Quantum state discrimination: A geometric approach

We analyze the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view, this problem is equivalent to that of embedding a simplex of points whose distances are maximal with respect to the Bures distance (or trace distance). We derive upper and lower bounds for the trace distance and for the fidelity between two quantum states, which imply bounds for the Bures distance between the unitary orbits of both states. We thus show that, when analyzing minimal and maximal distances between states of fixed spectra, it is sufficient to consider diagonal states only. Hence when optimal discrimination is considered, given freedom up to unitary orbits, it is sufficient to consider diagonal states. This is illustrated geometrically in terms of Weyl chambers.

Figure 1

Set $R$ of positive operators with $k+1=\left(\text{a}\right)2$ and (b) 3 distinguishable states ${\rho }_1,\dots ,{\rho }_{k+1}$ which form a $k$-dimensional simplex ${\Delta }_k$ of maximal side length $D^{\text{max}}$, with respect to the Bures (or the trace) metric.

Figure 2

Simplex of eigenvalues for $N=\left(\text{a}\right)2$ and (b) 3, split into $2!=2$ (left and right) and $3!=6$ (indicated by dotted lines) Weyl chambers, respectively. The minimal distance $m$ between the orbits of unitarily similar states stemming from two quantum states is equal to the distance between the corresponding spectra $a$ and $b$ belonging to the same Weyl chamber shown for $N=\left(\text{a}\right)2$ and (b) 3. The maximal distance $M$ is achieved for points $a$ and $b^{\prime }$ belonging to the opposite Weyl chambers.