Measuring nonclassical correlations of two-photon states

The threshold between classical and nonclassical two-qubit states is drawn at the place when these states can no longer be described by classical correlations, i.e., quantum discord or entanglement appear. However, to check if the correlations are classical (in terms of quantum discord and entanglement) it is sufficient to witness the lack of quantum discord because its zero value implies the lack of entanglement. We explain how the indicator of quantum discord introduced by Girolami and Adesso [Phys. Rev. Lett. 108, 150403 (2012)] can be practically measured in linear-optical systems using standard beam splitters and photon detectors. We study the efficiency of the setup assuming both ideal and real components and show that the efficiency of the proposed implementation is better than the full two-photon quantum tomography. Thus, we demonstrate that a class of experiments previously available on NMR platform can be implemented in optical systems.

Figure 1

Monte Carlo simulation outcomes for ${10}^6$ random two-qubit density matrices. Geometric quantum discord $D_i$ versus normalized quantum discord indicators $Q_i$ and $V_i$.

Figure 2

Diagram for measuring $M_{1,A}$ (left) and $M_{2,B}$ (right) for a two-qubit source emitting states $\rho$ with period $\tau$. For both qubits from a pair the delay varies depending on the pair number in the sequence. In the case of measuring $M_{i,1}$ the qubits in arm $A$ ($B$) are delayed by $2\tau$ or $\tau$ ($\tau$ or 0), whereas for $M_{i,2}$ the qubits in arm $A$ ($B$) are delayed by $4\tau$, $\tau$, or 0 ($\tau$ or 0). The difficulty of implementing the delay in a real experiment depends on the value of $\tau$ and on technical aspects depending on the physical properties of the qubits. For a very small $\tau$ it can be difficult to provide fast enough switching of the delay, whereas for a very large $\tau$ coherent storage of qubits could be challenging.

Figure 3

Diagram of measurement of $M_{2,A}$ using four photons (two two-qubit states $\rho$) emitted at intervals $\tau$. From left: $N$ iterations of the measurement procedure employing deterministic delay; a single iteration of the procedure in case of the probabilistic delay. In the latter case the time required for obtaining a single measurement outcome is twice as long as in the other case since in order to obtain the correct result one needs to ensure that photons left the delay lines. Thus, the deterministic scheme is $2/p^2$ times more efficient than the probabilistic one, where $p$ is the success probability of the delay. However, the deterministic scheme requires switching the duration of the delay with frequency $1/\tau$ equal to the repetition rate of the source. The measurement diagram for $M_{1,A}$ for a four-photon source is trivial since it does not require delays.

Figure 4

Setup for linear-optical measurement of $U_{mn}$ or $V_{mn}$ observables, i.e., the $U/V$ box. The setup consists of 50:50 asymmetric beam splitters (BSs) and conventional detectors ($D$). To measure $U_{mn}$ ($V_{mn}$) a value of $-4$ is assigned to coincidence $C_{23}$ at $D_2$ and $D_3$ as it implies singlet projection [see Eq. (19)] and value of 1 (2) if the coincidence $C_{14}$ implying $I_{mn}$ operation [see Eq. (12)] is registered by $D_1$ and $D_4$. The other combinations of detector clicks are irrelevant, thus we do not assign values to other detection events. The device works with probability $1/2$ because with this probability both photons reach unaltered the central BS or exit to detectors $D_1$ and $D_4$.

Figure 5

Optical setup for measuring $M_2$ using the probabilistic scheme from Fig. 3 with $p=1/4$. When the delay lines are removed, the setup measures $M_1$. At times $t=0$ and $t=\tau$ two photon pairs $\rho$ enter the setup. When $t=2\tau ,3\tau$ no new photons appear from sources $S_1$ and $S_2$ and only coincidences in the $U$ part of the setup are witnessed. The photons are delayed before entering the $U$ part. One of the photons is always delayed by $\tau$ and the second one randomly by $2\tau$ or 0, however the photon with probability $0.5$ can leak out of the system. In the successful cases ${\rho }_{A_1{B}_1}$ and ${\rho }_{A_2{B}_2}$ enter the setup at $t=0$ and $V_{B_1{B}_2}$ is measured. At the same time photons in the left part are delayed, the first from the left by $\tau$ and the second by $2\tau$ (with probability $1/4$), so at $t=0$ there is no photon detected in $U$. Photon pairs ${\rho }_{A_3{B}_3}$ and ${\rho }_{A_4{B}_4}$ enter the setup at $t=\tau$ and $V_{B_3{B}_4}$ is measured without delay. In the left part, the first photon is delayed by $\tau$, but the second one is not affected (with probability $1/4$). At the same time $U_{A_1{A}_4}$ is measured. Since there are no photons added at $t=2\tau$ only $U_{A_2{A}_3}$ is measured (for a photon provided at $t=0$ and delayed by $2\tau$ and a photon provided at $t=\tau$ delayed by $\tau$). The success rate of the delay procedure is $p^2=1/16$. The whole sequence is repeated until a good estimate of $M_2$ is obtained, e.g., we reach $N_2={10}^3$ successful iterations. No photons enter the setup at $t=3\tau$ to not affect the next measurement iteration.

Figure 6

Time $t$ in units of $\tau$ (the inverse of time between generation of the two two-photon states) needed for performing ${10}^3$ successful measurements of $M_1$ and $M_2$ versus the detection efficiency $\eta$. The success rate of generating two pairs of photons from a single pump pulse was assumed to be $1/100$. The solid curve corresponds to the probabilistic delay scheme whereas the dashed one corresponds to the deterministic delay scheme. The repetition rate of the pump can be set to $1/\tau =20\mathrm{MHz}$ ($\tau =50\mathrm{ns}$). The repetition rate is limited by the dead time of the detectors (typically about $50\mathrm{ns}$). Measuring both moments with ${10}^3$ iterations in a time of order of 1 s would be possible for perfect detectors and deterministic delay scheme. In a realistic case of $\eta =0.75$ and probabilistic delay the whole measurement would take less than 3 min. Note, however, that $N=100$ estimates for $M_1$ and $M_2$ would be available in a 10 times shorter time.

Figure 7

The measured value of $Q_A^{\prime }$ for the two sources $S_L$ and $S_R$ producing pairs of photons in states ${\rho }^L$ and ${\rho }^R$ versus the precise $Q_A$ value for $\rho =({\rho }^L+{\rho }^R)/2$. The blue data points are Monte Carlo results for ${10}^6$ randomly generated ${\rho }^R$ and ${\rho }^L$ states satisfying $F({\rho }^L,{\rho }^R)\ge 0.90$. The blue line indicates $Q_A^{\prime }=Q_A$.