Quantum deconvolution

Abstract

We propose a method for stably removing noise from measurements of a quantum many-body system. The question is cast to a linear inverse problem by using a quantum Fischer information metric as figure of merit. This requires the ability to compute the adjoint of the noise channel with respect to the metric, which can be done analytically when the metric is evaluated at a Gaussian (quasi-free) state. This approach can be applied effectively to n-point functions of a quantum field theory. For translation invariant noise, this yields a stable deconvolution method on the first moments of the field which differs from what one would obtain from a purely classical analysis.

Appendix

Here we derive Eq. (5). We do the calculation in terms of the dual metric

$$\begin{aligned} \langle \!\langle E,F \rangle \!\rangle _\rho = \mathrm{Tr}\,({E}^\dagger \Omega _\rho (F)), \end{aligned}$$

which we extended to non-hermitian operators. With $\Omega _\rho (X) = \sqrt{\rho }X \sqrt{\rho }$, this is

$$\begin{aligned} \langle \!\langle E,F \rangle \!\rangle _\rho = \rho (E^\dagger \rho ^{\tfrac{1}{2}} F \rho ^{-\tfrac{1}{2} }). \end{aligned}$$

We observe that if we write $\rho$ as a Gibbs state for a quadratic Hamiltonian H, i.e. as $\rho \propto \mathrm{e}^{-H}$, then

$$\begin{aligned} E \mapsto \rho ^s E \rho ^{-s} = \mathrm{e}^{-sH} E\, \mathrm{e}^{s H}. \end{aligned}$$

This is the imaginary time evolution generated by the Hamiltonian H. On a single canonical operator $\phi (f)$, we have $\rho ^{s} \phi (f) \rho ^{-s} = \phi (R_{s}^A f)$ where $R_{-it}^A$ is the linear phase-space evolution matrix corresponding to the quadratic Hamiltonian H for time t. Here A denotes the state’s covariance matrix, which also uniquely defines H.

The slight difficulty this introduces is that $R_{s}^A$ acts on a complexification of the phase space. We do not mean the usual technique of treating half of the phase-space variables as imaginary, which would yield an n-dimensional complex vector space from a 2n-dimensional real phase space. Instead, we simply allow all the vector coefficients to be complex, which yields a 2n-dimensional complex vector space.

In terms of the normal-ordered displacement operators $G_f^A$, this is

$$\begin{aligned} \rho ^{\tfrac{1}{2}} G_{f}^A \rho ^{-\tfrac{1}{2}} = G_{R_{\frac{1}{2}}^A f}^A. \end{aligned}$$

Therefore, the dual metric can be computed via

$$\begin{aligned} \langle \!\langle G_f^A,G_g^A \rangle \!\rangle _\rho = \rho \left( (G_f^A)^{\dagger } G_{R_{\frac{1}{2}}^A g}^A\right) . \end{aligned}$$

(13)

We need to be careful about how the displacement operators behave on the complexified phase space. We have

$$\begin{aligned} W_f^\dagger W_g = W_{g - \overline{f}} \, \mathrm{e}^{\tfrac{i}{2} (f ,\Delta g)}, \end{aligned}$$

where the phase-space scalar product is extended so as to be conjugate-symmetric. For instance, this implies

$$\begin{aligned} \rho (W_f) = \mathrm{e}^{\tfrac{1}{2} (\overline{f}, A f)}. \end{aligned}$$

where the overline denotes complex conjugation component-wise. It follows that

$$\begin{aligned} \rho ((G_f^A)^\dagger G_g^A) = \mathrm{e}^{\left( f, (A+ \tfrac{i}{2} \Delta ) g\right) }. \end{aligned}$$

(14)

The adjointness relation defining $\mathcal N_*$ reads

$$\begin{aligned} \langle \!\langle G_f^A,\mathcal N^\dagger (G_g^B) \rangle \!\rangle _\rho = \langle \!\langle \mathcal N^\dagger _*(G_f^A),G_g^B \rangle \!\rangle _{\mathcal N(\rho )} \end{aligned}$$

(15)

where A is the covariance matrix for $\rho$ and B is the covariance matrix for $\mathcal N(\rho )$, namely $B = X^\dagger A X + Y$. Using

$$\begin{aligned} \mathcal N^\dagger (G_g^B) = G_{Xg}^A, \end{aligned}$$

and Eq. (14), we obtain

$$\begin{aligned} \langle \!\langle G_f^A,\mathcal N^\dagger (G_g^B) \rangle \!\rangle _\rho = \mathrm{e}^{(f, (A+ \tfrac{i}{2} \Delta ) R_{\frac{1}{2}}^A Xg)} \end{aligned}$$

(16)

Hence, with the ansatz

$$\begin{aligned} \mathcal N_*^\dagger (G_f^A) = G_{X_*f}^B, \end{aligned}$$

Eq. (15) is satisfied if

$$\begin{aligned} \mathrm{e}^{(f, (A+ \tfrac{i}{2} \Delta ) R_{\frac{1}{2}}^A Xg)} = \mathrm{e}^{(X_* f, (B+ \tfrac{i}{2} \Delta ) R_{\frac{1}{2}}^B g)} \end{aligned}$$

(17)

for all f, g, which implies

$$\begin{aligned} (A+ \tfrac{i}{2} \Delta ) R_{\frac{1}{2}}^A X = X_*^\dagger (B+ \tfrac{i}{2} \Delta ) R_{\frac{1}{2}}^B, \end{aligned}$$

or

$$\begin{aligned} X_* = (B+ \tfrac{i}{2} \Delta )^{-1} (R_{-\frac{1}{2}}^B)^\dagger X^\dagger (R_{\frac{1}{2}}^A)^\dagger (A+ \tfrac{i}{2} \Delta ). \end{aligned}$$

To obtain the form of Eq. (5), we use the fact that $(R^A_s)^{-1} = R^A_{-s} = \overline{R^A_s}$, which can be seen for instance from the explicit expression Eq. (7). We also use the fact that the covariance matrix and symplectic structure are invariant under $R_s$, i.e. such that $(R^A_s)^T(A+ \tfrac{i}{2} \Delta ) R^A_s = (A+ \tfrac{i}{2} \Delta )$. From these properties it follows that

$$\begin{aligned} (A+ \tfrac{i}{2} \Delta ) R^A_s = (R^A_s)^\dagger (A+ \tfrac{i}{2} \Delta ), \end{aligned}$$

which yields

$$\begin{aligned} X_* = R_{-\frac{1}{2}}^B (B+ \tfrac{i}{2} \Delta )^{-1} X^\dagger (A+ \tfrac{i}{2} \Delta ) R_{\frac{1}{2}}^A. \end{aligned}$$