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arXiv:2607.25699

Counterfactual Quantum Sensing: What Interaction-Free Measurement Can and Cannot Buy

Christoph F. Wildfeuer

quant-phSubmitted Jul 28, 2026• Announced Jul 29, 2026• 0 citations

Abstract

Interaction-free measurement infers the presence of an absorbing object from a photon that, in the counterfactual sense, never interacted with it, and is widely described as a route to minimally invasive sensing. We ask what it actually buys, in estimation-theoretic terms. Written as a channel-estimation problem, the Elitzur-Vaidman interferometer carries exactly half the Fisher information about the object's transmissivity that direct transmission probing does, and the two schemes deliver identical Fisher information per absorbed photon. For measuring how transparent something is, the interferometer buys nothing. The advantage lies in discrimination, and we show that what it requires is not that the object be opaque but that the competing hypothesis be the object's absence. Against empty space the Chernoff information per absorbed photon grows almost linearly, as the number of Zeno cycles times its logarithm, even for a weakly absorbing object; between two partial transparencies it does not grow at all. Parasitic loss in the cycle caps the advantage. The number of conclusive interrogations per absorbed photon reaches a maximum inversely proportional to the loss per cycle, at an optimal cycle number that is likewise inversely proportional to it, which identifies the loss per cycle as the figure of merit governing how far interaction-free sensing can be pushed. Finally, the negative result is not special to the interferometer. For a single photon meeting a memoryless, non-dispersive object any number of times through arbitrary fixed optics, the accessible quantum Fisher information never exceeds that of the same incident flux spent on independent single-pass probes, and is generically far below it. This recovers the bound of Massar, Mitchison and Pironio for this class, by a short argument that also identifies when it is tight.

Comments: 10 pages, 5 figures

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Interaction-free measurement infers the presence of an absorbing object from a photon that, in the counterfactual sense, never interacted with it, and is widely described as a route to minimally invasive sensing. We ask what it actually buys, in estimation-theoretic terms. Written as a channel-estimation problem, the Elitzur-Vaidman interferometer carries exactly half the Fisher information about the object's transmissivity that direct transmission probing does, and the two schemes deliver identical Fisher information per absorbed photon. For measuring how transparent something is, the interferometer buys nothing. The advantage lies in discrimination, and we show that what it requires is not that the object be opaque but that the competing hypothesis be the object's absence. Against empty space the Chernoff information per absorbed photon grows almost linearly, as the number of Zeno cycles times its logarithm, even for a weakly absorbing object; between two partial transparencies it does not grow at all. Parasitic loss in the cycle caps the advantage. The number of conclusive interrogations per absorbed photon reaches a maximum inversely proportional to the loss per cycle, at an optimal cycle number that is likewise inversely proportional to it, which identifies the loss per cycle as the figure of merit governing how far interaction-free sensing can be pushed. Finally, the negative result is not special to the interferometer. For a single photon meeting a memoryless, non-dispersive object any number of times through arbitrary fixed optics, the accessible quantum Fisher information never exceeds that of the same incident flux spent on independent single-pass probes, and is generically far below it. This recovers the bound of Massar, Mitchison and Pironio for this class, by a short argument that also identifies when it is tight.

Interaction-free measurement is a quantum process where, in the ideal situation, an object can be detected as if no interaction took place with the probing photon. Here we show that the problem of interaction-free measurement can be regarded as a problem of quantum-channel discrimination. In particular, we look for the optimal photonic states that can minimize the detection error and the photon loss in detecting the presence or absence of the object, which is taken to be semitransparent, and the number of the interrogation cycle is assumed to be finite. Furthermore, we also investigated the possibility of minimizing the detection error through the use of entangled photons, which is essentially a setting of quantum illumination. However, our results indicate that entanglement does not exhibit a clear advantage; the same performance can be achieved with unentangled photonic states.

