Gaussian Channels

A Gaussian channel is a continuous-variable quantum channel that maps Gaussian states to Gaussian states. For bosonic modes, it is the natural channel class for linear optics, attenuation, phase-insensitive amplification, additive Gaussian noise, thermal oscillator damping, and many input–output models before conditioning on non-Gaussian measurement records.

The power of Gaussian channels is that their action on first moments and covariance matrices is finite-dimensional:

$$d \longmapsto Xd+d_0, \qquad V \longmapsto XVX^{\mathsf T}+Y.$$

The matrices $X$ and $Y$ encode the channel. The complete-positivity condition is a matrix inequality involving the symplectic form. This page is the open-systems entry point; the broader Gaussian-state background is in Gaussian States Preview.

Quadrature Convention

For one mode,

$$q = \frac{a+a^\dagger}{\sqrt2}, \qquad p = \frac{a-a^\dagger}{i\sqrt2}, \qquad [q,p]=i.$$

For $N$ modes, collect quadratures as

$$R = (q_1,p_1,\ldots,q_N,p_N)^{\mathsf T}.$$

The canonical commutation relations are

$$[R_j,R_k] = i\Omega_{jk},$$

with

$$\Omega = \bigoplus_{\ell=1}^N \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}.$$

The first moments and covariance matrix are

$$d_j=\langle R_j\rangle,$$

and

$$V_{jk} = \frac12 \left\langle \Delta R_j\Delta R_k+\Delta R_k\Delta R_j \right\rangle, \qquad \Delta R_j=R_j-d_j.$$

With this convention, the vacuum covariance of one mode is

$$V_0=\frac12 I_2,$$

and physical covariance matrices satisfy

$$V+\frac{i}{2}\Omega\ge0.$$

All formulas below use this convention. Other communities absorb factors of $2$ or $\hbar$ differently, so covariance noise terms must always be translated with care.

Definition by First and Second Moments

An $N$-mode Gaussian channel acts on first moments and covariance matrices as

$$d' = Xd+d_0,$$

and

$$V' = XVX^{\mathsf T}+Y,$$

where:

• $X$ is a real $2N\times2N$ matrix;
• $d_0$ is a real displacement vector;
• $Y$ is a real symmetric $2N\times2N$ matrix;
• the pair $(X,Y)$ obeys the complete-positivity condition below.

The displacement $d_0$ changes first moments but does not affect noise or complete positivity. The matrix $X$ describes attenuation, amplification, rotations, squeezing, and mode mixing. The matrix $Y$ is the added noise.

For Gaussian inputs, $d$ and $V$ determine the full output state. For non-Gaussian inputs, the channel still exists, but first and second moments alone do not determine the output state.

Complete Positivity

With the convention $V+i\Omega/2\ge0$, the Gaussian map is completely positive if and only if

$$Y+\frac{i}{2} \left( \Omega-X\Omega X^{\mathsf T} \right) \ge0.$$

This condition is the continuous-variable analogue of Choi positivity. It says that the added noise must be large enough to compensate for any change in the commutation-preserving structure caused by $X$.

Important special cases:

• If $X$ is symplectic, so $X\Omega X^{\mathsf T}=\Omega$, the condition reduces to $Y\ge0$.
• A Gaussian unitary has $X$ symplectic and $Y=0$.
• Attenuation and amplification are not arbitrary scalings; quantum mechanics fixes a minimum amount of noise.

The inequality is also the easiest way to see why deterministic noiseless phase-insensitive amplification is impossible.

Gaussian Unitary Channels

A Gaussian unitary is generated by a Hamiltonian at most quadratic in the quadratures. It acts as

$$d \longmapsto Sd+d_0, \qquad V \longmapsto SVS^{\mathsf T},$$

where $S$ is symplectic:

$$S\Omega S^{\mathsf T} = \Omega.$$

Examples include:

• phase-space displacements;
• phase rotations of a mode;
• beam splitters;
• single-mode squeezers;
• two-mode squeezers;
• general linear-optical transformations.

These channels are reversible and add no noise. They are the continuous-variable analogue of unitary channels in finite dimensions.

