Quantifying causal influence in quantum mechanics

We extend Pearl's definition of causal influence to the quantum domain, where two quantum systems $A, B$ with finite-dimensional Hilbert space are embedded in a common environment $C$ and propagated with a joint unitary $U$. For a finite-dimensional Hilbert space $C$, we find the necessary and sufficient condition on $U$ for a causal influence of $A$ on $B$ and vice versa. We introduce an easily computable measure of the causal influence and use it to study the causal influence of different quantum gates, its mutuality, and quantum superpositions of different causal orders. For two two-level atoms dipole-interacting with a thermal bath of electromagnetic waves, the space-time dependence of causal influence almost perfectly reproduces the one of reservoir-induced entanglement.

Figure 1

(a) CI (dimensionless) for some quantum gates. The environment for the three-qubit gates is initially in state ${|0\rangle }_C$ and, when required, $A$ is the control qubit and $B$, the target. (b) $E\left[I_{A\to B}\right]$ (dimensionless) as function of $d_A,d_B$ and $d_C$ [Eq. (7)], where $d_A$ is coded in the sequence of points for each $d_B, d_C$ with $2\le d_A\le 6$ from top to bottom ($3\le d_B\le 8$) or bottom to top ($d_B=2$, coinciding on this scale) in each sequence.

Figure 2

Histogram of $I_{B\to A}$ (dimensionless), for ${10}^4$ random Haar distributed unitary matrices of dimension $D=d_A{d}_B{d}_C$, for $D=2\times 2\times 2$ (light blue) and $D=3\times 2\times 2$ (dark blue), their mean values are 0.76 and 0.896, the standard deviations 0.19 and 0.095, and their expected values [Eq. (7)] $16/21\simeq 0.76$ and $128/143\simeq 0.895$, respectively.

Figure 3

(a) $I_{B\to A}$ (dimensionless) for two initially noninteracting DQD with $d=10$ nm coupled with the blackbody radiation at $T=2.73$ K with $y_m=4250$ ($\hslash {\omega }_m=1\mathrm{eV}$), parametrized by $x/c$ and $t$. The corresponding plot for the entanglement of formation $E$ looks almost identical (see Fig. 2 in Ref. [57]). Dotted line: $t={10}^{11}{\left(x/c\right)}^3$. Full line: light-cone $ct=x$. (b) Difference $\delta =0.795481I_{B\to A}-E$ (dimensionless), see text, $E$ for the initial state $\left(\right|0\rangle +|1\rangle )\otimes \left(\right|0\rangle +|1\rangle )/2$. Same parameters as in panel (a).

Figure 4

[50] Nonvanishing integrals $\mathcal{I}$ for $p=2$. Their respective values, from panel (a) to panel (e), are $\frac{1}{D^2-1}=:{\mathcal{I}}_{\left(a\right)}, \frac{-1}{D(D^2-1)}=:{\mathcal{I}}_{\left(b\right)}, \frac{1}{D(D+1)}=:{\mathcal{I}}_{\left(c\right)}, \frac{1}{D(D+1)}=:{\mathcal{I}}_{\left(d\right)}$, and $\frac{2}{D(D+1)}=:{\mathcal{I}}_{\left(a\right)}$.

Figure 5

$I_{A\to B}$ (dashed line) and $I_{B\to A}$ (continuous line), both dimensionless, of $U_{\sup }^1$ depending on the control qubit $|{\chi }^c\rangle =cos(\theta /2)|0\rangle +sin(\theta /2)|1\rangle$, parametrized by $\theta$.