Gaussian States Preview

A bosonic Gaussian state is a continuous-variable state whose Wigner function, or equivalently its Weyl characteristic function, is 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.

Helpful background. Density operators distinguishes pure and mixed states; EPR correlations explains the commuting combinations used to test two-mode entanglement. We assume finitely many bosonic modes and finite first and second moments.

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

$$v_qv_p \ge \frac14.$$

For a general one-mode covariance the condition is $\det V\ge1/4$ together with $V>0$. For example, $V=\operatorname{diag}(1/4,1)$ is physical and pure Gaussian, whereas $V=\operatorname{diag}(1/4,1/4)$ is positive as a classical covariance but violates quantum uncertainty.

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

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

Here the argument $R$ denotes an ordinary real phase-space coordinate, not the operator vector. A physical finite covariance is positive definite, so its inverse exists. Ideal infinite-squeezing limits do not define ordinary finite-covariance density operators.

The Wigner function of every Gaussian state is 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,

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

and arbitrary first moments $d$ (including $d=0$ for the vacuum). 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. More generally, its purity and entropy are

$$\operatorname{Tr}\rho^2=\frac{1}{\sqrt{\det(2V)}},\qquad S(\rho)=\sum_j g(\nu_j-1/2),$$

where $g(n)=(n+1)\log_2(n+1)-n\log_2n$, with $0\log_2 0=0$. The symplectic eigenvalues are the positive members of the paired spectrum $\{\pm\nu_j\}$ of $i\Omega V$; ordinary eigenvalues of $V$ are not a substitute. A one-mode squeezed pure state has ordinary eigenvalues $e^{\pm2r}/2$ but symplectic eigenvalue $\sqrt{\det V}=1/2$.

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, $C=0$ is equivalent to a product state across the $A,B$ mode split. First moments can always be removed by local displacements, which do not change entanglement. 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 a Gaussian bipartition of one mode against one mode, positivity of the partial transpose is also sufficient for separability (Simon, 2000). This extends to one mode against any finite number of modes, but fails in general for two modes against two modes: Gaussian PPT-entangled states exist (Werner and Wolf, 2001). These are claims about a specified bipartition, not a complete test for every form of multipartite separability.

For the squeezed vacuum, the partially transposed symplectic eigenvalues are $\widetilde\nu_-=e^{-2r}/2$ and $\widetilde\nu_+=e^{2r}/2$. Thus $\widetilde\nu_-<1/2$ for $r>0$, although the original physical covariance has both symplectic eigenvalues equal to $1/2$. For a two-mode Gaussian state the logarithmic negativity in bits is $\max\{0,-\log_2(2\widetilde\nu_-)\}$, giving $2r/\ln2$ for this example.

Another common two-mode witness uses EPR variances. With the present quadrature convention, every separable two-mode state with finite second moments obeys

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

for the symmetric choice of gains. The variance derivation does not assume Gaussianity. This particular unoptimized inequality is sufficient but not necessary for general Gaussian-state entanglement; local squeezing or unequal noise can require different gains or the full PPT test. 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.

Correlations that second moments can miss

The non-Gaussian pure state $(\lvert0,0\rangle+\lvert2,2\rangle)/\sqrt2$ is entangled, with two equal Schmidt probabilities. Nevertheless it has zero first moments and covariance $V=(3/2)I_4$, exactly the same covariance as two independent thermal modes with mean occupation one. Quadratures change occupation by one, so no cross-mode quadratic moment connects the $\lvert0,0\rangle$ and $\lvert2,2\rangle$ components.

A block-diagonal covariance therefore implies a product state only with the Gaussian assumption. Violation of a covariance uncertainty test after partial transposition can certify non-Gaussian entanglement too, but passing it cannot rule that entanglement out.

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.
• 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.
• R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Physical Review Letters 84, 2726–2729 (2000), doi:10.1103/PhysRevLett.84.2726.
• 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), doi:10.1103/RevModPhys.84.621.
• R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Physical Review Letters 86, 3658–3661 (2001), doi:10.1103/PhysRevLett.86.3658.

Exercises

1. Physical covariance or impossible noise? For each one-mode covariance $V_1=\operatorname{diag}(1/4,1)$ and $V_2=\operatorname{diag}(1/4,1/4)$, determine whether it can describe a quantum state. If physical and Gaussian, find its purity.

Solution

Both matrices are positive, but $\det V_1=1/4$ whereas $\det V_2=1/16<1/4$. Thus only $V_1$ satisfies quantum uncertainty. Its Gaussian purity is $1/(2\sqrt{\det V_1})=1$. A classical covariance can violate the additional quantum constraint.

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. A separable correlated Gaussian. Let a classical zero-mean Gaussian random variable $x$ have variance $s^2$. Prepare two vacuum modes and displace the $q$ quadrature of each by the same $x$. Find the covariance and explain why its correlations are not entanglement.

Solution

Each preparation is a product of two displaced vacua. Averaging those products is separable and Gaussian. In the ordering $(q_A,p_A,q_B,p_B)$,

$$V=\frac12I_4+s^2vv^T,\qquad v=(1,0,1,0)^T.$$

The cross block is $C=\operatorname{diag}(s^2,0)$, so it is nonzero for $s>0$. The common displacement cancels in $q_A-q_B$; both EPR variances remain one, saturating their separability bound. Nonzero covariance by itself does not prove entanglement.

4. Local squeezing can hide a fixed witness violation. Start with $V_{\rm TMSV}(r)$ at $r=(\ln2)/2$ and apply a local squeezing transformation to mode $A$:

$$V'=S V_{\rm TMSV}(r)S^T,\qquad S=\operatorname{diag}(3,1/3,1,1).$$

Compute the sum of the variances of $q_A-q_B$ and $p_A+p_B$ in the transformed state. Is a failed test evidence that the local operation removed entanglement? Compare with the variances of $q_A/3-q_B$ and $3p_A+p_B$.

Solution

Here $\cosh2r=5/4$ and $\sinh2r=3/4$, so

$$V'=\begin{pmatrix} 45/8&0&9/8&0\\ 0&5/72&0&-1/8\\ 9/8&0&5/8&0\\ 0&-1/8&0&5/8 \end{pmatrix}.$$

The fixed EPR variances are $45/8+5/8-2(9/8)=4$ and $5/72+5/8-2(1/8)=4/9$. Their sum $40/9$ does not violate the separable bound. This result is inconclusive: the applied squeezing is a local unitary, so it preserves entanglement. Indeed, the partially transposed symplectic eigenvalues remain $1/4$ and $1$, and the logarithmic negativity remains one bit.

Instead evaluate $u=q_A/3-q_B$ and $v=3p_A+p_B$. Their variances are each $1/2$, so $\operatorname{Var}u+\operatorname{Var}v=1<2$. The separable bound is still two because $[q_A/3,3p_A]=i$ and $[q_B,p_B]=i$; the same local uncertainty and mixing argument applies. These gains undo the known squeezing in the measured combinations. They recover a violation that the original fixed settings missed, without changing the state.

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

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