Sum Tomographic Entropy in Quantum and Imaging Tomography

• Sum tomographic entropy is defined as the sum of Shannon differential entropies computed from tomograms derived from measurement data, illustrating uncertainty in quantum and imaging contexts.
• It underpins entropic uncertainty relations, where canonical sums like S(θ)+S(θ+π/2) obey universal lower bounds, and is applicable to both continuous-variable and spin quantum tomography.
• In imaging inverse problems, the entropy sum serves as a regularizer that stabilizes reconstructions by reducing noise-driven artifacts and enforcing smooth, positive solutions.

Searching arXiv for the cited works and closely related papers on tomographic entropy and entropic inequalities. Sum tomographic entropy denotes a family of entropy constructions built from tomograms, i.e., probability distributions obtained directly from measurement data or from Radon-type transforms. In the tomographic-probability representation of quantum mechanics, its canonical form is the sum of Shannon differential entropies of conjugate tomographic slices, such as $S(\theta)+S(\theta+\pi/2)$ for rotated quadratures, with a universal lower bound fixed by entropic uncertainty relations. In spin and qubit tomography, the same expression also denotes sums of marginal tomographic entropies and entropy combinations constrained by subadditivity or strong subadditivity. In imaging inverse problems, by contrast, it refers to sums of pixelwise entropy penalties used as regularizers. These usages are related by their reliance on tomographic probability data, but they are not identical definitions (Man'ko et al., 2012, Kiktenko et al., 2013, Chernega et al., 2015, Prakash et al., 2017, Bao et al., 2020, Paul et al., 2023).

1. Definitions in tomographic representations

In continuous-variable quantum tomography, the basic tomograms are the optical tomogram

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$

and the symplectic tomogram

$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$

Both are normalized probability densities in $X$, and they are related by

$w(X,\theta)=M(X,\cos\theta,\sin\theta),$

together with the homogeneity relation

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$

For a qudit of spin $j$, the unitary spin tomogram is

$$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$

with $\sum_m w(m,u)=1$ for any unitary $u$ (Man'ko et al., 2012).

The corresponding tomographic entropies are Shannon entropies of these tomograms. For optical and symplectic tomography,

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$0

while for spin tomography,

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$1

Symplectic entropy inherits the tomogram homogeneity,

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$2

This scaling law is central to generalized entropy-sum bounds (Man'ko et al., 2012).

The literature uses the expression “sum tomographic entropy” in several distinct but structurally related senses:

Context	Tomographic object	Sum construction
CV optical/symplectic tomography	$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$3, $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$4	$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$5, $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$6
Two-qubit tomography	$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$7	$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$8
Single-qudit spin tomography	$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$9 with artificial partitions	sums constrained by subadditivity and SSA
TDLAS tomography	two-line absorbances $$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$0	$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$1
Optoacoustic tomography	image $$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$2	$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$3

The first row is the standard meaning in the tomographic-probability representation of quantum mechanics; the others are context-specific extensions or reuses of the term (Man'ko et al., 2012, Kiktenko et al., 2013, Chernega et al., 2015, Bao et al., 2020, Prakash et al., 2017).

2. Entropic sums for conjugate tomograms

The canonical sum tomographic entropy in continuous-variable systems is the sum of the entropies of two tomographic slices corresponding to conjugate quadratures. For optical tomograms, the entropic uncertainty relation is

$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$4

This is equivalent to the Hirschman–Bialynicki-Birula–Beckner relation in the wavefunction representation,

$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$5

Accordingly, the tomographic entropy sum encodes the same uncertainty tradeoff as the position-momentum entropic bound, but directly at the level of experimentally accessible tomograms (Man'ko et al., 2012).

Gaussian states saturate this bound. For a Gaussian quadrature distribution with variance $$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$6,

$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$7

and if the conjugate variance obeys

$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$8

then

$$M(X,\mu,\nu)=\left\langle \delta\!\left(X-\mu\hat q-\nu\hat p\right)\right\rangle.$$9

This includes coherent states and squeezed states. For a single-mode squeezed vacuum with squeezing parameter $X$0 and angle $X$1,

$X$2

and the conjugate pair still satisfies

$X$3

so the entropy sum is independent of $X$4 and again equals $X$5 (Man'ko et al., 2012).

For non-Gaussian states the inequality is generally strict. For the Fock state $X$6,

$X$7

which is independent of $X$8. Hence

$X$9

and the inequality holds strictly for $w(X,\theta)=M(X,\cos\theta,\sin\theta),$0 except for the Gaussian case $w(X,\theta)=M(X,\cos\theta,\sin\theta),$1 (Man'ko et al., 2012).

