Maximum-confidence measurement for qubit states

In quantum state discrimination, one aims to identify unknown states from a given ensemble by performing measurements. Different strategies such as minimum-error discrimination or unambiguous state identification find different optimal measurements. Maximum-confidence measurements (MCMs) maximize the confidence with which inputs can be identified given the measurement outcomes. This unifies a range of discrimination strategies including minimum-error and unambiguous state identification, which can be understood as limiting cases of MCM. In this work we investigate MCMs for general ensembles of qubit states. We present a method for finding MCMs for qubit-state ensembles by exploiting their geometry and apply it to several interesting cases, including ensembles of two and four mixed states and ensembles of an arbitrary number of pure states. We also compare MCMs to minimum-error and unambiguous discrimination for qubits. Our results provide interpretations of various qubit measurements in terms of MCM and can be used to devise qubit protocols.

Figure 1

Geometry of the MCM for qubit states shown on the Bloch sphere. The arrows represent Bloch vectors. For instance, Bloch vectors $OM$ and $OV$ on the sphere, i.e., pure states, denote orthogonal qubit states. An ensemble $\rho$ and a state of interest ${\rho }_x$ correspond to $OS$ and $OU$, respectively. A complementary state ${\sigma }_x$ is pure, i.e., lies on the sphere. Since $\rho$ is a convex combination of ${\rho }_x$ and a complementary state ${\sigma }_x$ [see Eq. (21)] the state ${\sigma }_x$ is immediately obtained as $OV$ by extending $US$. An optimal POVM element corresponds to $OM$. It holds that $OA+OB=OS$ and $OS+OC=OB\propto OV$.

Figure 2

Two states in Eq. (41) are depicted in the Bloch sphere. Here $OA$ and $OB$ denote the Bloch vectors of the states ${\rho }_0$ and ${\rho }_1$. Complementary states ${\sigma }_0$ and ${\sigma }_1$ on the sphere are found by extending $A_{0}B$ and $A_{1}B$ and consequently $O{S}_0$ and $O{S}_1$. By flipping them to their opposite directions, the MCM is obtained as $O{M}_0$ and $O{M}_1$, which coincide with $O{U}_0$ and $O{U}_1$ for $p=1$. Note that when two states are pure, unambiguous discrimination is possible and is implemented by a POVM containing $O{U}_0$ and $O{U}_1$. The optimal measurement for minimum error is given by $O{E}_0$ and $O{E}_1$. It is therefore found that the MCM, $O{M}_0$ and $O{M}_1$, is in between unambiguous and minimum-error discrimination.

Figure 3

Geometrically uniform pure states $O{A}_x$ form a circle defined by a radius $B{A}_x$ where $OB$ denotes the ensemble of the states with equal probabilities. For a state $O{A}_x$, its complementary state is found at $O{S}_x$ by extending $B{A}_x$. By rotating it with respect to the origin, an optimal POVM element $O{M}_x$ that is orthogonal to the state $O{S}_x$ is obtained. The half plane may be defined precisely by the collection of midpoints of $A_x{M}_x$. Note that the measurement for minimum-error discrimination contains $O{E}_x$ obtained by projecting states $O{A}_x$ onto the half plane. Alternatively, for those states in the half plane, an MCM coincides with a measurement for MED.

Figure 4

Ensemble of noisy tetrahedron states $O{A}_x$ for $x=0,1,2,3$, given as $\mathbb{I}/2$, corresponding to the origin $O$. Then, complementary states are found on the Bloch sphere in the opposite directions to given states: $O{S}_x$ for $x=0,1,2,3$. Optimal POVM elements are rank 1 and found by rotating $O{S}_x$ about $O$: $O{M}_x$ for $x=0,1,2,3$ are in the same direction as given states $O{A}_x$.

Figure 5

Three states $OA, O{A}_1$, and $O{A}_2$ are geometrically uniform. The first state is slightly tilted so that an ensemble of three states $O{A}_x$ for $x=0,1,2$ is considered. The ensemble is denoted by $OB$. Complementary states $O{S}_x$ are found by extending $A_{x}B$, and optimal POVM elements are found by inverting $O{S}_x$ with respect to $O$. None of the elements $O{M}_x$ are identical to states $O{A}_x$.

Figure 6

Three states $O{A}_0, O{A}_1$, and $O{A}_2$ are considered where $O{A}_1$ and $O{A}_2$ are orthogonal. Complementary states $O{S}_x$ are found on the sphere by extending $A_{x}B$. An optimal POVM consists of $O{A}_0, O{M}_1$, and $O{M}_2$. A measurement for minimum-error discrimination contains two POVM elements $O{A}_1$ and $O{A}_2$.