Scale-Space Entanglement and Factorization Caveats

Momentum bands, wavelet scales, and spatial regions define different subsystem questions. A scale-space entropy is meaningful only after a regulator supplies canonical mode algebras—or an operational algebra selected by a detector. It is generally basis dependent, can conflict with gauge constraints, and should not be identified with spatial entanglement or Wilsonian information loss without an explicit map.

Required background. Modes, regions, and factorization scales supplies the scale decompositions; separating scales distinguishes the limits that those decompositions probe.

Helpful background. Factorization failure and local algebras explains why continuum spatial subsystems need not be literal Hilbert factors.

The chapter overview separates the roles of its scales, places scale-space entropy in a comparison table, and lists the independent validity gates. Here the extra gate is the proposed subsystem itself: which canonical variables generate the retained algebra?

Canonical modes, not just basis labels

For $N$ regulated bosonic modes, a real linear transformation

$$Q=Wq, \qquad P=Wp$$

defines new canonical modes when $W W^{\mathsf T}=I$. Acting with the same orthogonal matrix on coordinates and momenta preserves $[Q_j,P_k]=i\delta_{jk}$. A split of the resulting mode algebras can then define a tensor factorization in this finite regulator.

That statement is more restrictive than “choose a basis.” A unitary change of coordinates within an already chosen low factor and within its high complement preserves the bipartition. A canonical transformation that mixes the two sides defines a new pair of commuting subalgebras and can change their entanglement. A general noncanonical transform does not even define oscillator subsystems.

For a real lattice field, real sine and cosine normal coordinates are often cleaner than treating complex Fourier coefficients as independent: the reality constraint relates the $k$ and $-k$ coefficients. In a free vacuum the normal-mode oscillators form a product, so momentum-band entropy can vanish while spatial entropy is nonzero. Interactions can entangle modes, but the answer still depends on the declared variables and band edge. Perturbative examples and their explicit mode-split dependence appear in Balasubramanian, McDermott, and Van Raamsdonk 2012, §§ II–IV.

Wavelets replace global momentum modes by variables localized jointly in position and scale. Orthogonal wavelet filters give canonical mode maps in the regulated bosonic problem, but the choice of filter is physical only if it models a preparation, detector, or coarse-graining architecture. The exact wavelet/MERA relation in Evenbly and White 2016, Eqs. (1)–(8) is a construction for particular free-fermion circuits, not a theorem that every wavelet entropy is universal. Likewise, the continuous real-space entanglement-renormalization ansatz of Haegeman et al. 2013, Eqs. (2)–(9) supplies a chosen variational architecture, not a canonical factorization for every QFT.

Two oscillators expose basis dependence

Start with the product ground state of two oscillators of frequencies $\omega_1=1$ and $\omega_2=4$ in units with $\hbar=1$. Its nonzero covariances are

$$\langle q_j^2\rangle=\frac{1}{2\omega_j}, \qquad \langle p_j^2\rangle=\frac{\omega_j}{2}.$$

Rotate both $q$ and $p$ through the same angle,

$$\begin{pmatrix}Q_1\\Q_2\end{pmatrix} =R(\theta)\begin{pmatrix}q_1\\q_2\end{pmatrix}, \qquad \begin{pmatrix}P_1\\P_2\end{pmatrix} =R(\theta)\begin{pmatrix}p_1\\p_2\end{pmatrix}.$$

The reduced covariance of the first rotated mode has symplectic eigenvalue

$$\nu_1(\theta)=\frac12\sqrt{ \left(\frac{\cos^2\theta}{\omega_1}+ \frac{\sin^2\theta}{\omega_2}\right) \left(\omega_1\cos^2\theta+ \omega_2\sin^2\theta\right)}.$$

Its entropy in nats is

$$s(\nu)=\left(\nu+\frac12\right)\log\left(\nu+\frac12\right) -\left(\nu-\frac12\right)\log\left(\nu-\frac12\right).$$

Rotation	$\nu_1$	Entropy $s(\nu_1)$	Interpretation
$\theta=0$	$0.50000000$	$0$	The original normal-mode factorization.
$\theta=\pi/8$	$0.56596157$	$0.24741932$	The new factors mix unequal frequencies.
$\theta=\pi/4$	$0.62500000$	$0.39243611$	The same pure state is entangled across the rotated split.