The symplectic counterpart follows from the relation between optical and symplectic tomograms together with entropy homogeneity. For normalized parameters $w(X,\theta)=M(X,\cos\theta,\sin\theta),$2,

$w(X,\theta)=M(X,\cos\theta,\sin\theta),$3

For general scaling,

$w(X,\theta)=M(X,\cos\theta,\sin\theta),$4

The additional logarithm is the direct entropy contribution of the symplectic rescaling (Man'ko et al., 2012).

A bipartite continuous-variable generalization appears in the form

$w(X,\theta)=M(X,\cos\theta,\sin\theta),$5

with the lower bound

$w(X,\theta)=M(X,\cos\theta,\sin\theta),$6

For a single mode, this reduces to $w(X,\theta)=M(X,\cos\theta,\sin\theta),$7, which is identical to $w(X,\theta)=M(X,\cos\theta,\sin\theta),$8 (Paul et al., 2023).

3. Family viewpoint, joint-distribution viewpoint, and the universal integral inequality

A central interpretational issue is whether a tomogram is regarded as a family of probability distributions indexed by an external parameter or as a single joint probability distribution of a random variable and a random parameter. In the first viewpoint, one works with $w(X,\theta)=M(X,\cos\theta,\sin\theta),$9 and defines

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$0

Then sums such as $$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$1 quantify uncertainty tradeoffs between conjugate settings; $$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$2 is the standard example (Man'ko et al., 2012).

In the second viewpoint, one introduces a normalized parameter distribution $$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$3 and forms a modified joint tomogram

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$4

with

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$5

The associated joint entropy is

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$6

For optical and symplectic tomograms this yields the modified entropies

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$7

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$8

The family viewpoint is the natural setting for entropy sums at fixed parameters, while the joint-distribution viewpoint is the natural setting for conditional entropies, subadditivity, and parameter-averaged inequalities (Man'ko et al., 2012).

From the optical entropic uncertainty relation and the explicit fractional Fourier transform kernel for pure-state tomograms,

$$M(\lambda X,\lambda\mu,\lambda\nu)=|\lambda|^{-1}M(X,\mu,\nu).$$9

one obtains a universal integral inequality valid for any normalized wavefunction $j$0: $j$1 This inequality is the zero Fourier component, in $j$2, of the entropic uncertainty relation. A plausible implication is that the tomographic formulation exposes an integral entropy constraint that is less transparent in the wavefunction representation (Man'ko et al., 2012).

4. Spin, qudit, and two-qubit entropy sums

In spin tomography, the principal entropy object is the discrete Shannon entropy

$j$3

or $j$4 for $j$5 rotations parametrized by a unit vector $j$6. For modified spin tomograms of the form

$j$7

the tomographic framework supports subadditivity and strong subadditivity statements formulated on the joint distribution over outcomes and settings. For two qudits, with

$j$8

the strong subadditivity relation is

$j$9

The same paper notes that Maassen–Uffink-type bounds for pairs of measurement bases are standard in the literature, but are not derived there; the emphasis is on subadditivity structures within tomography (Man'ko et al., 2012).

For single-qudit spin tomograms, a distinct construction uses artificial bipartitions or tripartitions of the tomographic outcome space. If $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$0 is written as $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$1 or $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$2, a bijection between the single outcome index and a multi-index creates “portrait” subsystems. The tomogram

$$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$3

then generates marginal probability vectors through stochastic maps. In this setting the tomograms satisfy a no signaling property under factorized unitaries, and the tomographic Tsallis $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$4-entropy

$$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$5

satisfies strong subadditivity for $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$6: $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$7 In the limit $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$8, this recovers the Shannon/von Neumann form (Chernega et al., 2015).

Two-qubit tomography introduces yet another sum construction. For local unitaries $$w(m,u)=\langle m|\,u\hat\rho\,u^\dagger\,|m\rangle,\qquad m=-j,-j+1,\dots,j,$$9 and $\sum_m w(m,u)=1$0, the tomographic probabilities are

$\sum_m w(m,u)=1$1

with marginal tomographic entropies

$\sum_m w(m,u)=1$2

and joint entropy

$\sum_m w(m,u)=1$3

The sum tomographic entropy is

$\sum_m w(m,u)=1$4

and it obeys Shannon subadditivity,

$\sum_m w(m,u)=1$5

Its gap from the joint entropy is the tomographic mutual information,

$\sum_m w(m,u)=1$6

This quantity underlies tomographic discord and asymmetry measures in two-qubit causal analysis (Kiktenko et al., 2013).

