How to observe and quantify quantum-discord states via correlations

Quantum correlations between parts of a composite system most clearly reveal themselves through entanglement. Designing, maintaining, and controlling entangled systems is very demanding, which raises the stakes for understanding the efficacy of entanglement-free, yet quantum correlations, exemplified by quantum discord. Discord is defined via conditional mutual entropies of parts of a composite system and its direct measurement is hardly possible even via full tomographic characterization of the system state. Here we design a simple protocol to detect and quantify quantum discord in an unentangled bipartite system. Our protocol relies on a characteristic of discord that can be extracted from repeated direct measurements of certain correlations between subsystems of the bipartite system. The proposed protocol opens a way of extending experimental studies of discord to electronic systems, but can also be implemented in quantum-optical systems.

Figure 1

Proposed setup of the bipartite system made of two Mach-Zehnder interferometers, ${\mathrm{MZI}}^A$, with the phase difference ${\varphi }_A$, and ${\mathrm{MZI}}^B$ with ${\varphi }_B$. The light-blue (light-brown) area represents the state-preparation (the state evolution and discord measurement) part of the protocol. Electrons from sources $S^A$ and $S^B$ enter beam splitters ${\mathrm{BS}}_0^A$ and ${\mathrm{BS}}_0^B$, whose random transparencies are synchronized by a classical computer, allowing the creation of mixed states of the form given by Eq. (1). The final state is controlled by transparencies of beam splitters ${\mathrm{BS}}_1^A$ and ${\mathrm{BS}}_1^B$ and phases ${\varphi }_{A,B}$, and is recorded at any pair of detectors $D_i^A$ and $D_i^B$ (with $i=1$ or 2). Varying the phase difference ${\varphi }_d$ in the third, detecting ${\mathrm{MZI}}^d$, would allow one to identify a state with no $A$ discord as one for which the interference pattern is suppressed for certain parameters of subsystem $A$ and remains suppressed for any tuning of subsystem $B$ (without adjusting $A$ any further), as illustrated below in Fig. 2.

Figure 2

A striking difference between (a) discorded and (b) nondiscorded states: zero-visibility (dark) lines are sensitive to changes in the state of passive subsystem $B$ in (a) and are independent of these changes in (b). Here we use the symmetric in-states: (a) ${\rho }^{AB}=\frac{1}{2}\left[|\uparrow \uparrow \rangle \langle \uparrow \uparrow |+\frac{1}{2}|++\rangle \langle ++|\right]$ and (b) ${\rho }^{AB}=\frac{1}{2}\left[|++\rangle \langle ++|+\frac{1}{2}|--\rangle \langle --|\right]$, with $|\pm \rangle \equiv \frac{1}{\sqrt{2}}\left(|\uparrow \rangle \pm |\downarrow \rangle \right)$. Since the states $|+\rangle$ and $|-\rangle$ are orthogonal whereas $|+\rangle$ and $|\uparrow \rangle$ are not, these density matrices describe (a) a discorded state and (b) a nondiscorded state, as explained after Eq. (6). Any continuous zero-visibility line in (b) can be chosen for a quantitative characteristic of discord, given by Eq. (11); cf. Fig. 4.

Figure 3

The visibility plots for ${\rho }^{AB}=\frac{1}{2}|\uparrow \uparrow \rangle |\uparrow \uparrow \rangle +\frac{1}{2}|\theta \theta \rangle \langle \theta \theta |$, where (a) $\theta =\frac{5}{6}\pi$ and (b) $\theta =\frac{1}{5}\pi$. These density matrices have similar discord, as can be seen from Fig. 4, yet their visibility landscapes look completely different: (a) an almost $\pi$ jump in zero-visibility lines over a small interval of $\beta$ followed by almost $\beta$-independent zero-visibility lines is equivalent to (b) zero-visibility lines with a small nonmonotonicity over a large interval; both signify weak sensitivity with respect to changes in a passive system for the states with a relatively small discord.

Figure 4

The standard definition of discord ${\mathcal{D}}_A$ (dashed blue line) vs the alternative quantifier of Eq. (11) ${\mathrm{\Delta }}_{\alpha }^2$ (solid red line) for the in-state with the density matrix ${\rho }_{\theta }^{AB}=\frac{1}{2}\left[|\uparrow \uparrow \rangle \langle \uparrow \uparrow |+|\theta \theta \rangle \langle \theta \theta |\right]$.

Figure 5

The raw visibility landscape for the two in-states used in Fig. 2 as a function of parameters $\alpha$ and ${\varphi }_A$ controlling, respectively, the transparency of ${\mathrm{BS}}_1^A$ and the phase difference in ${\mathrm{MZI}}^A$ (see Fig. 1). Here we keep fixed the values of the corresponding parameters in ${\mathrm{MZI}}^B$ ($\beta =\pi /3$ and ${\varphi }_B=0$). Here, zero-visibility points correspond to ${\varphi }_0=0mod\left(2\pi \right)$ as expected.

Figure 6

Visibility as a function of $\alpha$ and $\beta$ for nondiscorded states with ${\varphi }_A={\varphi }_{1,2}$. (a) ${\rho }^{AB}=1/5|++\rangle \langle ++|+4/5|--\rangle \langle --|$. (b) ${\rho }^{AB}=1/2|\uparrow \uparrow \rangle \langle \uparrow \uparrow |+1/2|+\downarrow \rangle \langle +\downarrow |$. (a) The “gridlike” visibility characteristic of a density matrix which is correlated between the $A$ and $B$ subsystems, but nondiscorded, with only the classical correlations between subsystems. (b) The “barcode” graph is a result of a density matrix which is completely uncorrelated between the $A$ and $B$ subsystems.

Figure 7

The visibility landscapes for the density matrix given by Eq. (B1) with ${\varphi }_2=\pi /2$. (a) Visibility as a function of $\alpha ,{\varphi }_A$ with $\beta =2\pi /3$. (b) Visibility as a function of $\alpha ,\beta$ with fixed ${\varphi }_A=arctan\left(\frac{2-\sqrt{3}}{3}\right)$ that corresponds to minimal visibility spots in the landscape plot (a).

Figure 8

(a) Visibility landscape for the density matrix given by Eq. (B1) with ${\varphi }_2=\pi /2$, where visibility lines are dependent on ${\varphi }_A$ and $\beta$ with fixed diagonalizing parameter $\alpha ={\alpha }_0=\pi /2$. (b) Discord (dashed blue line) and an alternative quantifier ${\mathrm{\Delta }}_{\varphi }^2$ (red solid line) for the state (B1) for a range of ${\varphi }_2$.

Figure 9

(a) Visibility landscape for the density matrix given by Eq. (B3) with $\theta =\pi /2$, with a characteristic signature of discord in the curviness of the zero-visibility lines. (b) The discord quantifier (red solid line) of Eq. (11) vs the standard discord (dashed blue line) for the density matrix given by Eq. (B3) for a range of $\theta$.