Quantum deconvolution

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.

This work was supported by the research fund of Hanyang University (HY-2016-2237).

Authors and Affiliations

1. Department of Applied Mathematics, Hanyang University (ERICA), 55 Hanyangdaehak-ro, Ansan, Gyeonggi-do, 426-791, Korea

   Cédric Bény

Corresponding author

Correspondence to Cédric Bény.

Appendix

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

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

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

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

Therefore, the dual metric can be computed via

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

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

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

The adjointness relation defining \(\mathcal N_*\) reads

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

and Eq. (14), we obtain

Hence, with the ansatz

Eq. (15) is satisfied if

for all f, g, which implies

or

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

which yields

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