Gaussian States Preview

Gaussian states are the continuous-variable states whose phase-space quasiprobability distributions are Gaussian. They are the workhorse states of harmonic oscillators, quantum optics, trapped ions, microwave modes, optomechanics, and continuous-variable quantum information.

The key simplification is severe but powerful: a Gaussian state is completely determined by its first moments and covariance matrix. This turns many infinite-dimensional questions into finite-dimensional matrix questions. The price is that Gaussian states are only a special family; many important states, measurements, and resources are non-Gaussian.

This page is a preview. It sets the notation for quadratures, covariance matrices, pure and mixed Gaussian states, and first Gaussian entanglement tests. It does not replace a full phase-space or quantum-optics treatment.

Quadratures

For one bosonic mode with

$$[a,a^\dagger]=1,$$

define dimensionless quadratures

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

They obey

$$[q,p]=i.$$

For $N$ modes, collect the quadratures into the column vector

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

The canonical commutation relations become

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

where

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

This symplectic matrix is the phase-space form of the canonical commutation relations. It is the continuous-variable analogue of the algebraic structure behind position and momentum.

The vacuum has

$$\langle q\rangle = \langle p\rangle = 0, \qquad \operatorname{Var}(q) = \operatorname{Var}(p) = \frac12.$$

This convention is used throughout this page and the EPR State Preview.

Covariance Matrices

The first moments are

$$d_j = \langle R_j\rangle.$$

The covariance matrix is the real symmetric matrix

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

For one mode,

$$V = \begin{pmatrix} \operatorname{Var}(q) & \operatorname{Cov}(q,p) \\ \operatorname{Cov}(p,q) & \operatorname{Var}(p) \end{pmatrix}.$$

The covariance matrix cannot be arbitrary. The uncertainty principle is the matrix condition

$$V+\frac{i}{2}\Omega \ge 0,$$

meaning that the Hermitian matrix on the left is positive semidefinite. For one mode with no $q$-$p$ covariance,

$$V = \begin{pmatrix} v_q & 0\\ 0 & v_p \end{pmatrix},$$

this reduces to

$$v_qv_p \ge \frac14.$$

That is the familiar uncertainty relation in covariance-matrix form.

Wigner Function Definition

A Gaussian state can be defined as a state whose Wigner function is a Gaussian in phase space. With the conventions above, an $N$-mode Gaussian state has

$$W(R) = \frac{1}{(2\pi)^N\sqrt{\det V}} \exp\!\left[ -\frac12 (R-d)^T V^{-1}(R-d) \right],$$

when $V$ is nonsingular. Singular limits are distributions and should be treated like the ideal EPR state: useful as limits, not as ordinary density operators.

The Wigner function of a Gaussian state can be everywhere nonnegative. That does not make the state classical. Noncommuting quadratures, measurement back-action, squeezing, and entanglement are still quantum features. Positivity of the Wigner function is not the same as separability or classical realism.

The most important examples are:

• coherent states;
• squeezed states;
• thermal oscillator states;
• displaced thermal states;
• two-mode squeezed states;
• Gaussian states produced by linear optics, squeezers, and thermal noise.

Pure and Mixed Gaussian States

A coherent state has the same covariance matrix as the vacuum,

$$V_{\rm coh} = \frac12 I_2,$$

but nonzero first moments $d$. Displacements move the center of the Wigner function without changing its covariance matrix.

A single-mode squeezed vacuum has

$$V_{\rm sq}(r) = \frac12 \begin{pmatrix} e^{-2r} & 0\\ 0 & e^{2r} \end{pmatrix}$$

in a phase convention where $q$ is squeezed. The determinant remains

$$\det V_{\rm sq} = \frac14,$$

so the state is pure even though one quadrature variance is below the vacuum value. The conjugate quadrature variance increases by the reciprocal factor.

A thermal oscillator state with mean occupation $\bar n$ has

$$V_{\rm th} = \left( \bar n+\frac12 \right)I_2.$$

It is mixed unless $\bar n=0$. The covariance matrix is larger than the vacuum covariance because thermal occupation adds isotropic noise in phase space.

For $N$ modes, Williamson’s theorem says that any physical covariance matrix can be written

$$V = S \left( \bigoplus_{j=1}^{N} \nu_j I_2 \right) S^T,$$

where $S$ is symplectic:

$$S\Omega S^T = \Omega.$$

The positive numbers $\nu_j$ are the symplectic eigenvalues. Physical states satisfy

$$\nu_j \ge \frac12.$$

A Gaussian state is pure exactly when all symplectic eigenvalues are $1/2$. Equivalently,

$$\det V = 2^{-2N}$$

for a pure $N$-mode Gaussian state.

