Minimum-error discrimination between two sets of similarity-transformed quantum states

Abstract

Using the equality form of the necessary and sufficient conditions introduced in Jafarizadeh (Phys Rev A 84:012102 (9 pp), 2011), minimum error discrimination between states of the two sets of equiprobable similarity transformed quantum qudit states is investigated. In the case that the unitary operators describing the similarity transformations are generating sets of two irreducible representations and the states fulfill a certain constraint, the optimal set of measurements and the corresponding maximum success probability of discrimination are determined in closed form. In the cases that they are generating sets of reducible representations, there exist no closed-form formula in general, but the procedure can be applied properly in each case provided that the states obey some constraints. Finally, we give the maximum success probability of discrimination and optimal measurement operators for some important examples of mixed quantum states, such as generalized Bloch sphere m-qubit states, qubit states and their three special cases.

Appendix

Here, we want to calculate maximal success probability given in Eq. (93). To this aim, first we express Eq. (13) in terms of the Bloch vectors components as

$$\begin{aligned}&\eta bn_{i}-\eta ^{\prime }b^{\prime }n^{\prime }_{i}=(p_{opt}-\eta ^{\prime })m^{\prime }_{i}-(p_{opt}-\eta )m_{i}\end{aligned}$$

(95)

$$\begin{aligned}&2^{m}\beta _{i}=\eta bn_{i}+(p_{opt}-\eta )m_{i} \end{aligned}$$

(96)

for any $i\in S_{I}\cap S^{\prime }_{I}$. From Eqs. (95) and (88), it is seen that for any $i\in S_{I}\cap S^{\prime }_{I}$, the signs of $\eta bn_{i}-\eta ^{\prime }b^{\prime }n^{\prime }_{i}$ and $m_{i}$ are opposite. In other words, if restrictions of $\hat{n}, \hat{n}^{\prime }, \hat{m}$ and $\hat{m}^{\prime }$ to the subspace corresponding to the index set $S_{I}\cap S^{\prime }_{I}$ are denoted by $\vec {n}_{0}, \vec {n}^{\prime }_{0}, \vec {m}_{0}$ and $\vec {m}^{\prime }_{0}$ respectively, then the vectors $\eta b\vec {n}_{0}-\eta ^{\prime }b^{\prime }\vec {n}^{\prime }_{0}$ and $\vec {m}_{0}$ point in the opposite directions and we can write

$$\begin{aligned} \vec {m}_{0}=-\sqrt{\sum _{i\in S_{I}\cap S^{\prime }_{I}}(m_{i})^{2}}\frac{\eta b\vec {n}_{0}-\eta ^{\prime }b^{\prime }\vec {n}^{\prime }_{0}}{|\eta b\vec {n}_{0}-\eta ^{\prime }b^{\prime }\vec {n}^{\prime }_{0}|}. \end{aligned}$$

(97)

Therefore, Eq. (96) takes the following vectorial form

$$\begin{aligned} 2^{m}\vec {\beta }=\eta b\vec {n}_{0}-(p_{opt}-\eta )\sqrt{\sum _{i\in S_{I}\cap S^{\prime }_{I}}(m_{i})^{2}}\frac{\eta b\vec {n}_{0}-\eta ^{\prime }b^{\prime }\vec {n}^{\prime }_{0}}{|\eta b\vec {n}_{0}-\eta ^{\prime }b^{\prime }\vec {n}^{\prime }_{0}|} \end{aligned}$$

(98)

Eqs. (88), (97) and (98) show that the vectors $\vec {n}_{0}, \vec {n}^{\prime }_{0}, \vec {\beta }, \vec {m}_{0}$ and $\vec {m}^{\prime }_{0}$ are coplanar. Next, to simplify the algebra, we choose an orthogonal coordinate system in the subspace corresponding to the index set $S_{I}\cap S^{\prime }_{I}$ in which the plane of $\vec {n}_{0}$ and $\vec {n}^{\prime }_{0}$ coincides with the plane defined by an arbitrary pair of coordinate axes. Also, $\vec {n}_{0}$ points toward the positive direction of one axis of the pair. Let us denote by $n^{\prime }_{0}, \beta _{0}, m_{0}$ and $m^{\prime }_{0}$ the components of $\vec {n}^{\prime }_{0}, \vec {\beta }, \vec {m}_{0}$ and $\vec {m}^{\prime }_{0}$ along an axis of the pair lying in the direction of $\vec {n}_{0}$ and by $n^{\prime }_{1}, \beta _{1}, m_{1}$ and $m^{\prime }_{1}$ their components along another axis, respectively. In the considered coordinate system, Eq. (88) and the third relations of Eqs. (91) and (92) are written as

$$\begin{aligned} \mu m_{0}&+ (1-\mu )m^{\prime }_{0}=0 \nonumber \\ \mu m_{1}&+ (1-\mu )m^{\prime }_{1}=0\end{aligned}$$

(99)

$$\begin{aligned} m_{0}&= \frac{2^{m}\beta _{0}-\eta bn_{0}}{p_{opt}-\eta }, \nonumber \\ m_{1}&= \frac{2^{m}\beta _{1}}{p_{opt}-\eta },\nonumber \\ m^{\prime }_{0}&= \frac{2^{m}\beta _{0}-\eta ^{\prime }b^{\prime }n^{\prime }_{0}}{p_{opt}-\eta ^{\prime }},\nonumber \\ m^{\prime }_{1}&= \frac{2^{m}\beta _{1}-\eta ^{\prime }b^{\prime }n^{\prime }_{1}}{p_{opt}-\eta ^{\prime }}, \end{aligned}$$

(100)

where we have introduced $\mu =\sum ^{n}_{j=1}\lambda _{j}$. We square both sides of the first two relations of Eqs. (91) and (92), then sum up over $i$ and use the fact that $\hat{n}, \hat{n}^{\prime }, \hat{m}$ and $\hat{m}^{\prime }$ are unit vectors, to obtain

$$\begin{aligned} 1-m^{2}_{0}-m^{2}_{1}&= \frac{\eta ^{2}b^{2}}{(p_{opt}-\eta )^{2}}(1-n^{2}_{0}) \\ 1-m^{\prime 2}_{0}-m^{\prime 2}_{1}&= \frac{\eta ^{\prime 2}b^{\prime 2}}{\big (p_{opt}-\eta ^{\prime }\big )^{2}}\big (1-n^{\prime 2}_{0}-n^{\prime 2}_{1}\big )\nonumber \end{aligned}$$

(101)

Finally, by composing Eqs. (99)–(101) we attain to Eq. (93) for $p_{opt}$.