Local quantum uncertainty and bounds on quantumness for orthogonally invariant class of states

Abstract

Local quantum uncertainty was introduced as a measure of quantum uncertainty in an quantum state as achievable on single local measurement. However, such quantity do satisfy all necessary criteria to serve as measure of discord-like quantum correlation and it has no closed formula except only for $2\otimes n$ system. Here, we consider orthogonal invariant class of states which includes both the Werner and Isotropic class of states and explore the possibility of closed form formula. Further, we extend our quest to the possibility of closed form of geometric discord and measurement-induced nonlocality for this class. We also provide a comparative study of the bounds of general quantum correlations with entanglement, as measured by negativity, for an interesting subclass of states.

Appendices

Appendix 1: Generators of SU($n$) and their algebra

For $n=2$, Pauli matrices can be used as the generators of SU(2). While for $n=3$, generally, Gell-Mann matrices are taken as the generators of SU(3). In this way, we can construct traceless, orthogonal generators (generalized Gell-Mann matrices) for SU($n$), containing $n^2-1$ elements as:

$$\begin{aligned} \lambda _{\alpha } \!=\! {\left\{ \begin{array}{ll} \sqrt{\frac{2}{\alpha (\alpha +1)}}\left( \sum _{k\,=\,1}^{\alpha }|k\rangle \langle k|\!-\!\alpha |\alpha \!+\!1\rangle \langle \alpha \!+\!1|\right) ,&{}\quad \alpha =1,\ldots ,n-1\\ |k\rangle \langle m|+|m\rangle \langle k|, &{}\quad 1\!\le \! k<m\!\le \! n, \alpha \!=\!n,\!\ldots ,\!\frac{n^2+n}{2}\!-\!1\\ \mathrm {i}( |k\rangle \langle m|-|m\rangle \langle k|), &{}\quad 1\!\le \! k\!<\!m\!\le \! n,\alpha \!=\!\frac{n(n+1)}{2},\ldots ,n^2-1\\ \end{array}\right. } \end{aligned}$$

(28)

Among the $(n^2-1)$ matrices, the first $(n-1)$ are mutually commutative, next $(n^2-1)/2$ are symmetric and rest $(n^2-1)/2$ are antisymmetric. The generators $\lambda _{\alpha }$ satisfy the orthogonality relation $\text {tr}(\lambda _{\alpha } \lambda _{\beta })=2 \delta _{\alpha \beta }$. The generators satisfy the following commutation and anti-commutation relations,

$$\begin{aligned} \begin{aligned} \left[ \lambda _i,\,\lambda _j\right] =&2\mathrm {i} \sum _k{f_{ijk}\,\lambda _k}\\ \{\lambda _i,\,\lambda _j\}=&2 \sum _k{d_{ijk}\,\lambda _k}+\frac{4}{n}\delta _{ij}\,\mathbb {I}_n \end{aligned} \end{aligned}$$

(29)

$\mathbb {I}_n$ is identity matrix of order $n$, $f_{ijk}$ are real antisymmetric tensors, and $d_{ijk}$ are real symmetric tensors. They are the structure constants of SU($n$). They are determined by the following relations,

$$\begin{aligned} \begin{aligned} f_{ijk}:=\frac{1}{4\mathrm {i}}\text {tr}\left( [\lambda _i,\lambda _j]\lambda _k\right) \\ d_{ijk}:=\frac{1}{4}\text {tr}\left( \{\lambda _i,\lambda _j\}\lambda _k\right) \end{aligned} \end{aligned}$$

(30)

From the relations (29), it follows,

$$\begin{aligned} \lambda _i\,\lambda _j=\mathrm {i} \sum _k{f_{ijk}\,\lambda _k}+ \sum _k{d_{ijk}\,\lambda _k}+\frac{2}{n}\delta _{ij}\,\mathbb {I}_n \end{aligned}$$

(31)

Appendix 2: LQU-a general approach

We can consider $d^2$ elements of SU($d$) as

$$\begin{aligned} u_{nm}:=\sum _{j=0}^{d-1} \exp \left[ \frac{2\pi \mathrm {i}jn}{d}\right] |j\rangle \langle j\oplus m\, \text {mod} \,d|;\quad n,m=0,\ldots ,d-1 \end{aligned}$$

(32)

Each $u_{nm}$(except $u_{00}$) has trace zero, and they all have eigenvalues in the form of $d$-th root of unity. Let us define a row vector $\mathbf {s}:=(s_0,s_1,s_2,\ldots ,s_{d-1})$. Now consider any general observable $K=\mathbf {s}.\varvec{\Theta }$ where $\varvec{\Theta }=(\theta _0,\theta _1,\ldots ,\theta _{d-1})$ and $\theta _{i}$’s are $d$ diagonal matrices of order $d$ with only single entry 1 at corresponding $ii$-th position. $\theta _{i}$‘s can be obtained from linear combination of $u_{n0}$’s. For example, in $3\otimes 3$ system we choose

$$\begin{aligned} \theta _0&= \frac{1}{3}\left( u_{00}+u_{10}+u_{20}\right) \end{aligned}$$