5. Sum entropic uncertainties and tomographic entanglement indicators

For bipartite continuous-variable systems, the tomographic program extends from uncertainty relations to entanglement diagnostics computable directly from homodyne data. The two-mode tomogram is

$\sum_m w(m,u)=1$7

with reduced tomograms obtained by marginalization. The corresponding tomographic entropies are

$\sum_m w(m,u)=1$8

$\sum_m w(m,u)=1$9

The entropic sum uncertainty is defined as

$u$0

and is computed directly from tomograms, without density-matrix reconstruction (Paul et al., 2023).

Two tomographic entanglement indicators are compared against this entropic sum. The entropy-based indicator is

$u$1

and the inverse-participation-ratio-based indicator is

$u$2

where

$u$3

Averaged versions,

$u$4

are formed by averaging over a quorum of slices (Paul et al., 2023).

The reported dynamical comparisons are model-dependent. In pure bipartite atom-field cases with weak nonlinearity, both tomographic entanglement indicators can show extrema where the entropic uncertainty relation does, but the detailed tracking depends on the time window. In strong-nonlinearity Fock-initial cases, $u$5 can reflect the entropic sum uncertainty well for large photon numbers. With a coherent-state field and weak nonlinearity, $u$6 generally tracks $u$7 better than $u$8. In a tripartite $u$9-atom model where the reduced bipartite field state is mixed, $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$00 is reported as the most robust tracker of entropic sum uncertainties, especially through collapse dynamics. Even the summed slice indicator

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$01

closely follows $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$02 in the collapse interval (Paul et al., 2023).

This literature places sum tomographic entropy at the interface of uncertainty quantification and entanglement diagnostics. A plausible implication is that the angular dependence of tomographic quantities, rather than full state reconstruction, is the relevant object when comparing dynamical uncertainty with accessible correlation structure.

6. Entropy-sum regularization in tomographic inverse problems

Outside quantum tomographic-probability theory, the expression “sum tomographic entropy” is used for entropy regularizers summed over image elements. In tunable diode laser absorption spectroscopy tomography, the RETRO method reconstructs two absorbance fields $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$03 by minimizing

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$04

subject to $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$05 and $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$06. The regularizer is a sum over pixels of a coupled relative-entropy term acting directly on the ratio $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$07. The paper explicitly distinguishes this from a sum of independent Kullback–Leibler divergences per line. In conic form, the convex function

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$08

is represented באמצעות the scalar relative-entropy cone

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$09

and the optimization is solved by interior-point methods. The operational role of the summed entropy term is to suppress noise-driven spikes in the two-line absorbance ratio, which controls the recovered temperature (Bao et al., 2020).

In optoacoustic tomography, the entropy sum enters as a maximum-entropy prior over the image. The reconstruction solves

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$10

with $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$11 in the reported experiments, so the entropy prior reduces to $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$12 up to a constant. The associated “sum tomographic entropy” is

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$13

Its gradient,

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$14

and Hessian,

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$15

show that the objective is strictly convex in the positive orthant. Here the entropy sum is not an uncertainty relation but a regularizer that enforces positivity implicitly and reduces negative artifacts in reconstructed images (Prakash et al., 2017).

These inverse-problem usages are conceptually distinct from the Shannon entropy sums of conjugate quantum tomograms. The common element is formal rather than interpretational: both are entropy functionals built from tomographic data and summed over a natural tomographic index, whether that index is a quadrature angle, a local measurement basis, or a pixel.

Practical computation of tomographic entropy sums depends on context. For optical homodyne data, one acquires quadrature samples at fixed $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$16 and $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$17, estimates $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$18 by histogramming or kernel density estimation, applies bias corrections for differential entropy, and computes

$$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$19

numerically. For bipartite CV systems, synchronized homodyne measurements yield joint tomograms from which $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$20, $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$21, and $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$22 are formed. For spin tomography, repeated measurements after unitary rotations provide empirical $$w(X,\theta)=\left\langle \delta\!\left(X-\hat q\cos\theta-\hat p\sin\theta\right)\right\rangle =\int W(q,p)\,\delta\!\left(X-q\cos\theta-p\sin\theta\right)\,\frac{dq\,dp}{2\pi},$$23, from which discrete Shannon or Tsallis tomographic entropies are computed. Differential-entropy estimates remain sensitive to binning and coarse-graining, and coarse-graining increases entropies and may loosen uncertainty bounds (Man'ko et al., 2012, Paul et al., 2023).

Across these domains, sum tomographic entropy functions as a compact descriptor of uncertainty, correlation, or regularization structure directly at the tomographic level. In continuous-variable quantum mechanics it is a sharp entropic uncertainty measure saturated by Gaussian states; in spin and qubit tomography it organizes subadditivity, no signaling, mutual information, and discord; and in imaging inverse problems it becomes a summed entropy prior that stabilizes reconstruction under ill-posedness and noise.