Maximum information gain of approximate quantum position measurement

Abstract

We perform a quantum information analysis for multi-mode Gaussian approximate position measurements, underlying noisy homodyning in quantum optics. The “Gaussian maximizer” property is established for the entropy reduction of these measurements which provides explicit formulas for computations of their maximum information gain or entanglement-assisted capacity. The case of one mode is discussed in detail.

Appendix

The kernel of a Gaussian density operator $\rho _{m,\alpha }$ with the mean $m_{q},m_{p}$ and the covariance matrix (7) in the Schrödinger representation has the form

$$\begin{aligned} \langle \xi |\rho _{\alpha }|\xi ^{\prime }\rangle =\frac{1}{\sqrt{\left( 2\pi \right) ^{s}\det \alpha _{qq}}}\exp \left[ \mu \left( m;\xi ,\xi ^{\prime }\right) -\frac{1}{2}\theta \left( \xi ,\xi ^{\prime }\right) \right] , \end{aligned}$$

(39)

where

$$\begin{aligned} \mu \left( m;\xi ,\xi ^{\prime }\right)= & {} \frac{1}{2}m_{q}^{t}\alpha _{qq}^{-1}\left( \xi ^{\prime }+\xi \right) -i\left( m_{p}-\alpha _{pq}\alpha _{qq}^{-1}m_{q}\right) ^{t}\left( \xi ^{\prime }-\xi \right) \nonumber \\&- \frac{1}{2}m_{q}^{t}\alpha _{qq}^{-1}m_{q}, \end{aligned}$$

(40)

$$\begin{aligned} \theta \left( \xi ,\xi ^{\prime }\right)= & {} \frac{1}{4}\left( \xi ^{\prime }+\xi \right) ^{t}\alpha _{qq}^{-1}\left( \xi ^{\prime }+\xi \right) +\left( \xi ^{\prime }-\xi \right) ^{t}\left( \alpha _{pp}-\alpha _{pq}\alpha _{qq}^{-1}\alpha _{qp}\right) \left( \xi ^{\prime }-\xi \right) \nonumber \\&+i\left( \xi ^{\prime }+\xi \right) ^{t}\alpha _{qq}^{-1}\alpha _{qp}\left( \xi ^{\prime }-\xi \right) . \end{aligned}$$

(41)

Proof

The quantum characteristic function of $\rho _{m,\alpha }$ is

$$\begin{aligned} \varphi (x,y)=\mathrm {Tr}\rho _{m,\alpha }W(x,y)=\exp \left[ im(x,y)-\frac{1}{2} \alpha (x,y)\right] , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} m(x,y)&= m_q x + m_p y ,\\ \alpha (x,y)&= x^t \alpha _{qq} x + x^t \alpha _{qp} y + y^t \alpha _{pq} x + y^t \alpha _{pp} y \end{aligned} \end{aligned}$$

and $W(x,y)=e^{ix^{t}y/2}e^{ix^{t}q}e^{iy^{t}p}$ is the Weyl operator. The kernel of the Weyl operator is

$$\begin{aligned} \langle \xi \vert W(x,y)\vert \xi '\rangle =\exp \left[ {i}\left( \frac{\xi +\xi ^{\prime }}{2} \right) ^{t}x\right] \delta (y-(\xi ^{\prime }-\xi ))=\exp \left( iu^{t}x\right) \delta (y-v). \end{aligned}$$

Here, we introduced the variables $u=\frac{\xi +\xi ^{\prime }}{2}$, $v=\xi ^{\prime }-\xi$. Using the inversion formula for the quantum Fourier transform, we readily compute the kernel of the Gaussian state:

$$\begin{aligned} \langle \xi \vert \rho _{m,\alpha }\vert \xi '\rangle= & {} \int \frac{d^{s}xd^{s}y}{(2\pi )^{s}} \varphi (x,y)\langle \xi \vert W(-x,-y)\vert \xi '\rangle \\= & {} \int \frac{d^{s}x}{(2\pi )^{s}}\exp \left[ -iu^{t}x+im(x,-v)-\frac{1}{2} \alpha (x,-v)\right] , \end{aligned}$$

the expression under the exponent expands as

$$\begin{aligned} -im_{p}^{t}v-\frac{1}{2}v^{t}\alpha _{pp}v-i(u^{t}-m_{q}^{t}+iv^{t}\alpha _{pq})x-\frac{1}{2}x^{t}\alpha _{qq}x. \end{aligned}$$

Taking the s-dimensional Gaussian integral yields

$$\begin{aligned} \langle \xi \vert \rho _{\alpha }\vert \xi '\rangle= & {} \frac{e^{-im_{p}^{t}v-\frac{1}{2} v^{t}\alpha _{pp}v}}{\sqrt{(2\pi )^{s}\det {\alpha _{qq}}}}\exp \left[ - \frac{1}{2}(u-m_{q}+i\alpha _{qp}v)^{t}\alpha _{qq}^{-1}(u-m_{q}+i\alpha _{qp}v)\right] \\= & {} \frac{1}{\sqrt{(2\pi )^{s}\det {\alpha _{qq}}}}\exp \left[ \mu (m;\xi ,\xi ^{\prime })-\frac{1}{2}\theta (\xi ,\xi ^{\prime })\right] , \end{aligned}$$

where $\mu (m;\xi ,\xi ^{\prime })$ and $\theta (\xi ,\xi ^{\prime })$ are given by (40) and (41).