Hierarchy in temporal quantum correlations

Einstein-Podolsky-Rosen (EPR) steering is an intermediate quantum correlation that lies in between entanglement and Bell nonlocality. Its temporal analog, temporal steering, has recently been shown to have applications in quantum information and open quantum systems. Here we show that there exists a hierarchy among the three temporal quantum correlations: temporal inseparability, temporal steering, and macrorealism. Given that the temporal inseparability can be used to define a measure of quantum causality, similarly the quantification of temporal steering can be viewed as a weaker measure of direct cause and can be used to distinguish between direct cause and common cause in a quantum network.

Figure 1

Beginning with quantum state ${\rho }_0$, one can construct three different temporal quantum correlations: (a) the temporal correlations scenario associated with nonmacrorealism, (b) the temporal steering scenario, and (c) constructing the pseudodensity matrix of a single system. Here the $x$ and $y$ denote two classical inputs at times $t_1$ and $t_2$ with classical outcomes $a$ and $b$, respectively. The quantum channel is denoted by $\mathrm{\Lambda }$. Quantum state tomography is denoted as QST.

Figure 2

The blue-dotted, red-solid, and black-dashed curves represent, respectively, the dynamics of the $f$, the TSR, and ${\langle B\rangle }_{\text{T-CHSH}}^{\text{max}}$ of a qubit undergoing the depolarizing channel Eq. (B1c). We can see that the order of the three quantifiers (from the earliest to the latest vanishing time) is ${\langle B\rangle }_{\text{T-CHSH}}$, TSR, and $f$, demonstrating the hierarchy relation proposed in this work. Here, $t$ is in units of depolarizing rate ${\gamma }_{\text{D}}$.

Figure 3

Schematic diagram for the quantum causality schemes considered here. (a) The quantum correlations of two qubits arise from a “black box.” The correlations may be due to (b) a common cause (space-like correlation) or (c) a direct cause (time-like correlation). (d) and (e) We propose to include an auxiliary qubit (qubit-3) coherently coupled to qubit-1. By examining the time-like steering of qubit-3 to qubit-2, one can infer whether the correlations are due to a common cause or a direct cause. This is because, if the correlations are from the common cause, there is no time-like steering of qubit-3 to qubit-2 (dashed line), while an oscillatory time-like steering (blue curve) exists if they are from a direct cause.