Direct method for measuring of purity, superfidelity, and subfidelity of photonic two-qubit mixed states

The Uhlmann-Jozsa fidelity (or, equivalently, the Bures distance) is a basic concept of quantum communication and quantum information, which however is very difficult to measure efficiently without recourse to quantum tomography. Here we propose a direct experimental method to estimate the fidelity between two unknown two-qubit mixed states via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method enables a direct measurement of the first- and second-order overlaps between two arbitrary two-qubit states. In particular, the method can be applied to measure the purity (or linear entropy) of a single two-qubit mixed state in a direct experiment. We also propose and critically compare several experimental strategies for measuring the sub- and superfidelities of polarization states of photons in various linear-optical setups.

Figure 1

Fidelity $F({\rho }_1,{\rho }_2)$ versus generalized mean $\overline{F}({\rho }_1,{\rho }_2)$, corresponding to (a) subfidelity $E({\rho }_1,{\rho }_2)$ for $m=-\infty$, (b) optimized mean for $m=-2.137$, (c) arithmetic mean for $m=1$, and (d) superfidelity $G({\rho }_1,{\rho }_2)$ for $m=+\infty$, as calculated in our Monte Carlo simulation for ${10}^7$ pairs of two-qubit states. The estimation error is given by $\Delta =\sqrt{\langle {(\overline{F}-F)}^2\rangle }$. Note that if the fidelity bounds $E$ and $G$ were tighter, then the area covered by the simulation outcomes would be smaller, converging to the $G=E$ line.

Figure 2

Method for a single iteration of the measurement of the overlap $\mathrm{Tr}\left({\rho }_1{\rho }_2\right)$ (and, in a special case, the purity $\chi$ and linear entropy $S_L=1-\chi$) between any two-qubit states ${\rho }_1$ and ${\rho }_2$ produced at constant time intervals $\tau$. The subsystem of the first (second) qubit is called $A$ ($B$). The measurement should be repeated until some large enough number $K$ of values is accumulated. The delay is implemented at times $t=2k\tau$, where $k=0,1,2,...,K-1$ and $2K\tau$ is the duration of the measurement. The delay can be implemented using fast switching between, e.g., two paths of the optical-length difference corresponding to delay $\tau$. This method works also if the states ${\rho }_1$ and ${\rho }_2$ are qubits, given that we removed the measurement $V_{B_1{B}_2}$ or $V_{A_1,A_2}$.

Figure 3

Setup implementing a direct measurement of the first-order overlap $O({\rho }_1,{\rho }_2)=\mathrm{Tr}\left({\rho }_1{\rho }_2\right)$ between arbitrary two-qubit mixed states ${\rho }_1$ and ${\rho }_2$. The setup consists of two $V$ blocks as in Ref. [31]. Due to the probabilistic nature of the path taken by photons after the BS interaction the setup can provide conclusive results in at most $1/4$ of the cases when ${\rho }_1$ and ${\rho }_2$ are delivered (as discussed in Fig. 2). This maximum performance is achieved when the photons are deterministically antibunched (i.e., always projected onto the singlet state) if both of them impinge on BS${}_3$. Note that the BSs are asymmetric, i.e., the photon is phase shifted by $\pi$ only when reflected from the black-shaded side of the BSs. If ${\rho }_1$ and ${\rho }_2$ are single photons then only one $V$ block is required and the measurement setup corresponds to the one used in Ref. [33].

Figure 4

Experimentally friendly methods for the measurement of the $V$ block as described in the text based on (a) a removable beam splitter, (b) a Mach-Zehnder interferometer, (c) a partially movable beam splitter, and (d) two-photon overlap alignment. BS, balanced beam splitter; PS, phase shifter; FC, fiber coupler; and D, detector. Motorized translation (double arrow) is used to tune the temporal delay between the photons in order to switch between the measurement regimes as explained in the text.

Figure 5

Proposal of the measurement setup for the second-order overlap $O^{\prime }({\rho }_1,{\rho }_2)$ for two-qubit states. The input is given as ${({\rho }_1\otimes {\rho }_1\otimes {\rho }_2\otimes {\rho }_2)}_{A_1{B}_1{A}_2{B}_2{A}_3{B}_3{A}_4{B}_4}$. If the states are single-qubit states then one of the blocks can be removed. Note that in our optimized methods (as described in Sec. 4), the number of required detectors is equal to the number of photons. Double arrows indicate the two-photon overlap alignment as in Fig. 4d.