Impact of quadrature measurement on quantum coherence

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.

💡 Research Summary

This paper investigates how quantum coherence defined in the continuous quadrature basis evolves when a quadrature measurement is performed on one output of a loss‑less beam splitter. The authors adopt the ℓ₁‑norm of coherence, extended to continuous variables, as their primary quantifier. Starting from a generic input state ρ in mode a and an auxiliary state ρ₀ in mode a₀, they describe the beam‑splitter transformation with real transmission t and reflection r (t²+r²=1), yielding the output quadrature operators X′=tX−rX₀ and X′₀=tX₀+rX. After the beam splitter, a homodyne‑type measurement of X′₀ is performed, producing an outcome x′₀; the conditional state of the other output mode is denoted ρ′(x′₀).

A key technical result is Eq. (11), which shows that the quadrature matrix elements of the unnormalized conditional state factorize into a product of matrix elements of the input and auxiliary states. This factorization demonstrates that the overall transformation is “incoherent” in the sense that a completely incoherent input (diagonal in the quadrature basis) remains incoherent after the process. Nevertheless, for partially coherent inputs the beam splitter together with the measurement can redistribute or even amplify coherence.

For Gaussian inputs the authors obtain closed‑form expressions. A centered Gaussian state with variance σ² and off‑diagonal width μ has ℓ₁‑coherence C=2√(2π)σμ/(2σ²+μ²). The auxiliary Gaussian state has analogous parameters σ₀, μ₀. Remarkably, the conditional output coherence C′(x′₀) is independent of the measurement result and obeys the simple relation

 1/C′² = t²/C² + r²/C₀²,

where C₀ is the coherence of the auxiliary state. Consequently, the average coherence over all measurement outcomes coincides exactly with the coherence of the reduced output state, confirming the general result that for the ℓ₁‑norm the average coherence does not depend on whether a measurement is performed. The output coherence lies between the larger and the smaller of the two input coherences; if the auxiliary state is much more coherent than the signal, the output coherence is amplified by a factor 1/t, whereas an incoherent auxiliary mode drags the output coherence down to C₀/r. Thus, by engineering the auxiliary mode one can control the coherence of the signal output.

The situation changes for number (Fock) states |n⟩. Their quadrature wavefunctions are Hermite‑Gaussian, and the ℓ₁‑coherence decreases with photon number. In this case the conditional output coherence C′(x′₀) does depend on the measurement outcome, and the average coherence is strictly smaller than the input coherence. Moreover, when the authors replace the ℓ₁‑norm with the relative‑entropy measure of coherence, the equality between average coherence and the coherence of the reduced state breaks down even for Gaussian inputs. This demonstrates that the “average‑coherence‑preserving” property is specific to the ℓ₁‑norm and does not hold for other legitimate coherence monotones.

Overall, the work provides a clear analytical framework for studying continuous‑variable coherence under basic linear optics and measurement. It highlights a distinctive feature of continuous‑variable coherence: incoherent transformations can nevertheless generate or redistribute coherence for partially coherent inputs, a phenomenon absent in discrete‑variable resource theories. The results suggest practical strategies for coherence engineering in quantum optics—by selecting an appropriate auxiliary state one can either protect, amplify, or suppress quadrature coherence of a signal. The authors conclude by proposing extensions to non‑linear optics, multi‑port interferometers, and realistic scenarios with loss and noise, pointing toward a broader understanding of coherence as a resource in continuous‑variable quantum technologies.