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For a two-dimensional positive-definite covariance matrix $\Sigma$, the centered covariance ellipse is the image of the unit circle under its positive definite square root of a matrix. Equivalently, it satisfies $x^T{\Sigma }^{-1}x=1$. Its semi-major and semi-minor axes have lengths given by the square roots of the larger and smaller eigenvalues, along the corresponding eigenvectors. The ellipse encodes a second-moment geometry; a probability-coverage interpretation needs additional information about the probability distribution and the chosen scale.

For a real random vector $X=(X_1,\dots ,X_p{)}^T$ with finite second moments, its covariance matrix collects all pairwise covariances:The diagonal entries are variances. The covariance matrix is a symmetric matrix and a positive semidefinite matrix, because for every real vector $a$,Thus one matrix describes the variance of every linear combination of the coordinates. It is a positive-definite matrix exactly when no nonzero linear combination is constant almost surely. A deterministic linear relation produces a zero eigenvalue.

For example,has unit coordinate variances, covariance $0.8$, and eigenvalues $1.8$ and $0.2$. Its unit eigenvectors $(1,1{)}^T/\sqrt{2}$ and $(1,-1{)}^T/\sqrt{2}$ describe directions of greatest and least projected variance. For a unit vector $a=(\cos \theta ,\sin \theta {)}^T$, direct multiplication gives $a^T\Sigma a=1+0.8\sin (2\theta )$: the variation in the sum direction is nine times that in the difference direction.

The spectral theorem supplies an orthonormal basis of these directions. This connects covariance matrices to principal component analysis: its leading eigenvector identifies the unit-direction projection with the largest variance. For a linear transformation $A$, the transformed random vector has covariance matrix $A\Sigma A^T$, so uncertainties propagate through a linear model by matrix multiplication.

For $X=(a+a^{†})/2$ and $Y=(a-a^{†})/(2i)$, $[X,Y]=i/2$, so $\Delta X\Delta Y\ge 1/4$. Vacuum and coherent states have variance $1/4$ in either optical quadrature. quadrature squeezing means that some rotated optical quadrature has variance below this reference, with compensating noise elsewhere. A reduced second moment about zero is not the correct test when the field's mean is displaced.

For $S(\zeta )=\exp [({\zeta }^{\ast }{a}^2-\zeta a^{†2})/2]$ and $\zeta =r{e}^{i\theta }$, the quiet principal axis of the optical squeezed vacuum is at angle $\theta /2$ under the squeeze-operator convention used above, andThe pure Gaussian quantum state saturates the uncertainty product. Displacement moves its centre without changing its covariance ellipse; squeezed thermal states instead have a larger uncertainty product. The optical squeezed vacuum has zero mean field, even photon-number parity and mean occupation ${\sinh }^{2}r$; it need not have sub-Poissonian photon statistics. number squeezing and quadrature squeezing are different conditions.

Degenerate parametric amplification generates single-mode squeezing; a two-mode squeezed vacuum instead has correlated occupations and joint quadrature squeezing. Either individual mode of an ideal two-mode state is thermal after the other is ignored, not individually squeezed. balanced homodyne detection measures the noise versus local oscillator phase. Reference phase drift admixes the anti-squeezed optical quadrature, while electronic noise and detector imbalance need separate checks.

For an optical loss channel with efficiency $\eta$ and vacuum entering the unused port, optical squeezing degradation by loss givesThe linear-loss limit follows directly by writing the output field as $\sqrt{\eta }a+\sqrt{1-\eta }v$ with independent vacuum mode $v$. Loss therefore places a floor under measured squeezing. Squeezed input can improve phase-sensitive measurements when aligned with the measured optical quadrature; it does not reduce every noise source or both conjugate uncertainties at once.

The Robertson-Schrödinger uncertainty relation keeps the real part discarded in the proof of the Robertson uncertainty principle. Write $A^{\prime }=A-⟨A⟩I$ and $B^{\prime }=B-⟨B⟩I$. Their quantum covariance is $C_{AB}=\mathrm{Re}⟨A^{\prime }\psi |B^{\prime }\psi ⟩$. On the same finite-moment and product operator domains used above, the Cauchy-Schwarz inequality givesFor position and momentum, form the real, symmetrized covariance matrixA larger coordinatewise product can therefore encode a tilted covariance ellipse, rather than a larger determinant. This explains why the equality case of the Heisenberg uncertainty relation is more restrictive than equality in the Robertson-Schrödinger uncertainty relation.