10 Jun 2026
VII: Non-Gaussian Mutual Information: Exact Entropy and Covariance Bounds
Quantum mutual information answers a concrete question: how much information is shared by two chosen subsystems? For continuous-variable systems it is tempting to answer from a covariance matrix alone. That shortcut is exact for Gaussian states, but it is generally wrong for non-Gaussian states.
This guide develops the distinction from the definitions. It requires only the basic ideas of a wavefunction, an inner product, and matrix eigenvalues. The central task is to derive the exact correlation measure and then identify, mathematically, the more limited statement supplied by a covariance matrix.
All logarithms are natural logarithms, so entropies are measured in nats. Divide by $\ln 2$ to convert nats to bits.
1. Physics carried forward
This is the information-theory step of a progressive calculation. Three results have already been established:
- The magnetic Hamiltonian and its stability domain give two rotated canonical modes $(q_\pm,p_\pm)$.
- Exceptional-Hermite preparation gives a normalized product input \(\lvert m_+,\ell_+\rangle\otimes\lvert m_-,\ell_-\rangle\).
-
Spectral propagation gives cutoff-tested coefficients
\[|\Psi(t)\rangle =\sum_{r,s}c_{rs}(t) |m_+,r\rangle\otimes|m_-,s\rangle.\]
Those derivations are not repeated here. The new problem is to convert the coefficient array $c_{rs}(t)$ into an exact correlation measure and to determine precisely what can—and cannot—be inferred from second moments.
The two-mode Hilbert space is
\[\mathcal H=\mathcal H_A\otimes\mathcal H_B, \qquad A=+,\quad B=-.\]The labels $A$ and $B$ are physical choices, not decorative notation.
If a rotation relates $(q_+,q_-)$ to laboratory coordinates $(x,y)$, then the partition $+:-$ need not equal the partition $x:y$. A transformation that mixes the two sides changes what is meant by a subsystem and can change the mutual information. Every calculation must therefore state the partition before quoting a number.
2. Pure states, mixed states, and the partial trace
A normalized pure state is represented by a ket \(\lvert\Psi\rangle\) satisfying
\[\langle\Psi|\Psi\rangle=1.\]Its density operator is
\[\rho_{AB}=|\Psi\rangle\langle\Psi|, \qquad \rho_{AB}^2=\rho_{AB}.\]Observer $A$ does not have access to measurements on $B$. The state available to that observer is obtained by tracing out $B$:
\[\rho_A=\operatorname{Tr}_B\rho_{AB}.\]Similarly, \(\rho_B=\operatorname{Tr}_A\rho_{AB}\). Even when \(\rho_{AB}\) is pure, $\rho_A$ and $\rho_B$ are usually mixed. This mixedness records entanglement between the two modes.
For a positive, unit-trace density matrix $\rho$ with eigenvalues ${\lambda_k}$, the von Neumann entropy is
\[S(\rho)=-\operatorname{Tr}(\rho\ln\rho) =-\sum_k\lambda_k\ln\lambda_k.\]The convention $0\ln0=0$ follows from the limit $\lim_{x\to0^+}x\ln x=0$. A pure state has eigenvalues $(1,0,0,\ldots)$ and therefore zero entropy.
3. Derive mutual information from its definition
For any bipartite state,
\[\boxed{ I(A:B)_\rho =S(\rho_A)+S(\rho_B)-S(\rho_{AB}) }.\]It vanishes for a product state $\rho_{AB}=\rho_A\otimes\rho_B$ and is non-negative for every valid quantum state. It includes both classical and quantum correlations.
Unitary time evolution preserves global purity:
\[\rho_{AB}(t)=U(t)\rho_{AB}(0)U^\dagger(t), \qquad U(t)=e^{-iHt}.\]Thus $S(\rho_{AB}(t))=0$ when the initial state is pure. A Schmidt decomposition,
\[|\Psi(t)\rangle =\sum_k\sqrt{\lambda_k(t)} \,|u_k(t)\rangle_A|v_k(t)\rangle_B,\]shows that $\rho_A$ and $\rho_B$ have the same nonzero eigenvalues. Hence
\[S(\rho_A)=S(\rho_B)\]and the pure-state formula is
\[\boxed{I(A:B)_\rho=2S(\rho_A)}.\]The factor of two is important: the entanglement entropy is $S(\rho_A)$, whereas the mutual information of a globally pure bipartite state is twice that value.
