A Gaussian quantum state is an infinite-dimensional state that can nevertheless be described completely by a finite set of first and second moments. That compression is what makes continuous-variable quantum information mathematically tractable.
For n bosonic modes, a generic density operator requires infinitely many Fock-basis matrix elements. A Gaussian state avoids that complexity because its Wigner function is Gaussian in a 2n-dimensional phase space. Once the displacement vector and covariance matrix are known, the entire state is fixed.
This guide keeps the same convention as Guide 37: [x_j,p_k]=iδ_jk, so each vacuum quadrature has variance 1/2. Under this convention physical covariance matrices satisfy V+iΩ/2≥0. If another source uses vacuum variance one, every covariance formula carries corresponding factors of two.
Infinite-dimensional state → phase-space moments → covariance matrix → symplectic geometry → squeezing, mixing and entanglement.
1. Collect all quadratures into one vector
For n modes define
R=(x₁,p₁,x₂,p₂,…,x_n,p_n)^T.
The canonical commutation relations can be written compactly as
[R_j,R_k]=iΩ_jk
where
Ω=⊕_{m=1}^n [[0,1],[-1,0]].
Ω is the standard symplectic form. It is the continuous-variable counterpart of the canonical structure that tells us which coordinates are conjugate.
2. Displacement vector
The first moments form the displacement vector
d_j=⟨R_j⟩.
For one mode, d=(⟨x⟩,⟨p⟩). It locates the centre of the Gaussian Wigner function in phase space.
Changing d without changing the covariance moves the state but does not change its uncertainty ellipse. Coherent states differ from vacuum primarily by such a displacement.
3. Covariance matrix
Define centred operators ΔRj=Rj−dj. The covariance matrix is
V_jk=(1/2)⟨ΔR_jΔR_k+ΔR_kΔR_j⟩.
V is real and symmetric. Its diagonal entries are quadrature variances. Off-diagonal entries are symmetrised covariances.
For a one-mode vacuum or coherent state,
V_vac=I₂/2.
4. A Gaussian state is fixed by d and V
The Wigner function of an n-mode Gaussian state has the form
W(r)=[(2π)^n√detV]^{-1} exp[-(1/2)(r-d)^T V^{-1}(r-d)]
under the phase-space normalisation compatible with this guide’s covariance convention.
The exact front factor changes across Wigner-function conventions, but the structural statement does not: a Gaussian state is completely determined by its mean and covariance.
5. Quantum uncertainty constrains V
Not every real positive matrix is a physical quantum covariance matrix. The Robertson–Schrödinger uncertainty relation becomes
V+iΩ/2≥0.
For one mode with diagonal covariance V=diag(vx,vp), this implies
v_xv_p≥1/4.
Thus the covariance matrix already contains the Heisenberg principle.
6. Worked physicality test
Consider one-mode diagonal matrices
V₁=diag(0.25,1);V₂=diag(0.2,1).
For V₁, the product is 0.25, exactly the minimum 1/4, so it is compatible with a pure squeezed state.
For V₂, the product is 0.20<0.25. It violates the uncertainty principle and cannot be the covariance of a physical quantum state under this convention.
7. Symplectic transformations
A real matrix S is symplectic when
SΩS^T=Ω.
This condition preserves canonical commutation relations. Gaussian unitaries generated by Hamiltonians at most quadratic in the quadrature operators act affinely:
d→Sd+s;V→SVS^T.
The vector s describes a displacement; S describes rotations, squeezing and multimode linear mixing.
8. Phase rotation as a symplectic matrix
A one-mode phase rotation by φ acts as
S_R(φ)=[[cosφ,sinφ],[-sinφ,cosφ]].
Direct multiplication confirms S_RΩS_R^T=Ω.
On a covariance ellipse, this rotates the principal axes without changing its eigenvalues, determinant or purity.
9. Single-mode squeezing
An ideal squeezing operation with real parameter r can be represented by
S_sq(r)=diag(e^{-r},e^{r}).
Applied to vacuum,
V=(1/2)diag(e^{-2r},e^{2r}).
One quadrature variance decreases while the conjugate variance increases. Their product remains 1/4, so ideal squeezed vacuum is pure and saturates the uncertainty bound.
10. Squeezing in decibels
Experimental squeezing is often quoted relative to vacuum variance. The squeezed-variance ratio is e−2r.
The noise reduction in decibels is
-10log₁₀(e^{-2r})≈8.686r dB.
For r=0.5, the squeezed variance is e−1≈0.3679 times vacuum, corresponding to approximately 4.34 dB of ideal squeezing.
The antisqueezed variance is e+1≈2.718 times vacuum.
11. Why squeezing costs energy
For squeezed vacuum, the mean photon number is
n̄=sinh²r.
