}|\otimes|-\frac{1}{2}\rangle\langle\frac{1}{2}|)$
			$\displaystyle+b_{-}^{2}|m+\frac{1}{2}\rangle\langle m+\frac{1}{2}|\otimes|-\frac{1}{2}\rangle\langle-\frac{1}{2}|)$
			$\displaystyle+\frac{1-F}{2(j+1)}\sum_{m=-j-\frac{1}{2}}^{j+\frac{1}{2}}(a_{+}^{2}|m-\frac{1}{2}\rangle\langle m-\frac{1}{2}|\otimes|\frac{1}{2}\rangle\langle\frac{1}{2}|$
			$\displaystyle+a_{+}b_{+}(|m-\frac{1}{2}\rangle\langle m+\frac{1}{2}|\otimes|\frac{1}{2}\rangle\langle-\frac{1}{2}|$
			$\displaystyle+|m+\frac{1}{2}\rangle\langle m-\frac{1}{2}|\otimes|-\frac{1}{2}\rangle\langle\frac{1}{2}|)$
			$\displaystyle+b_{+}^{2}|m+\frac{1}{2}\rangle\langle m+\frac{1}{2}|\otimes|-\frac{1}{2}\rangle\langle-\frac{1}{2}|),$

where $a_{\pm}=\pm\sqrt{\frac{j+\frac{1}{2}\pm m}{2j+1}}$ and $b_{\pm}=\sqrt{\frac{j+\frac{1}{2}\mp m}{2j+1}}$.

Any von Neumann measurement on the spin-$\frac{1}{2}$ subsystem can be written as $B_{k}=V\Pi_{k}V^{{\dagger}}$, $k=0,1,$ where $\Pi_{k}=|k\rangle\langle k|$, $|k\rangle$ is the computational basis, $V=tI+i\vec{y}\cdot\vec{\sigma}\in SU(2)$, $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ with $\sigma_{1},\sigma_{2},\sigma_{3}$ being Pauli matrices, $t\in\Re$ and $\vec{y}=(y_{1},y_{2},y_{3})\in\Re^{3}$, $t^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1$. After the measurement, $\rho^{ab}$ has the ensemble of post-measurement states $\{\rho_{k},p_{k}\}$, with the post-measurement states $\rho_{k}$ and the corresponding probabilities $p_{k}$,

	$\displaystyle p_{k}\rho_{k}$	$\displaystyle=$	$\displaystyle(I\otimes B_{k})\rho^{ab}(I\otimes B_{k})=(I\otimes V\Pi_{k}V^{{\dagger}})\rho^{ab}(I\otimes V\Pi_{k}V^{{\dagger}})$
		$\displaystyle=$	$\displaystyle(I\otimes V)(I\otimes\Pi_{k})(I\otimes V^{{\dagger}})\rho^{ab}(I\otimes V)(I\otimes\Pi_{k})(I\otimes V^{{\dagger}}).$

Denote

	$\displaystyle N=\sum_{m=-j}^{j}$		$\displaystyle(z_{3}\frac{m(2Fj+F-j)}{j(j+1)(2j+1)}|m\rangle\langle m|$
			$\displaystyle+(z_{1}+iz_{2})\frac{\sqrt{j(j+1)-m(m+1)}(2Fj+F-j)}{2j(j+1)(2j+1)}|m\rangle\langle m+1|$
			$\displaystyle+(z_{1}-iz_{2})\frac{\sqrt{j(j+1)-m(m+1)}(2Fj+F-j)}{2j(j+1)(2j+1)}|m+1\rangle\langle m|),$

where $z_{1}=2(-ty_{2}+y_{1}y_{3})$, $z_{2}=2(ty_{1}+y_{2}y_{3})$, $z_{3}=t^{2}+y_{3}^{2}-y_{1}^{2}-y_{2}^{2}$ with $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1$. By using the transformation properties of Pauli matrices in luo ; cakmak , we have the probabilities $p_{0}=p_{1}=\frac{1}{2}$, and the corresponding post-measurement states

$\displaystyle\rho_{0}=(\frac{1}{2j+1}(\sum_{m=-j}^{j}|m\rangle\langle m|-N)\bigotimes V\Pi_{0}V^{\dagger}$