4. Turn a propagated wavefunction into a density spectrum
Expand the state in any orthonormal product basis:
\[|\Psi(t)\rangle =\sum_{r,s=0}^{\infty}c_{rs}(t) |m_+,r\rangle\otimes|m_-,s\rangle.\]Insert this expansion into \(\rho_A=\operatorname{Tr}_B\lvert\Psi\rangle\langle\Psi\rvert\). Orthogonality on $B$ gives
\[\rho_A =\sum_{r,r',s}c_{rs}c_{r's}^* |m_+,r\rangle\langle m_+,r'|.\]If $C=[c_{rs}]$ is the coefficient matrix, this is simply
\[\boxed{\rho_A=CC^\dagger}, \qquad \boxed{\rho_B=C^\dagger C}.\]The nonzero eigenvalues of $CC^\dagger$ and $C^\dagger C$ are the squared singular values of $C$. Therefore a singular-value decomposition
\[C=U\,\operatorname{diag}(s_k)\,V^\dagger\]immediately gives
\[\lambda_k=s_k^2, \qquad \sum_k\lambda_k=1.\]For the propagated pure state,
\[\boxed{ I(A:B)_\rho(t) =-2\sum_k\lambda_k(t)\ln\lambda_k(t) }.\]This formula contains all orders of the state, not only its first and second moments.
Numerically, singular values are preferable to diagonalizing a rounded $CC^\dagger$: they preserve non-negativity more reliably. The identities \(\lVert C\rVert_F^2=1\) and $\sum_k\lambda_k=1$ must hold within the numerical tolerance. A substantial negative density-matrix eigenvalue is not something to clip away; it signals an error in propagation, reshaping, normalization, or the partial trace.
5. What a covariance matrix knows
Collect the four quadratures into
\[\boldsymbol R=(q_+,p_+,q_-,p_-)^T\]and define centered operators
\[\delta R_j=R_j-\langle R_j\rangle.\]The covariance matrix is
\[\sigma_{jk} =\frac12\langle\delta R_j\delta R_k+\delta R_k\delta R_j\rangle.\]It stores means, variances, and pair correlations. It does not store higher moments such as $\langle q^4\rangle$, nor the full shape of a non-Gaussian Wigner function.
For one mode $X$, with
\[\sigma_X= \begin{pmatrix} \langle(\delta q_X)^2\rangle & \tfrac12\langle\{\delta q_X,\delta p_X\}\rangle\\ \tfrac12\langle\{\delta q_X,\delta p_X\}\rangle & \langle(\delta p_X)^2\rangle \end{pmatrix},\]the symplectic eigenvalue is
\[\nu_X=\sqrt{\det\sigma_X}.\]Because $[q_X,p_X]=i$, the uncertainty principle requires
\[\nu_X\ge\frac12.\]If a computation gives $\nu_X<1/2$ by more than rounding error, the state, operator matrices, or covariance convention is inconsistent.
6. The Gaussian entropy function
A Gaussian state is completely determined by its first and second moments. The one-mode Gaussian state having symplectic eigenvalue $\nu$ has entropy
\[h(\nu) =\left(\nu+\frac12\right)\ln\left(\nu+\frac12\right) -\left(\nu-\frac12\right)\ln\left(\nu-\frac12\right).\]Useful checks are
\[h\!\left(\frac12\right)=0, \qquad h(\nu)>0\quad\text{for}\quad\nu>\frac12.\]For a non-Gaussian state, $h(\nu_X)$ is not automatically $S(\rho_X)$. It is the entropy of the Gaussian state with the same local covariance.
7. Two quantities that must not be confused
7.1 Mutual information of the globally matched Gaussian state
Let $\tau_{AB}$ be the Gaussian state with the same global first moments and the same $4\times4$ covariance matrix as $\rho_{AB}$. Its marginals $\tau_A$ and $\tau_B$ match the local covariances of $\rho_A$ and $\rho_B$.