Two-Mode Squeezed Covariance

The two-mode squeezed vacuum is the standard Gaussian entangled state. In the ordering

$$R = (q_A,p_A,q_B,p_B)^T,$$

a common covariance matrix convention is

$$V_{\rm TMSV}(r) = \frac12 \begin{pmatrix} \cosh 2r & 0 & \sinh 2r & 0\\ 0 & \cosh 2r & 0 & -\sinh 2r\\ \sinh 2r & 0 & \cosh 2r & 0\\ 0 & -\sinh 2r & 0 & \cosh 2r \end{pmatrix}.$$

From this matrix,

$$\operatorname{Var}(q_A-q_B) = e^{-2r}, \qquad \operatorname{Var}(p_A+p_B) = e^{-2r}.$$

Thus the same covariance matrix language captures the EPR-like correlations discussed in the EPR State Preview, the number-basis Schmidt expansion discussed in Two-Mode Entanglement, and the operator viewpoint developed in Squeezed States as Entangled Modes.

The one-mode reduction of a two-mode squeezed vacuum is thermal:

$$V_A = V_B = \frac12 \cosh 2r\,I_2.$$

Its mean occupation is

$$\bar n = \sinh^2 r.$$

This is the covariance-matrix version of a general entanglement fact: a pure entangled bipartite state has mixed reduced states.

Gaussian Entanglement Preview

For a bipartition into modes $A$ and $B$, write the covariance matrix in block form:

$$V = \begin{pmatrix} V_A & C\\ C^T & V_B \end{pmatrix}.$$

Here $V_A$ and $V_B$ are local covariance matrices, while $C$ contains cross-correlations. For Gaussian states, if $C=0$ and the first moments split as $d=(d_A,d_B)$, then the state is a product across the $A,B$ mode split. Nonzero $C$ signals correlations, but not necessarily entanglement; mixed Gaussian states may contain classical correlations.

One central Gaussian diagnostic is partial transposition. In phase space, transposing subsystem $B$ corresponds to flipping the signs of the $B$ momenta:

$$V^{\Gamma_B} = \Lambda_B V\Lambda_B,$$

where $\Lambda_B$ leaves $q$ quadratures unchanged and sends each transposed $p_B$ to $-p_B$.

If

$$V^{\Gamma_B} +\frac{i}{2}\Omega \not\ge 0,$$

then the state is entangled. For two-mode Gaussian states, this positive-partial-transpose test is also sufficient for separability. For larger multipartite Gaussian systems, partial transpose tests remain important but can be inconclusive.

Another common two-mode witness uses EPR variances. With the present quadrature convention, a separable two-mode state obeys a bound of the form

$$\operatorname{Var}(q_A-q_B) + \operatorname{Var}(p_A+p_B) \ge 2$$

for the symmetric choice of gains. The two-mode squeezed vacuum gives $2e^{-2r}$, violating the bound for any $r>0$.

What Gaussian Methods Do and Do Not Give

Gaussian methods are powerful because many operations preserve Gaussianity:

• displacements;
• phase shifts and mode rotations;
• beam splitters;
• single-mode and two-mode squeezing;
• thermal noise and loss channels;
• homodyne detection with Gaussian conditioning.

Within this controlled family, covariance matrices support efficient calculations of reduced states, entropies, EPR correlations, and many entanglement criteria.

The open-system version of this statement is developed in Gaussian Channels, where attenuation, amplification, additive noise, and thermal loss act directly on first moments and covariance matrices.

Gaussian methods do not describe everything. Number states beyond the vacuum, photon subtraction, most projective measurements, Schrödinger-cat-like superpositions, and many resource states for quantum advantage are non-Gaussian. A correct continuous-variable analysis must therefore say whether it is staying inside the Gaussian sector or using Gaussian states only as a starting point.

The companion page Squeezed States as Entangled Modes focuses on the physical squeeze operators and how two-mode squeezing creates entanglement.

Common Mistakes

• Treating a nonnegative Wigner function as proof of classicality.
• Forgetting the quadrature convention, especially whether vacuum variance is $1/2$ or $1$.
• Calling every correlated Gaussian state entangled.
• Applying the two-mode PPT criterion as if it solved all multipartite Gaussian separability questions.
• Ignoring first moments when comparing states, even though first moments do not affect covariance-based entanglement tests.
• Forgetting that finite squeezing gives finite EPR correlations, not exact delta correlations.
• Treating Gaussian methods as universal for continuous-variable quantum systems.