The total state and its full symplectic spectrum have not changed. The entropy changed because the subsystem algebras changed. Equal frequencies provide the control: when $\omega_1=\omega_2$, the covariance is rotation invariant and $\nu_1=1/2$ for every $\theta$.

Fourier and Haar reconstruction of one covariance

The first application compares two exact mode maps on the same regulated state. Consider six sites with periodic boundary conditions,

$$H=\frac12\sum_{j=0}^{5}\left[ p_j^2+m^2q_j^2+(q_{j+1}-q_j)^2 \right], \qquad m=0.5,$$

where $q_6=q_0$ and $[q_j,p_k]=i\delta_{jk}$. Its frequencies are

$$\omega_k=\sqrt{m^2+4\sin^2(k/2)}, \qquad k=\frac{2\pi n}{6},\qquad n=0,\ldots,5,$$

and the ground-state covariances are

$$X_{ij}=\frac1{12}\sum_k\frac{\cos[k(i-j)]}{\omega_k}, \qquad P_{ij}=\frac1{12}\sum_k\omega_k\cos[k(i-j)].$$

There are no $q$–$p$ correlations. We compare:

• the real Fourier matrix $F$, retaining the constant mode and the sine/cosine pair at $k=2\pi/6$;
• the one-level Haar matrix $H_{\rm Haar}$, retaining the three pair averages $Q_j=(q_{2j}+q_{2j+1})/\sqrt2$ and discarding the three pair differences.

Both transforms act identically on $q$ and $p$. The covariance is transformed as $X'=W XW^{\mathsf T}$, $P'=W PW^{\mathsf T}$ and reconstructed as $X=W^{\mathsf T}X'W$, $P=W^{\mathsf T}P'W$. The following double-precision table is obtained from the displayed finite sums; “reconstruction” is the largest absolute entrywise residual in $X$ or $P$.

Mode map and retained algebra	$\lVert WW^{\mathsf T}-I\rVert_{\max}$	Reconstruction residual	Largest full-spectrum $\lvert\nu_j-1/2\rvert$	Retained symplectic eigenvalues	Entropy (nats)
Real Fourier; three lowest normal coordinates	$3.33\times10^{-16}$	$7.77\times10^{-16}$	$4.44\times10^{-16}$	$0.5,0.5,0.5$	$0$
One-level Haar; three pair averages	$2.22\times10^{-16}$	$3.33\times10^{-16}$	$3.33\times10^{-16}$	$0.5,0.51078793,0.51078793$	$0.11941594$

This table provides three independent controls. Orthogonality checks that the variables are canonical; reconstruction checks that both descriptions encode the same covariance; and the full symplectic spectrum checks that neither transform changed the global pure state. Only the retained subalgebra differs. The Fourier vacuum factorizes over exact normal modes, whereas pair averages and differences are slightly entangled.

Claim	Typed input	Licensed conclusion	Failure mode
Fourier-band entropy	Regulated canonical normal modes plus a declared band	Entanglement across that mode algebra	Not spatial entropy; reality and degeneracies must be handled.
Haar scale entropy	Orthogonal filter bank plus retained/detail split	Entanglement across that multiresolution algebra	Filter dependent; compact support broadens momentum response.
Detector-defined scale entropy	Smeared observable algebra and state	Operationally accessible correlations	Depends on aperture, time window, and response function.
Wilsonian shell integration	Action, cutoff, and matching prescription	Effective dynamics for retained variables	No state-channel or entropy interpretation without extra structure.

Locality, gauge constraints, and continuum limits

A compactly supported wavelet has broad momentum response, so it cannot implement a perfectly sharp $k>\Lambda$ trace. Conversely, a sharp Fourier shell is spatially nonlocal. Their finite entropies answer different questions even before taking a continuum limit.

Gauge-invariant observables add another obstruction. Gauss law couples would-be factors, and choices of center or edge algebra change the entropy. The algebra must be fixed before a trace or entropy is defined; imposing constraints afterward can count gauge redundancy as information. Casini, Huerta, and Rosabal 2014, §§ II–IV exhibit this algebra dependence explicitly in lattice gauge theories.

Continuum scale-space statements therefore need a convergent family of regulated algebras and observables. Agreement of one number at one lattice spacing is not enough. One should vary lattice size, boundary conditions, filter family, and band edge while holding the operational response fixed. No general basis-independent theorem identifies momentum- or wavelet-space entropy with a spatial entropic $c$-, $F$-, or $a$-function.