The two global symplectic eigenvalues $\nu_1,\nu_2$ are obtained from the positive eigenvalues of \(\lvert i\Omega\sigma\rvert\), where
\[\Omega= \begin{pmatrix} 0&1&0&0\\ -1&0&0&0\\ 0&0&0&1\\ 0&0&-1&0 \end{pmatrix}.\]Then
\[S(\tau_{AB})=h(\nu_1)+h(\nu_2)\]and
\[\boxed{ I(A:B)_\tau =h(\nu_A)+h(\nu_B)-h(\nu_1)-h(\nu_2) }.\]Even if $\rho_{AB}$ is pure, its covariance-matched Gaussian state $\tau_{AB}$ can be mixed. Consequently,
\[I(A:B)_\tau\ne2h(\nu_A)\]in general.
7.2 Local-covariance maximum-entropy bound
Among all states with a fixed covariance, the Gaussian state has the largest entropy. Therefore
\[S(\rho_A)\le S(\tau_A)=h(\nu_A).\]For a globally pure state,
\[I(A:B)_\rho=2S(\rho_A),\]so
\[\boxed{I(A:B)_\rho\le2h(\nu_A)}.\]The right-hand side is a local-covariance upper bound. It is not generally the exact mutual information and is not generally $I(A:B)_\tau$.
8. Why the gap measures non-Gaussianity
The quantum relative entropy is
\[D(\rho\Vert\tau) =\operatorname{Tr}\!\left[\rho(\ln\rho-\ln\tau)\right].\]When $\tau$ is the Gaussian state matching the first and second moments of $\rho$, $\ln\tau$ is at most quadratic in the quadratures. Moment matching therefore gives
\[\operatorname{Tr}(\rho\ln\tau) =\operatorname{Tr}(\tau\ln\tau).\]It follows that
\[\delta_{\mathrm{NG}}(\rho) \equiv D(\rho\Vert\tau) =S(\tau)-S(\rho)\ge0.\]Apply this relation to $AB$, $A$, and $B$:
\[S(\rho_X)=S(\tau_X)-\delta_{\mathrm{NG}}(\rho_X).\]Substitution into the definition of mutual information gives the exact identity
\[\boxed{ I(A:B)_\rho =I(A:B)_\tau +\delta_{\mathrm{NG}}(\rho_{AB}) -\delta_{\mathrm{NG}}(\rho_A) -\delta_{\mathrm{NG}}(\rho_B) }.\]For a general mixed state, the correction has no fixed sign. A covariance-only reconstruction may overestimate or underestimate the true mutual information.
For a globally pure state, the local form is especially transparent:
\[S(\rho_A)=h(\nu_A)-\delta_{\mathrm{NG}}(\rho_A).\]Therefore
\[\boxed{ 2h(\nu_A)-I(A:B)_\rho =2\delta_{\mathrm{NG}}(\rho_A) =2D(\rho_A\Vert\tau_A) }.\]This proves exactly what the covariance bound misses: the gap is twice the non-Gaussianity of the reduced state.
The algebraic step can be checked directly in Maxima without inserting any model-dependent numbers:
/* S(rho_X) = S(tau_X) - delta_X */
SrA : StA - dA$
SrB : StB - dB$
SrAB : StAB - dAB$
Irho : SrA + SrB - SrAB$
Itau : StA + StB - StAB$
ratsimp(Irho - (Itau + dAB - dA - dB));
/* 0 */
/* Pure global state: SrAB=0 and SrA=SrB. */
Ipure : 2 * SrA$
ratsimp(2 * StA - Ipure - 2 * dA);
/* 0 */
9. A zero-correlation example that catches the common mistake
Suppose the initial state is a product of two pure non-Gaussian factors:
\[|\Psi(0)\rangle=|\phi_+\rangle\otimes|\phi_-\rangle.\]Then
\[\rho_A(0)=|\phi_+\rangle\langle\phi_+|, \qquad S(\rho_A(0))=0,\]and hence
\[I(A:B)_\rho(0)=0.\]However, a pure non-Gaussian one-mode state may have $\det\sigma_A>1/4$, so $\nu_A>1/2$ and
\[2h(\nu_A)>0.\]There is no contradiction. The exact mutual information is zero because the modes are uncorrelated. The positive covariance bound is the entropy of a different state—a mixed Gaussian state having the same covariance as the pure non-Gaussian factor.
This is a useful unit test: any method that reports nonzero exact mutual information for a normalized product input is computing the wrong quantity or using the wrong subsystem partition.