Cross-Links

• Continuous-Variable Systems
• Mode Decompositions
• Two-Mode Entanglement
• EPR State Preview
• Position-Space Two-Particle States
• Squeezed States as Entangled Modes
• Partial Trace
• Subsystem Entropy
• Entanglement Entropy
• Field Operators
• Entanglement in Quantum Optics
• Formula Sheet
• Gaussian Wave Packets
• Quantum Harmonic Oscillator
• Gaussian Channels

References

• S. L. Braunstein and P. van Loock, “Quantum information with continuous variables”, Reviews of Modern Physics 77, 513-577, 2005, doi:10.1103/RevModPhys.77.513.
• C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information”, Reviews of Modern Physics 84, 621-669, 2012, doi:10.1103/RevModPhys.84.621.
• A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press, 2017.
• A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, De Gruyter, 2012.
• R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems”, Physical Review Letters 84, 2726-2729, 2000, doi:10.1103/PhysRevLett.84.2726.
• L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability Criterion for Continuous Variable Systems”, Physical Review Letters 84, 2722-2725, 2000, doi:10.1103/PhysRevLett.84.2722.
• G. Adesso, S. Ragy, and A. R. Lee, “Continuous Variable Quantum Information: Gaussian States and Beyond”, Open Systems & Information Dynamics 21, 1440001, 2014, doi:10.1142/S1230161214400010.

Exercises

1. Quadrature commutator. Starting from $[a,a^\dagger]=1$, show that $q=(a+a^\dagger)/\sqrt2$ and $p=(a-a^\dagger)/(i\sqrt2)$ obey $[q,p]=i$.

Solution

Compute

$$\begin{aligned} [q,p] &= \frac{1}{2i} [a+a^\dagger,a-a^\dagger] \\ &= \frac{1}{2i} \left( -[a,a^\dagger]+[a^\dagger,a] \right) \\ &= \frac{1}{2i} (-1-1) \\ &= i. \end{aligned}$$

2. Single-mode uncertainty. For $V=\operatorname{diag}(v_q,v_p)$, use $V+i\Omega/2\ge0$ to derive $v_qv_p\ge1/4$.

Solution

For one mode,

$$V+\frac{i}{2}\Omega = \begin{pmatrix} v_q & i/2\\ -i/2 & v_p \end{pmatrix}.$$

Positive semidefiniteness requires nonnegative principal minors. The diagonal entries require $v_q,v_p\ge0$, and the determinant condition gives

$$v_qv_p-\frac14 \ge 0.$$

Thus $v_qv_p\ge1/4$.

3. Squeezed purity check. Show that the covariance matrix $V_{\rm sq}(r)=\operatorname{diag}(e^{-2r}/2,e^{2r}/2)$ has determinant $1/4$.

Solution

The determinant is

$$\det V_{\rm sq} = \frac{e^{-2r}}{2} \frac{e^{2r}}{2} = \frac14.$$

The product of variances is fixed even though one variance is squeezed below the vacuum value.

4. EPR variances from the two-mode covariance. Use $V_{\rm TMSV}(r)$ to compute $\operatorname{Var}(q_A-q_B)$.

Solution

From the covariance matrix,

$$\operatorname{Var}(q_A) = \operatorname{Var}(q_B) = \frac12\cosh2r,$$

and

$$\operatorname{Cov}(q_A,q_B) = \frac12\sinh2r.$$

Therefore

$$\begin{aligned} \operatorname{Var}(q_A-q_B) &= \operatorname{Var}(q_A) + \operatorname{Var}(q_B) -2\operatorname{Cov}(q_A,q_B) \\ &= \cosh2r-\sinh2r \\ &= e^{-2r}. \end{aligned}$$

5. Product Gaussian state. Suppose a two-mode Gaussian state has $d=(d_A,d_B)$ and covariance matrix $V=V_A\oplus V_B$. Explain why it is a product state.

Solution

A Gaussian state is completely determined by its first moments and covariance matrix. If the first moments split into local parts and the covariance matrix is block diagonal, then the Wigner function factorizes:

$$W_{AB}(R_A,R_B) = W_A(R_A)W_B(R_B).$$

Thus the density operator factorizes as

$$\rho_{AB} = \rho_A\otimes\rho_B.$$