At r=0.5, sinh²(0.5)≈0.2715 photons on average. At r=1, the value rises to about 1.381.
Reducing one quadrature below vacuum is therefore not free: the state becomes more energetic while uncertainty is redistributed.
12. Beam splitter as a two-mode symplectic transformation
For transmissivity η, a lossless beam splitter mixes two modes. Ignoring a convention-dependent reflection phase, a real quadrature form is
S_BS=[[√η I₂, √(1−η) I₂],[-√(1−η) I₂, √η I₂]].
It preserves Ω and total quadratic vacuum noise.
At η=1/2 it mixes the two input phase-space coordinates equally. Guide 39 develops its action directly on optical mode operators and photon states.
13. Gaussian displacement does not change entanglement or purity
A displacement changes d but leaves V unchanged.
Therefore quantities depending only on covariance—such as symplectic eigenvalues, Gaussian purity and Gaussian entanglement tests after local displacements—are unaffected.
This is why coherent states all have the same vacuum covariance despite having different mean field amplitudes.
14. Symplectic eigenvalues
Ordinary eigenvalues of V change under squeezing, so they do not capture the invariant quantum noise content.
The symplectic eigenvalues νk are obtained from the spectrum of |iΩV|. Physicality requires
ν_k≥1/2
for every mode under this guide’s convention.
Pure Gaussian states have every νk=1/2.
15. Williamson’s theorem
Every positive definite covariance matrix can be symplectically diagonalised:
V=S[⊕_k ν_k I₂]S^T.
This is Williamson’s theorem.
It says every Gaussian covariance is a symplectic transformation of independent thermal modes whose effective variances are the symplectic eigenvalues.
Ordinary diagonalisation uses rotations; Williamson diagonalisation uses canonical phase-space transformations.
16. Purity from det V
For an n-mode Gaussian state in this convention, purity is
μ=Tr(ρ²)=1/√det(2V).
For one-mode vacuum, 2V=I, so μ=1.
For a one-mode thermal state with covariance V=(n̄+1/2)I,
μ=1/(2n̄+1).
As thermal occupation grows, purity falls.
17. Entropy from symplectic eigenvalues
The von Neumann entropy of a Gaussian state is the sum of thermal-mode entropy functions of its symplectic eigenvalues:
S(ρ)=Σ_k g(ν_k−1/2)
where
g(n)=(n+1)log₂(n+1)−nlog₂n.
A pure Gaussian state has ν=1/2 and therefore zero entropy globally.
18. Two-mode squeezed vacuum
A two-mode squeezed vacuum is the canonical Gaussian entangled state. In quadrature ordering (x₁,p₁,x₂,p₂), its covariance is
V=(1/2)[[cI₂,sZ],[sZ,cI₂]]
where c=cosh(2r), s=sinh(2r) and Z=diag(1,−1).
The EPR-like combinations satisfy
Var(x₁−x₂)=Var(p₁+p₂)=e^{-2r}.
At r=0 these variances equal one, the value from two independent vacua. Increasing r suppresses both commuting EPR combinations simultaneously.
19. Local states of a two-mode squeezed vacuum are thermal
Tracing out mode 2 leaves mode 1 with covariance
V_1=(cosh2r/2)I₂.
This is a thermal covariance with mean occupation
n̄=sinh²r.
The global state is pure while each subsystem is mixed. This is the continuous-variable version of entanglement entropy: local thermal uncertainty can arise from tracing out an entangled partner rather than from a literal heat bath.
20. Partial transpose is a phase-space mirror
For a bipartite continuous-variable state, partial transpose of the second subsystem corresponds in phase space to reversing one momentum coordinate, for example
p₂→−p₂.
At the covariance level, apply a diagonal matrix Λ that flips the sign of p₂:
V^PT=ΛVΛ.
If VPT violates the quantum uncertainty condition, the original state is entangled. Simon’s criterion makes this phase-space test exact for the two-mode Gaussian setting and, more generally, PPT remains a central Gaussian separability diagnostic. [2]
21. Worked partial-transpose test for two-mode squeezing
For the two-mode squeezed vacuum, the smallest symplectic eigenvalue after partial transpose is
ν_tilde_min=e^{-2r}/2.
At r=0, ν̃=1/2 and there is no entanglement. For every r>0, e−2r/2<1/2, so the partially transposed covariance violates the physical uncertainty bound and certifies entanglement.
The logarithmic negativity is then proportional to max(0,-log₂(2ν̃_min)), giving a simple monotone for this state family.
22. Gaussian channels act affinely on moments
A general Gaussian channel maps
d→Xd+s;V→XVX^T+Y.
The matrices X and Y cannot be arbitrary; complete positivity imposes a matrix inequality linking Y, Ω and XΩXT.