(33)

$$\begin{aligned} \theta _1&= \frac{1}{3}\left( u_{00}+\omega u_{10}+\omega ^2 u_{20}\right) \end{aligned}$$

(34)

$$\begin{aligned} \theta _2&= \frac{1}{3}\left( u_{00}+\omega ^2 u_{10}+\omega u_{20}\right) \end{aligned}$$

(35)

In general, $\theta _j$’s $(j=0,1,\ldots ,d-1)$ can be written as

$$\begin{aligned} \theta _j=\frac{1}{d}\left( \sum _{k=0}^{d-1}\exp \left[ \frac{2 \pi \mathrm {i}(d-kj)}{d}\right] u_{k0}\right) \end{aligned}$$

(36)

Hence

$$\begin{aligned} \begin{aligned} K=&\mathbf {s}.\varvec{\Theta }\\ =&\sum _{j=0}^{d-1} s_j \theta _j\\ =&\sum _{j=0}^{d-1} s_j \frac{1}{d}\sum _{k=0}^{d-1}\exp \left[ \frac{2 \pi \mathrm {i}(d-jk)}{d}\right] u_{k0}\\ =&\sum _{k=0}^{d-1}\left( \frac{1}{d}\sum _{j=0}^{d-1}s_j\exp \left[ \frac{2 \pi \mathrm {i}(d-jk)}{d}\right] \right) u_{k0}\\ =&\sum _{k=0}^{d-1}t_k u_{k0} \end{aligned} \end{aligned}$$

(37)

where we define $t_k=\left( \frac{1}{d}\sum _{j=0}^{d-1}s_j\exp [\frac{2 \pi \mathrm {i}(d-jk)}{d}]\right)$. For any unitarily connected observable with same spectrum,

$$\begin{aligned} \begin{aligned} VKV^\dag =t_{0}\mathbb {I}+\sum _{i=1}^{d-1}t_{i}\left( \sum _{j=0}^{d^2-1}\chi ^{i}_{j}\tilde{\lambda }_{j}\right) ;\,\, \text {where}\, Vu_{i0}V^\dag =\left( \sum _{j=0}^{d^2-1}\chi ^{i}_{j} \tilde{\lambda }_{j}\right) \end{aligned} \end{aligned}$$

(38)

Now we define, $\Lambda =(\tilde{\lambda }_0,\tilde{\lambda }_1,\ldots ,\tilde{\lambda }_{d^2-1})$ with $\tilde{\lambda }_{0}=u_{00}=\mathbb {I}_d$($d$-th order unit matrix), $\tilde{\lambda }_i$’s are remaining $d^2-1$ elements (32) of SU($d$) and another $d^2-1$ dimensional vector $\mathbf {m}=(m_1,m_2,\ldots ,m_{d^2-1})$. In terms of these quantities we can express (38) as

$$\begin{aligned} \begin{aligned} VKV^\dag =&\mathbf {m}.\varvec{\Lambda }+m_{0}\mathbb {I}_d \,\,\text {with}\, m_j=\sum _{i=1} ^{d-1}t_i \chi _j^i,\,j=1,2,\ldots , d^2-1\; \text {and}\\ m_0=&t_0+\sum _{i=1} ^{d-1}t_i \chi _0^i\\ =&\parallel \mathbf {m}\parallel \hat{\mathbf {m}}.\varvec{\Lambda }+m_{0}\mathbb {I} \end{aligned} \end{aligned}$$

(39)

In the last step, we have decomposed the vector $\mathbf {m}$ into an unit vector $\hat{\mathbf {m}}$ and modulus $\parallel \mathbf {m}\parallel$. Hence, we can safely choose any observable(maximally informative) as $\hat{\mathbf{m}}.\varvec{\Lambda }$ and perform the optimization over all unit vector $\hat{\mathbf{m}}$. The amount of LQU are proportional on all such orbits. In fact, the optimization problem (2) turns out to be

$$\begin{aligned} \begin{aligned} \mathcal {U}^{\Lambda }_{A}(\rho )=\min _{\hat{m}_i,\sum \hat{m}_i^2=1} g(\hat{m}_i,f,\hat{f}) \end{aligned} \end{aligned}$$