(7)

and

$\displaystyle\rho_{1}=(\frac{1}{2j+1}\sum_{m=-j}^{j}|m\rangle\langle m|+N)\bigotimes V\Pi_{1}V^{\dagger}.$

(8)

By means of Eqs. (2), we have

	$\displaystyle\rho_{A|0}$	$\displaystyle=$	$\displaystyle\mbox{tr}_{B}\rho_{0}$
		$\displaystyle=$	$\displaystyle\frac{1}{2j+1}(\sum_{m=-j}^{j}|m\rangle\langle m|-N$
		$\displaystyle=$	$\displaystyle\frac{1}{2j+1}I_{2j+1}-N,$

and

	$\displaystyle\rho_{A|1}$	$\displaystyle=$	$\displaystyle\mbox{tr}_{B}\rho_{1}$
		$\displaystyle=$	$\displaystyle\frac{1}{2j+1}\sum_{m=-j}^{j}|m\rangle\langle m|+N$
		$\displaystyle=$	$\displaystyle\frac{1}{2j+1}I_{2j+1}+N.$

Furthermore, we have

$\displaystyle p_{0}\rho_{A|0}+p_{0}\rho_{A|0}=\frac{I_{2j+1}}{2j+1},$

(11)

and

$\displaystyle S(\sum_{i}p_{i}\rho_{A|i})=\log(2j+1).$

(12)

According to the properties of tensor product, the eigenvalues of $\rho_{A|0}$ and $\rho_{A|1}$ being equal to the eigenvalues of $\rho_{0}$ and $\rho_{1}$ wang1 are the same as follow

$\displaystyle\lambda_{n}^{\pm}=\frac{1}{2j+1}\pm\frac{j-n}{j(j+1)(2j+1)}\lvert(F(2j+1)-j)\rvert,$

(13)

where $n=0,\cdots,\lfloor j\rfloor$, and $\lfloor j\rfloor$ denotes the largest integer that is less or equal to $j$. It is obvious that the eigenvalues are independent of the measurement. Consequently we have

	$\displaystyle\sum_{i}p_{i}S(\rho_{A|i}))$	$\displaystyle=$	$\displaystyle\frac{1}{2}S(\rho_{A|0})+\frac{1}{2}S(\rho_{A|1})$		(14)
		$\displaystyle=$	$\displaystyle-\sum_{n=0}^{\lfloor j\rfloor}\lambda_{n}^{\pm}\log\lambda_{n}^{\pm}.$

From Eqs. (1), (12) and (14), we have

[Theorem] The Holevo quantity of the $SU(2)$-invariant states is given by

$\displaystyle\chi\{\rho_{AB}|\{\Pi^{B}_{i}\}\}=\log(2j+1)+\sum_{n=0}^{\lfloor j\rfloor}\lambda_{n}^{\pm}\log\lambda_{n}^{\pm}.$

(15)

Fig. 1 shows Holevo quantity as a function of the system parameter $F$ for different $j$. Interestingly, the state $\rho^{ab}$ is separable for $F_{s}\leq 2j/(2j+1)$ Schliemann , which is just twice $F_{d}=j/(2j+1)$ where the Holevo quantity vanishes. As the dimension of the system increases, the Holevo quantity increases in the region $F<F_{d}$ and decreases in the region $F>F_{d}$. The maximum value of the Holevo quantity is attained at $F=1$ for all dimensional systems. Moreover, as $j$ increases, the maximum value of the Holevo quantity tends to decrease, so does the Holevo quantity, see Fig. 1(a). Furthermore, when $j$ tends to infinite, the Holevo quantity is symmetric around the point $F=1/2$ where Holevo quantity is exactly zero, while its maximum values obtained at $F=0$ and $F=1$ keep unchange, see Fig. 1(b).

We have analytically calculated the Holevo quantity of $SU(2)$-invariant states in bipartite spin-$j$ and spin-$\frac{1}{2}$ systems, with the projective measurements performed on the spin-$\frac{1}{2}$ subsystem. It has been shown that the Holevo quantity increases in the region $F<F_{d}$ and decreases in the region $F>F_{d}$ when $j$ increases, and the maximum value of the Holevo quantity is attained at $F=1$ for all $j$. The Holevo quantity and the $SU(2)$-invariant states have particular importance in quantum information processing. Our results give explicit relations between the Holevo quantity and the $SU(2)$-invariant states.