10. Add the information layer to the propagated state
The propagation procedure and its cutoff diagnostics were constructed in the preceding guide. Its output at each time is a normalized coefficient array $C_N(t)$ together with the one-mode quadrature matrices. The singular values of $C_N$ give the exact finite-cutoff information
\[I_N(t)=-2\sum_k s_k^2(t)\ln s_k^2(t).\]Independently, the quadrature matrices give $\sigma_A$, then $\nu_A=\sqrt{\det\sigma_A}$ and the bound $2h(\nu_A)$. These two calculations must remain separate: one uses the complete coefficient matrix and the other discards everything beyond second moments.
For a square cutoff, the boundary population
\[P_\partial(t)= \sum_{\substack{r=N-1\\\text{or }s=N-1}}|c_{rs}(t)|^2\]helps expose probability reaching the truncation edge. The information curve, the covariance bound, and the low moments must also be compared at increasing $N$.
Norm preservation alone is not a convergence test: unitary evolution inside an inadequate truncated space preserves the norm perfectly.
11. Worked benchmark and interpretation
Consider
\[g=0,\qquad m_+=m_-=2,\qquad \ell_+=\ell_-=0,\]with
\[\Omega_+=\frac54,\qquad \Omega_-=\frac{17}{20},\qquad \omega_c=\frac34,\qquad \omega_0=1.\]The bare-confinement margins are positive:
\[\Omega_+^2-\omega_c^2=1, \qquad \Omega_-^2-\omega_c^2=\frac4{25}.\]For this example, one-mode matrix elements were evaluated with 360-point Gauss-Hermite quadrature. The projected Hamiltonian was propagated on $0\le t\le12$ using square cutoffs $N=30,34,38,42$. Differences below are maximum absolute differences from the $N=42$ curve over the whole interval:
| $N$ | \(\max\lvert\Delta I\rvert\) | \(\max\lvert\Delta[2h(\nu_A)]\rvert\) | $\max P_\partial$ |
|---|---|---|---|
| 30 | $3.49\times10^{-4}$ | $1.74\times10^{-3}$ | $1.01\times10^{-6}$ |
| 34 | $2.39\times10^{-4}$ | $1.20\times10^{-3}$ | $5.15\times10^{-7}$ |
| 38 | $1.58\times10^{-4}$ | $1.34\times10^{-3}$ | $1.59\times10^{-7}$ |
| 42 | numerical reference | numerical reference | $1.04\times10^{-7}$ |
The covariance-bound differences are not perfectly monotonic. Reporting them is more honest than inferring convergence from a small norm error. At $N=42$, the overlap-matrix error was $1.71\times10^{-12}$ and the maximum norm error was $5.77\times10^{-15}$; neither number alone certifies observable convergence.
At $t=0$,
\[I(A:B)_\rho\simeq2.2\times10^{-15},\]which is numerical zero, while
\[2h(\nu_A)=0.174840637013\ldots\]and
\[\delta_{\mathrm{NG}}(\rho_A) =0.0874203185065\ldots.\]Thus
\[2h(\nu_A)-I=2\delta_{\mathrm{NG}}(\rho_A)\]already holds at the product-state starting point.
Over the stated time window, the $N=42$ calculation gives
\[\max_t I(A:B)_\rho(t)=0.0433166\]near $t=9.188$, while
\[0.173412\lesssim2h(\nu_A)\lesssim0.230489.\]The reduced non-Gaussianity lies approximately between $0.079683$ and $0.102965$ nats. The maximum numerical residual in
\[2h(\nu_A)-I-2\delta_{\mathrm{NG}}(\rho_A)\]is about $10^{-13}$. This is a strong internal consistency check, not a proof of infinite-cutoff convergence.
12. Conceptual result
The reduced density spectrum is the exact source of two-mode mutual information. For a pure state,
\[I(A:B)=2S(\rho_A).\]A covariance matrix has a narrower role. It defines a globally matched Gaussian comparison state and a local maximum-entropy bound, but those are different objects. The precise loss of information in the local covariance description is
\[2h(\nu_A)-I(A:B)=2\delta_{\mathrm{NG}}(\rho_A).\]In a truncated calculation these identities are exact for the propagated finite-dimensional state. Their interpretation as statements about the original continuous system depends on the convergence analysis carried forward from the spectral step.
Discussion