(40)

where $g$ is a real valued function of the parameters $\hat{m}_i$, $f$, $\hat{f}$. In 3$\otimes 3$ scenario, for orthogonal invariant class, we obtain the real valued function $g$ in the form,

$$\begin{aligned} \begin{aligned} g(\hat{m}_i,f,\hat{f})&=\!6(3a+b+c)(\hat{m}_1\hat{m}_2\!+\!\hat{m}_3\hat{m}_6\!+\!\omega \hat{m}_3\hat{m}_7\!+\!\omega ^2 \hat{m}_4\hat{}m_8)\!-\!3(2b_1c_1(\hat{m}_1^2\!+\!\hat{m}_2^2)\\&\qquad \quad +(6a_1^2+4a_1b_1+4a_1c_1)(\hat{m}_1\hat{m}_2+\hat{m}_3\hat{m}_6+\omega \hat{m}_5\hat{m}_7+\omega ^2 \hat{m}_4\hat{m}_8)\\&\qquad \quad + 4b_1c_1(\hat{m}_4\hat{m}_7+\hat{m}_5\hat{m}_8+\hat{m}_3\hat{m}_6)) \end{aligned} \end{aligned}$$

the parameters $a,b,c,a_1,b_1,c_1$ are related to $f,\hat{f}$ by the relations (15) and (18). Indeed, this function is real valued. Substituting all parameters in terms of $f$ and $\hat{f}$ justifies the claim.

Appendix 3: Bloch vector and correlation matrix elements for orthogonal invariant class

Elements of Bloch vector can be written as

$$\begin{aligned} \begin{aligned} x_{k}=&\frac{n}{2}\text {tr}(\rho \lambda _k\otimes \mathbb {I}_n)\\ \end{aligned} \end{aligned}$$

(41)

Since the operators $\mathbb {F}$, $\mathbb {F}^2$ and $\mathbb {I}$ have maximally mixed marginals, it is straight forward to claim that

$$\begin{aligned} x_{k}=0\quad \text {for all}\quad k=1,2,\ldots ,n^2-1 \end{aligned}$$

(42)

The correlation matrix elements,

$$\begin{aligned} \begin{aligned} t_{kl}=&\frac{n^2}{4}\text {tr}(\rho \lambda _k \otimes \lambda _l)\\ =&\frac{n^2}{4}\text {tr}\left[ a \,\sum _{i,j} |i\rangle \langle i|\lambda _k \otimes |j\rangle \langle j|\lambda _l+ b \,\sum _{i,j} |i\rangle \langle j|\lambda _k \otimes |j\rangle \langle i|\lambda _l + c \,\sum _{i,j} |i\rangle \langle j|\lambda _k \otimes |i\rangle \langle j|\lambda _l\right] \\ =&\frac{n^2}{4}\text {tr}\left[ b\, \sum _{i,j} |i\rangle \langle j|\lambda _k \otimes |j\rangle \langle i|\lambda _l+ c\, \sum _{i,j} |i\rangle \langle j|\lambda _k \otimes |i\rangle \langle j|\lambda _l\right] \\ \end{aligned} \end{aligned}$$

(43)

Whenever $k\ne l$,

$$\begin{aligned} \begin{aligned} t_{kl}=&\frac{n^2}{4}\left[ b\,\sum _{i,j}\langle j|\lambda _{k}|i\rangle \langle i|\lambda _{l}|j\rangle + c\,\sum _{i,j}\langle j|\lambda _{k}|i\rangle \langle j|\lambda _{l}|i\rangle \right] \\ =&\frac{n^2}{4}\left[ c\,\sum _{i,j}(\lambda _{k})_{ji}(\lambda _{l})_{ji}\right] =0\\ \end{aligned} \end{aligned}$$

(44)

By $(\lambda _{k})_{ij}$, we denote the $ij$-th element of $\lambda _{k}$. The second equality follows from the fact that according to the construction of SU($n$) generators (28), any two $\lambda _{k}$ and $\lambda _{l}, k\ne l$ have no element in common at any position (or conjugate position) in their respective matrix form in computational basis. This claims that $T$ is in a fact diagonal matrix. Diagonal form of $T$ can also be claimed easily by the same logic as in $x_k$’s. Whenever $k=l,$

$$\begin{aligned} t_{kk}=\frac{n^2}{4}\left[ 2b+c\,\sum _{i,j} (\lambda _{k})_{ji}^{2}\right] \end{aligned}$$

(45)

There are $n^2-1$ generators of SU($n$) and among them, $\sum _{i,j} (\lambda _{k})_{ji}^{2}= 2$ for $k=1,\ldots ,\frac{n^2+n-2}{2}$ and $\sum _{i,j} (\lambda _{k})_{ji}^{2}= -2$ for $k=\frac{n^2+n}{2},\ldots ,n^2-1$. Hence,

$$\begin{aligned} t_{kk}= \frac{n^2}{2} {\left\{ \begin{array}{ll} (b+c)\quad \text {for}\, k=1,2,\ldots ,\frac{n^2+n-2}{2}\\ (b-c)\quad \text {for}\, k=\frac{n^2+n}{2},\ldots ,n^2-1\\ \end{array}\right. } \end{aligned}$$

(46)