Probability-based comparison of quantum states

We address the following state comparison problem: is it possible to design an experiment enabling us to unambiguously decide (based on the observed outcome statistics) on the sameness or difference of two unknown state preparations without revealing complete information about the states? We find that the claim “the same” can never be concluded without any doubts unless the information is complete. Moreover, we prove that a universal comparison (that perfectly distinguishes all states) also requires complete information about the states. Nevertheless, for some measurements, the probability distribution of outcomes still allows one to make an unambiguous conclusion regarding the difference between the states even in the case of incomplete information. We analyze an efficiency of such a comparison of qudit states when it is based on the swap measurement. For qubit states, we consider in detail the performance of special families of two-valued measurements, enabling us to successfully compare at most half of the pairs of states. Finally, we introduce almost-universal comparison measurements which can distinguish almost all nonidentical states (up to a set of measure zero). The explicit form of such measurements with two and more outcomes is found in any dimension.

Figure 1

Illustration of probability-based comparison. If the observed probability distribution belongs to $P_{\mathsf{E}}^{-}\setminus P_{\mathsf{E}}^{+}$, then states $\varrho$ and $\xi$ are for sure different.

Figure 2

Body $B\left({\varkappa }_0\right)$ [i.e., the region of parameters $({\varkappa }_1,{\varkappa }_2,{\varkappa }_3)$] when (15) is a true POVM effect. Parameter ${\varkappa }_0$ takes values $\frac{1}{4}$, $\frac{3}{8}$, $\frac{1}{2}$, and $\frac{3}{4}$ for figures from left to right. The union ${\cup }_{{\varkappa }_0\in [0,1]}B\left({\varkappa }_0\right)$ is a rhombododecahedron and is depicted by solid lines. POVM effects $E_{\mathrm{asym}}$, $E_{\mathrm{sym}}$, $E_{z\pm }$ are vertices and POVM effects $E_{xy\pm }$ are face centers of this convex polytope.

Figure 3

States $\varrho$ in Bloch ball (determined by vectors $\mathbf{r}$) which can be distinguished from a fixed state $\xi$ (given by vector $\mathbf{k}$) by using measurement ${\mathsf{E}}_{\mathrm{SWAP}}$ (a), ${\mathsf{E}}_{xy}$ (b), and ${\mathsf{E}}_z$ (c).

Figure 4

Densities of states $N_{E,\mu }^{\mathrm{diff}}\left(p\right)$ (solid) and $N_{E,\mu }^{\mathrm{same}}\left(p\right)$ (dashed) for different POVM effects: $E_{\mathrm{asym}}$ (left column), $E_{xy-}$ (middle column), and $E_{z-}$ (right column); and for different measures: the Hilbert-Schmidt measure (top row) and the Bures measure (bottom row).

Figure 5

States $\varrho$ inside Bloch ball (determined by vectors $\mathbf{r}$) which can be distinguished from a particular fixed state $\xi =\mathrm{diag}({\Xi }_1,{\Xi }_2)$ [given by vector $\mathbf{k}=(0,0,{\Xi }_1-{\Xi }_2)$] by using the nondiagonal POVM effect $E=|{\Xi }_2\otimes {\Xi }_1\rangle \langle {\Xi }_2\otimes {\Xi }_1|\equiv \mathrm{diag}(0,0,1,0)$ for the following cases: $\xi =\mathrm{diag}(1,0)$ (a), $\xi =\mathrm{diag}(\frac{1}{2},\frac{1}{2})$ (b), $\xi =\mathrm{diag}(\frac{1}{3},\frac{2}{3})$ (c), and $\xi =\mathrm{diag}(\frac{1}{4},\frac{3}{4})$ (d).

Figure 6

Almost-universal comparison measurement (for qubits) with $n$ outcomes: (a) $n=2$, the images $P_{\text{2-out}}^{\pm }$ on 1 simplex and the densities $N_{E,\mu }^{\mathrm{diff}}\left(p\right)$ for Hilbert-Schmidt (HS) and Bures (B) measures, with $\delta$ peak being ascribed to $N_{E,\mu }^{\mathrm{same}}\left(p\right)$; (b) $n=3$, the images $P_{\text{3-out}}^{\pm }$ on 2 simplex. In both cases the states $\varrho$ and $\xi$ can be distinguished whenever their Bloch vectors satisfy $r_z\ne k_z$.