Attenuation and Pure Loss

The one-mode attenuator with transmissivity $\eta$ has

$$0\le\eta\le1, \qquad X=\sqrt{\eta}\,I_2.$$

If the environment mode has covariance $V_E$, the covariance map is

$$d \longmapsto \sqrt{\eta}\,d, \qquad V \longmapsto \eta V+(1-\eta)V_E.$$

For a vacuum environment,

$$V_E=\frac12 I_2,$$

so

$$V' = \eta V+\frac{1-\eta}{2}I_2.$$

This is the quantum-limited pure-loss channel. It is the covariance-matrix form of the beam-splitter loss model discussed in Erasure and Loss Channels.

The added vacuum term is not optional. If one tried to map $V\mapsto\eta V$ with no environmental contribution, the vacuum would be sent to $\eta I_2/2$, violating the uncertainty principle for $\eta\lt1$.

Serial pure-loss channels multiply transmissivities; the discrete-time composition viewpoint is collected in Channel Composition and Fixed Points.

Thermal-Loss Channel

For a thermal environment with mean occupation $\bar n_E$,

$$V_E = \left( \bar n_E+\frac12 \right)I_2.$$

The thermal-loss channel is therefore

$$V' = \eta V + (1-\eta) \left( \bar n_E+\frac12 \right)I_2.$$

Quantum Illumination uses two such noisy attenuation hypotheses to formulate target absence versus presence. That page owns the retained-idler receiver, error-exponent bounds, and experimental comparator; the channel mechanics remain canonical here.

It has the thermal covariance as a fixed point:

$$V=V_E \quad\Longrightarrow\quad V'=V_E.$$

In a Markovian oscillator damping model,

$$\eta(t)=e^{-\kappa t}.$$

The bath occupation $\bar n_E$ is set by the environment temperature and oscillator frequency. At optical frequencies, vacuum loss is often a good approximation; at microwave frequencies, thermal occupation can be important.

Phase-Insensitive Amplification

A phase-insensitive amplifier with gain $G\ge1$ has

$$X=\sqrt G\,I_2.$$

The covariance map is

$$d \longmapsto \sqrt G\,d, \qquad V \longmapsto GV+(G-1)V_E',$$

where $V_E'$ is the covariance of the auxiliary mode entering the amplifier dilation. For a quantum-limited amplifier,

$$V_E'=\frac12I_2,$$

and therefore

$$V' = GV+\frac{G-1}{2}I_2.$$

The added-noise term is required by complete positivity. Without it, the map would scale both quadratures by $\sqrt G$ while also amplifying commutators incorrectly. Physically, a phase-insensitive amplifier must add at least vacuum noise.

Phase-sensitive amplification is different: squeezing can amplify one quadrature while deamplifying the conjugate quadrature through a Gaussian unitary. The no-free-lunch statement above is for deterministic phase-insensitive gain.

Additive Gaussian Noise

An additive Gaussian-noise channel leaves the first moments unchanged and adds classical displacement noise:

$$d \longmapsto d, \qquad V \longmapsto V+N,$$

with

$$N=N^{\mathsf T}\ge0.$$

This is the Gaussian channel with $X=I$ and $Y=N$. The complete-positivity condition reduces to $N\ge0$.

A physical picture is random displacement:

$$\rho \longmapsto \int d^{2N}\xi\, P(\xi) D(\xi)\rho D(\xi)^\dagger,$$

where $P(\xi)$ is a classical Gaussian distribution over phase-space displacements. If $P$ has covariance $N$, the covariance matrix gains $N$ in the same convention.

Additive noise models technical displacement noise, classical Gaussian backgrounds, and some effective limits of thermal or electronic noise. It is not the same as photon loss: loss attenuates first moments, while additive noise does not.

Multimode Form

For several modes, the same formula

$$V' = XVX^{\mathsf T}+Y$$

allows mode mixing, correlated noise, and loss to several environments. A passive linear-optical network is symplectic and orthogonal, so it has $Y=0$. Independent loss on each mode has

$$X = \bigoplus_{j=1}^N \sqrt{\eta_j}\,I_2,$$

and

$$Y = \bigoplus_{j=1}^N (1-\eta_j) \left( \bar n_j+\frac12 \right)I_2.$$

Correlated Gaussian noise has off-diagonal blocks in $Y$. Such correlations matter in multimode sensing, communication, and oscillator networks.

Physical Interpretation

Gaussian channels arise from three ingredients:

1. a system of bosonic modes;
2. a Gaussian environment state;
3. a joint Gaussian unitary, usually generated by a Hamiltonian at most quadratic in quadratures.

