Counterfactual quantum measurements

Abstract

Counterfactual reasoning plays a crucial role in exploring hypothetical scenarios, by comparing some consequent under conditions identical except as results from a differing antecedent. David Lewis' well-known analysis evaluates counterfactuals using a hierarchy of desiderata. These were, however, built upon a deterministic classical framework, and whether it could be generalized to indeterministic quantum theory has been an open question. In this Letter, we propose a formalism for quantum counterfactuals in which antecedents are measurement settings. Unlike other approaches, it non-trivially answers questions like: "Given that my photon-detector, observing an atom's fluorescence, clicked at a certain time, what would I have seen using a field-quadrature detector instead?"

Figures (6)

• Figure 1: A counterfactual in the CHSH scenario. Alice and Bob make measurements on a shared singlet state. (a) Alice knows her outcome $A$ for the setting $X$, and also Bob's setting (blue, evidence). (b) Alice wonders about her outcome $A'$ (golden, consequent) if she had used a different setting $X'$ (purple, antecedent). In both cases Bob's outcomes $B$ (brown) is unknown to Alice but fixed. The black lines show a future light cone originating at an event in space-time.
• Figure 2: Space-time diagram for the fluorescence counterfactual. Alice's actual record of photo-detections $\overleftrightarrow{N}$ (blue, evidence) is space-like separated from Bob's record of photo-detections $\overleftrightarrow{M}$ (brown, fixture). Alice's counterfactual measurement is homodyne, with record $\overleftrightarrow{Y}$ (golden, consequent). The photons (cyan wiggles, not classical events but included to aid intuition) are emitted by the atom asymmetrically in accordance with the chosen parameters for the simulation: $\eta_A=0.2$, $\eta_B=0.8$, and $\Omega=2\gamma^{-1}$, for an interval $[0,10\gamma^{-1})$.
• Figure 3: Dynamics of various estimates by the Alice of Fig. \ref{['fig:SpaceTimediagram']}. The expectation of $\hat{\sigma}_{y}$ is shown for the unconditioned atomic state (gray dashed line) and Alice's filtered state conditioned on her actual record (a photo-detection at $t_A=6.25\gamma^{-1}$) (blue dotted line). The gray dashed line is also the unconditioned expectation ${\mathbb E}[Y_t]$ of Alice's Y-homodyne current (\ref{['homodynecurrent']}). Finally, the green solid curve curve is the suspectation ${\mathbb S}[Y_t]$ for (\ref{['homodynecurrent']}), conditioned on her actual record $\overleftrightarrow{n}$, as in Eq. (\ref{['defsuspect']}).
• Figure 4: States on the Bloch ball, with colours following from Fig.\ref{['fig:sidebyside']}. (a) Actual world: Alice's outcome $\uparrow$ and Bob's possible outcomes (probabilities $\approx0.15$ and $\approx0.85$) using his basis $Y=$origin=c]135$\updownarrow$ in the actual world.(b) Counterfactual world: Fixing Bob's outcomes, Alice's possible outcomes in the basis $X'=\leftrightarrow$ (probabilities $=0.75$ and $=0.25$) in the counterfactual world. Note that the arrows have thickness corresponding to their actual or counterfactual probabilities, with the thickness of $\uparrow$ corresponding to probability one.
• Figure 5: Tendency of $\langle\hat{\sigma}_y\rangle$ to be positive. The blue curve, with light blue filling, is the normalized probability distribution of the conditioned state ${\rho}_t^{\overleftarrow{M},\overleftarrow{Y}}$ at time $t=t_A$. This state lives on the $y$--$z$ plane of the Bloch sphere and is thus parametrized by the single variable $\theta$. It is conditioned on both Alice's Y-homodyne measurement ($\overleftarrow{Y}$) and Bob's photo-counts ($\overleftarrow{M}$). We take $\overleftarrow{M}$ to be a fixed $\overleftarrow{m}$, with a jump at time $t=4.71\gamma^{-1} = t_A - 1.54\gamma^{-1}$, and no jumps in the interval $(4.71\gamma^{-1},t_A)$. This is the most characteristic record for Bob ($\overleftarrow{m}^\chi$) since the rate of his jumps, conditioned on $\overleftrightarrow{N}=\delta_{t,t_A}$, has a maximum $\approx 0.6\gamma$ at $t=4.71\gamma^{-1}$; see the SM SM. The total probability (from integrating the curve) in the region $\theta > 0$ is $\approx 90 \%$. This corresponds to the region where $\langle\hat{\sigma}_y\rangle > 0$, as seen from the magenta dot-dashed curve which shows the sinusoidal curve which is $\langle\hat{\sigma}_y\rangle$ as a function of $\theta$.