Unitary Gaussian evolution is the special noiseless case Y=0 with X symplectic. Guide 40 develops attenuator, amplifier and additive-noise examples.
23. Gaussian measurement
Homodyne detection measures one quadrature. Heterodyne detection jointly estimates both conjugate quadratures by coupling to an auxiliary vacuum mode and paying additional measurement noise.
For Gaussian input states, these measurement outcome distributions are Gaussian, and conditional post-measurement states remain Gaussian under the standard ideal measurement model.
This closure under common optical operations and measurements explains why the Gaussian toolbox is so analytically efficient.
24. Gaussian closure is also a computational limitation
Gaussian states, Gaussian unitaries and Gaussian measurements form a highly structured family. Their evolution can often be simulated efficiently by tracking finite-dimensional moments rather than infinite Fock amplitudes.
Universal continuous-variable quantum computation therefore requires an appropriate non-Gaussian resource, measurement or interaction in standard architectures.
Likewise, Gaussian entanglement cannot generally be distilled using only Gaussian local operations and classical communication. [4]
25. Common misconception: a Gaussian state means the Fock-number probabilities are Gaussian
“Gaussian” refers to the state’s phase-space quasiprobability or characteristic function. A coherent state’s photon-number distribution is Poisson, not Gaussian in general.
26. Common misconception: covariance matrix eigenvalues are the symplectic eigenvalues
Ordinary eigenvalues change under squeezing. Symplectic eigenvalues come from |iΩV| and remain invariant under symplectic transformations.
27. Common misconception: squeezing beats uncertainty
Squeezing lowers noise in one quadrature by increasing it in the conjugate quadrature. Ideal pure squeezing preserves the minimum uncertainty product.
28. Worked synthesis problem
A one-mode Gaussian state has zero displacement and covariance
V=diag(0.30,1.20).
Step 1: Physicality. The determinant is 0.36, larger than 1/4, so the one-mode uncertainty product is satisfied.
Step 2: Symplectic eigenvalue. For one mode, ν=√detV=0.6.
Step 3: Purity.
μ=1/√det(2V)=1/(2√0.36)=1/1.2≈0.8333.
Step 4: Squeezing direction. x variance 0.30 is below the vacuum variance 0.50, so x is squeezed relative to vacuum. p is antisqueezed.
Step 5: Mixedness. Because detV>1/4 and μ<1, this is a mixed squeezed Gaussian state rather than a pure squeezed vacuum.
29. Practice set
- Write the multimode quadrature vector R.
- What matrix encodes the canonical commutation relations?
- Define the covariance matrix.
- What is vacuum covariance in this guide’s convention?
- State the covariance-matrix uncertainty condition.
- Define a symplectic matrix.
- How does V transform under a Gaussian unitary?
- What covariance results from squeezing vacuum by r?
- What is the physicality lower bound on symplectic eigenvalues?
- State Williamson’s covariance decomposition.
- Write Gaussian purity in terms of V.
- What phase-space operation represents partial transpose of one mode?
Answers
(x₁,p₁,…,x_n,p_n)^T.- The symplectic form Ω.
V_jk=(1/2)⟨{ΔR_j,ΔR_k}⟩.I/2per mode.V+iΩ/2≥0.- A real S satisfying
SΩS^T=Ω. V→SVS^T.diag(e^{-2r},e^{2r})/2.ν_k≥1/2.V=S(⊕ν_kI₂)S^T.1/√det(2V).- A momentum sign flip on the transposed subsystem, implemented by a diagonal mirror matrix Λ.
Sources and further study
[1] Christian Weedbrook and colleagues, Gaussian Quantum Information, Reviews of Modern Physics 84, 621 (2012). A standard review of Gaussian states, covariance matrices, symplectic transformations, channels and measurements.
[2] R. Simon, Peres–Horodecki Separability Criterion for Continuous Variable Systems. The phase-space partial-transpose criterion for Gaussian entanglement.
[3] F. Nicacio, Williamson theorem in classical, quantum, and statistical physics. A modern linear-algebra-focused account of symplectic diagonalisation and its physical applications.
[4] J. Eisert, S. Scheel and M. B. Plenio, On the impossibility of distilling Gaussian states with Gaussian operations. A foundational Gaussian entanglement-distillation limitation.
Continue through Quantum Mathematics
Guide 37: Bosonic Modes, Ladder Operators, Fock States and Quadratures supplies the operator foundation. Guide 39: Quantum Optics, Coherent States, Beam Splitters, Interferometry and Photodetection turns Gaussian and non-Gaussian states into optical experiments. Guide 40: Continuous-Variable Quantum Information, Gaussian Channels, Homodyne Detection and Teleportation develops communication and processing protocols.