Evaluation of counterfactuality in counterfactual communication protocols

We provide an in-depth investigation of parameter estimation in nested Mach-Zehnder interferometers (NMZIs) using two information measures: the Fisher information and the Shannon mutual information. Protocols for counterfactual communication have, so far, been based on two different definitions of counterfactuality. In particular, some schemes have been based on NMZI devices, and have recently been subject to criticism. We provide a methodology for evaluating the counterfactuality of these protocols, based on an information-theoretical framework. More specifically, we make the assumption that any realistic quantum channel in MZI structures will have some weak uncontrolled interaction. We then use the Fisher information of this interaction to measure counterfactual violations. The measure is used to evaluate the suggested counterfactual communication protocol of H. Salih et al. [Phys. Rev. Lett. 110, 170502 (2013)]. The protocol of D. R. M. Arvidsson-Shukur and C. H. W. Barnes [Phys. Rev. A 94, 062303 (2016)], based on a different definition, is evaluated with a probability measure. Our results show that the definition of Arvidsson-Shukur and Barnes is satisfied by their scheme, while that of Salih et al. is only satisfied by perfect quantum channels. For realistic devices the latter protocol does not achieve its objective.

Figure 1

Sketch of an optical circuit of the form of Eq. (4), which is described in the text.

Figure 2

The chained nested Mach-Zehnder interferometer suggested for CFC in Ref. [23]. Alice inputs a photon in the upper left path and Bob has the choice of introducing detectors in his part of the device. His choice governs the statistics of the final output detections at $D_1$ and $D_2$.

Figure 3

The chained Mach-Zehnder interferometer as used in the CFC scheme of Ref. [34]. As in Fig. 2, Alice inputs a photon in the upper left path and Bob governs the statistics of the final output detections at $D_1$ and $D_2$ by inserting or not inserting detectors, respectively.

Figure 4

A single-photon state is incident on a weak polarization rotator. The photon is then measured in its number state and polarization state.

Figure 5

A single-photon state is sent as the input to one of the ports of a NMZI. There are six possible detection outcomes (three spatial outcomes, each having a polarization degree of freedom). The green barred lines and blue arrowed lines represent mirrors and nonpolarizing beam splitters, respectively.

Figure 6

A polarization rotation is added to one, but only one, of the positions 1–5 in the nested Mach-Zehnder interferometer from Fig. 5.

Figure 7

The mutual information between the polarization rotation, ${\vartheta }_w$, and the measurement outcomes, $\left\{M_i\right\}$, as a function of beam-splitter transmission, $t_1$, as described in the text. The solid black line shows the true curve of Eq. (66), the thick gray line shows the Padé approximation (virtually indistinguishable from the true curve), and the red dashed line shows the second-order Taylor expansion.

Figure 8

Probability of detection at $D_1$ (a) and $D_2$ (b) if Bob unblocks and blocks his path in Fig. 2, respectively. The success probabilities are expressed as functions of the beam-splitter numbers $N$ and $M$.

Figure 9

The nested Mach-Zehnder interferometer as used in the communication scheme presented in the text.

Figure 10

This figure shows the quantum evolution [time steps (a) to (f)] of the probability density distribution of a spin-$\frac{1}{2}$ particle in a nested Mach-Zehnder interferometer with and without a weak spin-rotation interaction (exaggerated for visibility) in Bob's laboratory. The dotted red and solid blue curves indicate spin-up and spin-down components of the wave function, respectively. The dashed green curves show the potentials. Beam splitters are denoted with vertical yellow lines. The spatial components are indicated with the vertical dashed gray lines.

Figure 11

The logical-0 process for the chained nested Mach-Zehnder interferometer suggested for CFC in Ref. [23]. The weak polarization interactions mimic realistic systematic errors in the quantum channels of Bob's laboratory.

Figure 12

The spatially conditioned type-I counterfactual violation strength as a function of the beam-splitter numbers $N$ and $M$ for the scenario of Bob not introducing his detectors in Fig. 2 or 11.

Figure 13

The logical-0 process of the chained Mach-Zehnder interferometer CFC scheme of Ref. [34]. Again, the weak polarization interactions mimic realistic systematic errors in the quantum channels of Bob's laboratory.

Figure 14

The type-II logical-0 (see Ref. [34]) counterfactual violation strength, $P_{\mathit{A}}$ (dashed lines read on the left $y$ axis), and the spatially conditioned Fisher information, $F_{\mathit{A}}$ (solid line read on the right $y$ axis), as functions of the beam-splitter number, $N$.