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arXiv:2507.13735 · Physical Review A

Impact of quadrature measurement on quantum coherence

Lucía Álvarez, Alfredo Luis

Published Jan 13, 2026• 0 citations
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Abstract

We examine the behavior of quadrature coherence under the measurement of the same field quadrature. This is carried out with the help of a beam splitter, which implies the contribution of an auxiliary field state impinging at the other input port. To this end we consider the linear input-output transformation of a lossless beam splitter to relate input and output coherences, measured in terms of the $l_1$-norm. After obtaining a general input-output relation between coherences we apply the result to Gaussian and number states. For Gaussian states we obtain that coherence does not depend on the measurement outcome, and that the average coherence always equals the coherence of the reduced state, showing no average effect on coherence of the measurement. On the other hand, for number states the output coherence depends on the measurement, decreasing the relative coherence with increasing photon number. Finally, we consider relative-entropy as a measure of coherence to show that for number states and coherence measures other than the $l_1$-norm the average coherence no longer equals the coherence of the output reduced state.

DOI
10.1103/8j2m-fsy5

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We examine the behavior of quadrature coherence under the measurement of the same field quadrature. This is carried out with the help of a beam splitter, which implies the contribution of an auxiliary field state impinging at the other input port. To this end we consider the linear input-output transformation of a lossless beam splitter to relate input and output coherences, measured in terms of the l1l_1l1​-norm. After obtaining a general input-output relation between coherences we apply the result to Gaussian and number states. For Gaussian states we obtain that coherence does not depend on the measurement outcome, and that the average coherence always equals the coherence of the reduced state, showing no average effect on coherence of the measurement. On the other hand, for number states the output coherence depends on the measurement, decreasing the relative coherence with increasing photon number. Finally, we consider relative-entropy as a measure of coherence to show that for number states and coherence measures other than the l1l_1l1​-norm the average coherence no longer equals the coherence of the output reduced state.
A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr\"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.
The second-order coherence of optical fields at output ports of a beam splitter is investigated. The properties of the second-order coherence are not only strongly dependent on the input states but also quite involved with the nature of the beam splitter. When parameters of both the input states and the beam splitter are chosen appropriately, the fields at the output ports will exhibit quantum behaviour. Between the two output modes, there stands mode–mode correlation which can even be anticorrelation. Some nonclassical properties, such as the photon antibunching effect of the input fields, may be destroyed by the beam splitter.
Abstract The aim of this work is to answer the question of how much quantum coherence a beam splitter is able to produce. To this end, we consider as the variables under study both the amount of coherence of the input states as well as the beam splitter characteristics. We conclude that there is an optimal combination of these factors making the gain of coherence maximum. In addition, the two-mode squeezed vacuum arises as the studied state most capable of gaining coherence when passing through a beam splitter. These results are qualitatively equivalent for the l1-norm of coherence and the relative entropy of coherence.
We investigate the photon statistics at the outputs of a four-port beam splitter, when a strong coherent field is incident on one input port and various other fields enter at the second inner port. Some non-classical interference effects are encountered. In particular, there is a large difference between the cases when a one photon field enters at input port 2 and when a field in a coherent state with an average of one photon enters at port 2. In the former case the probability P(m) of the difference m between the detected photons at output ports 3 and 4 has a minimum at m = 0 whenever the quantum efficiency exceeds one third. As the quantum efficiency falls from one to zero, the form of the probability distribution P(m/1) changes drastically.
We study multiphoton quantum-interference effects at a beam splitter and its connection to the prospect of attaining interferometric phase-shift measurements with noise levels below the standard quantum limit. Specifically, we consider the mixing of the most classical states of light coherent states with the most nonclassical states of light number states at a 50 : 50 beam splitter. Multiphoton quantum-interference effects from mixing photon-number states of small photon numbers with coherent states of arbitrary amplitudes are dramatic even at the level of a single photon. For input vacuum and coherent states, the joint photon-number distribution after the beam splitter is unimodal, a product of Poisson distributions for each of the output modes but with the input of a single photon, the original distribution is symmetrically bifurcated into a bimodal distribution. With a two-photon-number state mixed with a coherent state, a trimodal distribution is obtained, etc. The obtained distributions are shown to be structured so as to be conducive for approaching Heisenberg-limited sensitivities in photon-number parity-based interferometry. We show that mixing a coherent state with even a single photon results in a significant reduction in noise over that of the shot-noise limit. Finally, based on the results of mixing coherent light with single photons, we consider the mixing coherent light with the squeezed vacuum and the squeezed one-photon states and find the latter yields higher sensitivity in phase-shift measurements for the same squeeze parameter owing to the absence of the vacuum state.