"Interaction-free measurements" (IFM) originate from the latest quantum interferometric technologies. The latest research of quantum optics demonstrates, by using the complementary wavelike and particlelike natures of photons, it is possible to make interaction-free measurements by which the presence of an object can be determined with no photons being absorbed. The paper introduces the concept of "Interaction-free Measurement" (IFM), the original Elitzur-Vaidman scheme, the "High Efficiency Interaction-free Measurement", the application of quantum Zeno effect and the improved scheme proposed by Kwiat et al. The EV scheme is implemented in a Mach-Zehnder interferometer. Theoretically this paper also draws a conclusion in IFM feasibility by analyzing the wave functions of photon at various locations in the interferometer.

Multi-pass schemes send a photon through a sample several times to learn more about it. When the sample rather than the light is scarce, the natural figure of merit is the information gained per photon the object absorbs. We show that one constant fixes the loss-limited optimum of every fixed scheme in which a single photon passes repeatedly through the sample and is detected once at the end. Three properties suffice: information that grows as the square of the pass number, a fixed survival probability per pass, and no dose from a photon already lost. They force a single trade-off function

h(x)=x^2/(e^x-1)

, where

x

is the number of passes times the loss per pass. Its maximum, 0.648, sits at

x_{\mathrm{opt}}=1.594

. The loss-limited ceiling of interaction-free interrogation and the multi-pass phase optimum of Yu et al. are two instances. Phase sensing of a weakly absorbing object is a third, optimal at

m_{\mathrm{opt}}=x_{\mathrm{opt}}/(ε+α)

passes; the absorption

α

and the parasitic loss

ε

enter the optimum only through their sum, but the damage counts only

α

. N00N states and their loss-robust generalisations do worse per absorbed photon, and optimising over photon number and pass number together returns a single recycled photon. Two further problems are priced in the same measure. Absorption estimation gains nothing, for any scheme. Detecting a faint companion below the Rayleigh limit costs the companion a dose that does not depend on its faintness, while under direct imaging the dose grows without bound. For a fragile sample the gentlest measurement is also the simplest one: a single photon, recycled.

The phenomenon of quantum interrogation allows one to optically detect the presence of an absorbing object, without the measuring light interacting with it. In an application of the quantum Zeno effect, the object inhibits the otherwise coherent evolution of the light, such that the probability that an interrogating photon is absorbed can in principle be arbitrarily small. We have implemented this technique, achieving efficiencies of up to

73%

, and consequently exceeding the

50%

theoretical maximum of the original ``interaction-free'' measurement proposal. We have also predicted and experimentally verified a previously unsuspected dependence on loss.

Summary form only given.The purpose of an interaction-free measurement is to detect the presence of an object without letting any energy quanta be absorbed by the object. In this work we present interaction-free measurement results using a Fabry-Perot interferometer as a measurement apparatus. If the cavity of the interferometer is empty, an incoming photon will be transmitted through the interferometer. If an absorbent is placed between the mirrors, the photon will be reflected by the first mirror. By detecting which way the photon takes, we will gain information about the presence of the absorbent without losses/interaction. For a perfect measurement, the figure of merit will be equal to the mirror reflectivity R. Thus, in theory, one can approach the perfect interaction-free measurement simply by increasing R.

An Elitzur–Vaidman bomb tester demonstrates interaction-free sensing: detecting an object without absorbing a photon. The original protocol, however, has a success probability capped at 25%. We present a scalable quantum circuit that harnesses the quantum Zeno effect to overcome this limit. A single probe qubit undergoes repeated cycles of a small rotation followed by a controlled-NOT (CNOT) with an ancilla qubit representing the obstacle. The circuit requires only two qubits, regardless of the number of cycles N and has depth linear in N, compatible with near-term processors that support mid-circuit measurement and conditional reset. We simulate the circuit under ideal conditions and under a depolarizing noise model with gate errors typical of superconducting qubits (CNOT error 2% and single-qubit gate error rate 0.1%). Under noise, the success probability peaks around N=14 (reaching 72.0%), outperforming the original 25% limit by nearly a factor of three, with a well-defined optimal region. The protocol requires approximately 4.4 times fewer photons than the original Elitzur–Vaidman scheme to achieve 99.9% confidence. Its simple repetitive structure and well-defined optimal regime make it a practical benchmark for quantifying gate errors and decoherence on noisy intermediate-scale quantum hardware for quantum sensing applications.