Common pitfalls

Calling a mode rotation harmless. It is harmless for the global state, but it changes entanglement if it mixes the two proposed factors.

Calling momentum-shell entropy spatial entropy. These bipartitions have different algebras and can disagree even in a free vacuum.

Imposing gauge constraints after tracing. Define the physical observable algebra, its center, and any edge prescription first.

Exercises

1. Entropy created by a canonical rotation

Derive $\nu_1(\theta)$ for the two-oscillator example. Evaluate it at $\theta=\pi/4$ for $\omega_1=1$, $\omega_2=4$, and explain why the answer returns to $1/2$ when the frequencies coincide.

Solution

The first rotated coordinate is $Q_1=q_1\cos\theta+q_2\sin\theta$, so independence of the original modes gives

$$\langle Q_1^2\rangle =\frac12\left(\frac{\cos^2\theta}{\omega_1} +\frac{\sin^2\theta}{\omega_2}\right).$$

Similarly,

$$\langle P_1^2\rangle =\frac12\left(\omega_1\cos^2\theta +\omega_2\sin^2\theta\right),$$

and $\langle\{Q_1,P_1\}\rangle/2=0$. Therefore $\nu_1=\sqrt{\langle Q_1^2\rangle\langle P_1^2\rangle}$, which is the stated formula. At $\theta=\pi/4$,

$$\nu_1=\frac14\sqrt{(1+1/4)(1+4)}=\frac58=0.625,$$

giving $s=0.39243611$ nats. If $\omega_1=\omega_2=\omega$, the two parentheses multiply to $1$, so $\nu_1=1/2$ for every angle. The state is then invariant under the rotation.

2. Why the Fourier split has zero entropy

Diagonalize the six-site Hamiltonian in real Fourier coordinates and show that tracing any collection of complete normal coordinates from its ground state leaves a pure product state on the rest.

Solution

The real Fourier transform is orthogonal and diagonalizes the circulant stiffness matrix. The Hamiltonian becomes

$$H=\frac12\sum_r\left(P_r^2+\omega_r^2Q_r^2\right),$$

where the sine and cosine coordinates at the same nonzero momentum have the same frequency but are independent oscillators. Its ground state is therefore

$$\lvert\Omega\rangle=\bigotimes_r\lvert0_{\omega_r}\rangle.$$

Tracing any subset removes product factors and leaves a tensor product of pure vacua. Equivalently, every retained symplectic eigenvalue is $1/2$, so the Gaussian entropy is zero. One must use independent real coordinates—or impose the complex-field reality relation—rather than double count $k$ and $-k$.

3. Which transformations preserve a fixed bipartition?

Let the one-particle mode space be ${\cal K}_L\oplus{\cal K}_H$. Show that an orthogonal canonical transform preserves the corresponding low/high algebras exactly when its matrix is block diagonal (up to a swap of equally labelled whole factors). Relate this to the two-oscillator adversarial test.

Solution

A low canonical generator after the transform is a linear combination of old $q$ and $p$ generators with coefficients from the low rows of $W$. It belongs to the old low algebra only if those rows have no high components. Applying the same reasoning to the inverse requires the high rows to have no low components. Thus

$$W=\begin{pmatrix}W_L&0\\0&W_H\end{pmatrix}$$

relative to ${\cal K}_L\oplus{\cal K}_H$. Such a transform merely changes coordinates within each factor and leaves its entropy invariant. The rotation $R(\theta)$ has off-diagonal entries whenever $\sin\theta\ne0$, so it defines new factors; the nonzero entropies in the benchmark are therefore allowed, not a contradiction with unitary invariance.

References

• Balasubramanian, Vijay, Michael B. McDermott, and Mark Van Raamsdonk. “Momentum-Space Entanglement and Renormalization in Quantum Field Theory.” Physical Review D 86 (2012): 045014. DOI.
• Casini, Horacio, Marina Huerta, and José Alejandro Rosabal. “Remarks on Entanglement Entropy for Gauge Fields.” Physical Review D 89 (2014): 085012. DOI.
• Evenbly, Glen, and Steven R. White. “Entanglement Renormalization and Wavelets.” Physical Review Letters 116 (2016): 140403. DOI.
• Haegeman, Jutho, Tobias J. Osborne, Henri Verschelde, and Frank Verstraete. “Entanglement Renormalization for Quantum Fields in Real Space.” Physical Review Letters 110 (2013): 100402. DOI.