Impact of quadrature measurement on quantum coherence

TL;DR

The paper investigates how quadrature coherence, defined via a continuous-variable extension of the $l_1$-norm, behaves when a quadrature measurement is performed on one output of a lossless beam splitter that mixes a signal state ρ with an auxiliary state ρ0, described by the relation $X' = t X - r X_0$ with $t^2 + r^2 = 1$. For Gaussian inputs the output coherence is independent of the measurement outcome and the averaged coherence equals the reduced-state coherence, with a compact relation $1/\mathcal{C}'^2 = t^2/\mathcal{C}^2 + r^2/\mathcal{C}_0^2$, yielding bounds between the input and auxiliary coherences. In contrast, for number-state inputs the outcome-dependent coherence can vary with $x'_0$, and the averaged coherence is reduced relative to the input; using relative entropy coherence, the equality between average and reduced-state coherences fails and the average can exceed the reduced-state value, highlighting a genuine quantum backaction. Altogether, the work deepens understanding of coherence dynamics under measurement in continuous-variable optics and has implications for coherence-resource theories and quantum metrology that exploit beam-splitter–measurement interactions.

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.

Figures (9)

• Figure 1: Scheme for the system transformation $\rho \rightarrow \rho^\prime$ induced by a measurement of quadrature $X^\prime_0$ with outcome $x_0^\prime$. The measurement is carried out with the help of a beam splitter BS that mixes the state $\rho$ of the signal mode with the field state $\rho_0$ in the auxiliary mode impinging at the other input port.
• Figure 2: Transformation of coherence expressed in the form $\mathcal{C}^\prime-\mathcal{C}$, where $\mathcal{C}$ and $\mathcal{C}^\prime$ are the input and output coherence, respectively. This is plotted as a function of the transmission coefficient $t$ for the vacuum state in the auxiliary mode. The state in the signal mode is pure and squeezed, with quadrature uncertainties $\Delta X =1/4$ (solid line) and $\Delta X =1$ (dashed line).
• Figure 3: Plot of the quadrature coherence $\mathcal{C}$ for number states as a function of the number of photons $n$.
• Figure 4: Plot of the output coherence $\mathcal{C}^\prime (x^\prime_0)$ relative to the input coherence $\mathcal{C}$ as a function of the outcome $x^\prime_0$ of the quadrature measurement. This is plotted for input number states with different photon numbers $n$, that is for $n=1$ (solid line), $n=2$ (dashed line), and $n=3$ (dotted line).
• Figure 5: Plot of the probability $p(x^\prime_0)$ for the outcome $x^\prime_0$ of the quadrature measurement as a function of the outcome $x^\prime_0$ of the quadrature measurement for input number states with different photon numbers $n$, that is $n=1$ (solid line), $n=2$ (dashed line), and $n=3$ (dotted line).