After the environment is traced out, the system undergoes a Gaussian channel. This is the continuous-variable version of a Stinespring representation.

Common laboratory readings include:

• optical fiber or cavity loss as attenuation;
• microwave resonator damping as thermal loss;
• imperfect detection as loss before measurement;
• linear amplifiers as phase-insensitive gain plus required noise;
• classical amplitude or phase noise as additive displacement noise;
• beam splitters and squeezers as Gaussian unitaries.

The same formal channel can have different experimental meanings depending on which modes are retained, which environment is monitored, and what convention is used for quadrature normalization.

What Gaussian Channels Do Not Cover

Gaussian channels do not include every important continuous-variable process. Non-Gaussian processes include:

• photon counting with conditioning;
• photon subtraction or addition;
• Kerr nonlinearities;
• threshold detectors as state-update maps;
• finite-dimensional truncation artifacts;
• loss conditioned on a number-resolved environment record;
• non-Gaussian noise distributions.

Also, a Gaussian channel acting on a non-Gaussian input generally produces an output whose full state is not determined by $d$ and $V$ alone. Moment formulas remain useful, but they do not replace the density operator.

Continuous-Variable Quantum Computation records when a declared Gaussian channel enters a computation, propagates moments only while Gaussian closure is licensed, charges its noise and resources, and states any non-Gaussian completion. This page retains the $(X,Y,d_0)$ representation, complete-positivity condition, channel taxonomy, and capacities.

Common Mistakes

• Using covariance formulas without stating whether the vacuum covariance is $I/2$ or $I$.
• Calling $V\mapsto\eta V$ pure loss; the vacuum noise term is required.
• Treating phase-insensitive amplification as noiseless.
• Confusing additive noise with attenuation.
• Assuming a Gaussian channel makes every input state Gaussian.
• Applying a covariance-only calculation to a non-Gaussian input and forgetting higher moments.
• Ignoring thermal occupation in microwave or mechanical modes.
• Treating a postselected photon-counting event as a Gaussian channel.

References

• A. S. Holevo, Quantum Systems, Channels, Information, De Gruyter (2012).
• A. S. Holevo and R. F. Werner, “Evaluating capacities of bosonic Gaussian channels,” Physical Review A 63, 032312 (2001).
• C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Reviews of Modern Physics 84, 621-669 (2012).
• A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press (2017).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• M. M. Wilde, Quantum Information Theory, Cambridge University Press, 2nd ed. (2017).

Exercises

1. Pure-loss vacuum check. Show that the pure-loss channel maps the vacuum covariance $I_2/2$ to itself.

Solution

For pure loss,

$$V' = \eta V+\frac{1-\eta}{2}I_2.$$

Set $V=I_2/2$:

$$V' = \eta\frac{I_2}{2} + \frac{1-\eta}{2}I_2 = \frac12I_2.$$

Thus vacuum is a fixed point of quantum-limited attenuation.

2. Amplifier noise floor. Use the complete-positivity condition to explain why the phase-insensitive amplifier with $X=\sqrt G I_2$ needs added noise for $G>1$.

Solution

For one mode,

$$X\Omega X^{\mathsf T} = G\Omega.$$

The complete-positivity condition is

$$Y+\frac{i}{2}(1-G)\Omega\ge0.$$

If $Y=0$ and $G>1$, the Hermitian matrix $i(1-G)\Omega/2$ has one negative eigenvalue. Thus the map is not completely positive. The quantum-limited choice

$$Y=\frac{G-1}{2}I_2$$

just saturates the required noise bound.

3. Thermal fixed point. Show that the thermal-loss map has $V_E=(\bar n_E+1/2)I_2$ as a fixed covariance matrix.

Solution

The thermal-loss covariance map is

$$V' = \eta V+(1-\eta)V_E.$$

Set $V=V_E$:

$$V' = \eta V_E+(1-\eta)V_E = V_E.$$

Thus the channel relaxes covariance matrices toward the environment covariance.

4. Additive noise is not loss. Compare the transformations of first moments under pure loss and additive Gaussian noise.

Solution

Pure loss gives

$$d\mapsto\sqrt{\eta}\,d,$$

so displacements are attenuated. Additive Gaussian noise gives

$$d\mapsto d,$$

so the mean displacement is unchanged. Both can increase uncertainty, but only attenuation reduces the signal amplitude.