arXiv:0807.4108 · Physical Review A

Phase variance of squeezed vacuum states

E. Bagán, Alex Monràs, R. Muñoz-Tapia

Published Oct 31, 2008• 9 citations

Abstract

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number $n$, as dictated by the Heisenberg limit, i.e., as ${n}^{\ensuremath{-}2}$, only for $N>4$. For $N\ensuremath{\leqslant}4$ this fundamental scaling breaks down and it becomes ${n}^{\ensuremath{-}N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling ${n}^{\ensuremath{-}2}$ is recovered for large samples.

DOI: 10.1103/physreva.78.043829

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arXiv:0807.4108 · Physical Review A

Phase variance of squeezed vacuum states

E. Bagán, Alex Monràs, R. Muñoz-Tapia

Published Oct 31, 2008• 9 citations

arXiv HTML PDF

Abstract

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number

N

of uncorrelated copies. We find that it scales with the mean photon number

n

, as dictated by the Heisenberg limit, i.e., as

{n}^{\ensuremath{-}2}

, only for

N>4

. For

N\ensuremath{\leqslant}4

this fundamental scaling breaks down and it becomes

{n}^{\ensuremath{-}N∕2}

. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling

{n}^{\ensuremath{-}2}

is recovered for large samples.

DOI: 10.1103/physreva.78.043829

Related papers

Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum

Takafumi Ono, Holger Friedrich Hofmann

Two-path interferometry with coherent states and a squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information $F$ of the state. In the limit of high squeezing, the ratio $(F--N)/{N}^{2}$ of Fisher information above shot noise to the square of the average photon number $N$ depends only on the average number of photons lost, ${n}_{\mathrm{loss}}$ , and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by $(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})$ . The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.Two-path interferometry with coherent states and a squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information $F$ of the state. In the limit of high squeezing, the ratio $(F--N)/{N}^{2}$ of Fisher information above shot noise to the square of the average photon number $N$ depends only on the average number of photons lost, ${n}_{\mathrm{loss}}$, and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by $(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})$. The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.

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We consider a two-mode bosonic state with fixed photon number $n \in \mathbb {N}$ , whose upper and lower modes pick up a phase $\phi$ and $-\phi$ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating $\phi$ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider $\phi$ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider $n=1$ and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.We consider a two-mode bosonic state with fixed photon number $$n \in \mathbb {N}$$ , whose upper and lower modes pick up a phase $$\phi $$ and $$-\phi $$ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating $$\phi $$ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider $$\phi $$ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider $$n=1$$ and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.

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Mach-Zehnder Interferometry at the Heisenberg Limit with Coherent and Squeezed-Vacuum Light

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We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where ${N}_{T}$ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter $r$ and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where ${N}_{T}$ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter $r$ and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.

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Optimal phase measurements with pure Gaussian states

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We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisenberg limit for large samples. This proves that this limit can be attained with local projective von Neuman measurements.We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisenberg limit for large samples. This proves that this limit can be attained with local projective von Neuman measurements.

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A recently proposed phase-estimation protocol that is based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that Cramer-Rao bound sensitivity can be obtained [P. M. Anisimov, et al., Phys. Rev. Lett. {\bf104}, 103602 (2010)]. This sensitivity, however, is expected in the case of an infinite number of parity measurements made on an infinite number of photons. Here we consider the case of a finite number of parity measurements and a finite number of photons, implemented with photon-number-resolving detectors. We use Bayesian analysis to characterize the sensitivity of the phase estimation in this scheme. We have found that our phase estimation becomes biased near 0 or $\pi/2$ phase values. Yet there is an in-between region where the bias becomes negligible. In this region, our phase estimation scheme saturates the Cramer-Rao bound and beats the shot-noise limit.A recently proposed phase-estimation protocol that is based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that Cramer-Rao bound sensitivity can be obtained [P. M. Anisimov, et al., Phys. Rev. Lett. {\bf104}, 103602 (2010)]. This sensitivity, however, is expected in the case of an infinite number of parity measurements made on an infinite number of photons. Here we consider the case of a finite number of parity measurements and a finite number of photons, implemented with photon-number-resolving detectors. We use Bayesian analysis to characterize the sensitivity of the phase estimation in this scheme. We have found that our phase estimation becomes biased near 0 or $\pi/2$ phase values. Yet there is an in-between region where the bias becomes negligible. In this region, our phase estimation scheme saturates the Cramer-Rao bound and beats the shot-noise limit.

arXiv:1110.1587Civil War Book ReviewOct 7, 20111 citation

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Daniela Denot, Thorsten C. Bschorr, Matthias Freyberger

We discuss an adaptive interferometric approach to quantum estimation of phase shifts. In a first step we find optimized two-mode input states for a generalized Mach-Zehnder interferometer. These two states show a small phase variance at the so-called Heisenberg limit. It turns out that they can be generated by entangling a coherent state and a squeezed vacuum at a beam splitter. Moreover, the interferometric setup allows us to describe an adaptive estimation scheme for arbitrary phase shifts if we assume a fixed mean photon number as a quantum resource.We discuss an adaptive interferometric approach to quantum estimation of phase shifts. In a first step we find optimized two-mode input states for a generalized Mach-Zehnder interferometer. These two states show a small phase variance at the so-called Heisenberg limit. It turns out that they can be generated by entangling a coherent state and a squeezed vacuum at a beam splitter. Moreover, the interferometric setup allows us to describe an adaptive estimation scheme for arbitrary phase shifts if we assume a fixed mean photon number as a quantum resource.

Physical Review AJan 30, 20068 citations

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number

N

of uncorrelated copies. We find that it scales with the mean photon number

n

, as dictated by the Heisenberg limit, i.e., as

{n}^{\ensuremath{-}2}

, only for

N>4

. For

N\ensuremath{\leqslant}4

this fundamental scaling breaks down and it becomes

{n}^{\ensuremath{-}N∕2}

{n}^{\ensuremath{-}2}

is recovered for large samples.

F

of the state. In the limit of high squeezing, the ratio

(F--N)/{N}^{2}

of Fisher information above shot noise to the square of the average photon number

N

depends only on the average number of photons lost,

{n}_{\mathrm{loss}}

, and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by

(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})

. The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.

We consider a two-mode bosonic state with fixed photon number

n \in \mathbb {N}

, whose upper and lower modes pick up a phase

\phi

and

-\phi

, respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating

\phi

is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider

\phi

to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider

n=1

and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.

{N}_{T}

is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter

r

and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.

\pi/2

phase values. Yet there is an in-between region where the bias becomes negligible. In this region, our phase estimation scheme saturates the Cramer-Rao bound and beats the shot-noise limit.