Bose-Einstein Quantum Relative Entropy

• Bose-Einstein quantum relative entropy is a family of entropy measures in bosonic settings that vary by framework and incorporate Bose-Einstein statistics.
• It encompasses diverse formulations—including Bregman divergences for unbounded positive operators, Umegaki entropy for Gaussian states, and Araki-Uhlmann entropy in QFT—with unique thermodynamic and algebraic implications.
• The constructions address optimization challenges and physical interpretations, establishing properties like restricted monotonicity and energy flux relations through bosonic coherent states.

“Bose-Einstein quantum relative entropy” designates a family of relative-entropy constructions that arise in bosonic settings rather than a single universally fixed functional. Recent arXiv work uses the phrase in at least three distinct ways: as the Bregman divergence generated by the negative Bose-Einstein entropy on the unbounded positive semidefinite cone, as standard Umegaki relative entropy evaluated on bosonic Gaussian or bosonic many-body states, and as Araki-type relative entropy in bosonic algebraic quantum field theory (Minervini et al., 26 May 2026, Parthasarathy, 2021, Guo et al., 9 May 2026, Dorau et al., 28 Oct 2025). The common thread is the bosonic structure—occupation numbers, CCR/Weyl algebras, Gaussian or coherent states, or Bose-Einstein kinetic statistics—while the precise entropy notion depends on the ambient framework.

1. Terminological scope and principal formulations

The recent literature separates several non-equivalent objects that may all be described as bosonic or Bose-Einstein relative entropies. In semidefinite optimization over the unbounded positive semidefinite cone, the relevant object is the Bose-Einstein quantum relative entropy

$D_{\mathrm{BE}(X\|Y) \coloneqq -S_{\mathrm{BE}(X) + \Tr[(X+I)\ln(Y+I) - X\ln Y],$

generated by the negative Bose-Einstein entropy

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$

on positive semidefinite operators (Minervini et al., 26 May 2026).

In finite-mode bosonic continuous-variable systems, the relative entropy of two $n$-mode Gaussian states is instead taken to be the standard Umegaki quantity

$$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$

evaluated on boson Fock space $\Gamma(\mathbb C^n)$ and expressed through Gaussian normal forms, transformed means, and one-mode covariance blocks (Parthasarathy, 2021).

In bosonic mean-field dynamics, the same Umegaki entropy is used for symmetric $N$-body density matrices and a factorized Hartree reference state: $S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$ The bosonic content there comes from permutation symmetry and comparison with $\gamma_t^{\otimes N}$, not from a modified bosonic entropy formula (Guo et al., 9 May 2026).

In algebraic quantum field theory for free scalar bosons, the entropy is the relative (Araki-Uhlmann) entropy associated with type III local algebras. For coherent horizon excitations, it is computed through modular theory rather than through von Neumann entropy (Dorau et al., 28 Oct 2025). Closely related work on the Rindler wedge computes the Araki relative entropy between the vacuum and a coherent state and identifies it with the entropy of the underlying classical Klein-Gordon wave (Ciolli et al., 2019).

At the kinetic level, Bose-Einstein relative entropy appears as the Bregman divergence associated with the scalar entropy density

$\gamma(f)=f\log f-(1+f)\log(1+f),$

for a phase-space density $f$, rather than as an operator-theoretic quantum-information quantity (Alexandre et al., 2013).

Setting	Relative-entropy notion	Bosonic content
Unbounded PSD cone	$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$0	Bose-Einstein entropy and occupation numbers
Boson Fock Gaussian states	Umegaki $S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$1	Gaussian bosonic modes
Symmetric many-body mean field	Umegaki $S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$2	Permutation-symmetric bosons
Algebraic QFT	Araki-Uhlmann relative entropy	CCR/Weyl algebra, coherent states
Bose kinetic theory	Bregman divergence of $S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$3	Bose-Einstein statistics

This variety suggests that the phrase is framework-dependent rather than canonical.

2. Bose-Einstein entropy as a Bregman generator on the unbounded positive semidefinite cone

The most explicit recent use of the phrase appears in “Bose-Einstein thermal operators for semidefinite optimization” (Minervini et al., 26 May 2026). There the scalar bosonic entropy is

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$4

and its operator extension is

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$5

The generator is the convex trace functional

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$6

with

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$7

The associated Bregman divergence is

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$8

equivalently

$S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$9

where

$n$0

The intended domain is the unbounded positive semidefinite cone $n$1. The divergence is finite iff

$n$2

since otherwise the $n$3 term diverges (Minervini et al., 26 May 2026). In scalar form,

$n$4

and the operator spectral expansion is

$n$5

Several structural properties are proved. The divergence is nonnegative, with equality iff $n$6; it is unitarily invariant; it is strictly convex in the first argument; it is additive under direct sums; and it is not jointly convex (Minervini et al., 26 May 2026). The paper also states that it does not satisfy the generalized data-processing inequality under arbitrary CPTP maps, precisely because joint convexity fails. What survives is a restricted monotonicity theorem: if $n$7 satisfy

$n$8

then

$n$9

This applies to affine occupation-number maps modeling attenuator, amplifier, and additive-noise bosonic Gaussian channels (Minervini et al., 26 May 2026).

The thermodynamic interpretation is central. The regularized primal problem

$$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$0

has optimal primal variable

$$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$1

the Bose-Einstein thermal operator (Minervini et al., 26 May 2026). The derivation uses

$$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$2

so the minimizer is obtained by the positivity of $$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$3. This ties the divergence directly to bosonic free-energy geometry on unnormalized PSD operators.

3. Umegaki relative entropy in bosonic Gaussian and many-body settings

In the boson Fock-space Gaussian setting, the operative entropy is the Umegaki relative entropy, not a modified Bose-Einstein operator divergence. For $$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$4-mode Gaussian states $$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$5 on $$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$6,

$$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$7

is computed by first reducing $$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$8 to a product thermal normal form using a Gaussian unitary (Parthasarathy, 2021). If

$$S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma),$$9

then the transformed state

$\Gamma(\mathbb C^n)$0

has transformed annihilation mean $\Gamma(\mathbb C^n)$1 and covariance $\Gamma(\mathbb C^n)$2, and one obtains a modewise formula for $\Gamma(\mathbb C^n)$3 involving the thermal parameters of $\Gamma(\mathbb C^n)$4 and $\Gamma(\mathbb C^n)$5, the transformed one-mode covariance blocks $\Gamma(\mathbb C^n)$6, and the transformed mean components $\Gamma(\mathbb C^n)$7 (Parthasarathy, 2021).

The paper emphasizes a decomposition into a “classical” thermal part and a “quantum” correction. The classical contribution is

$\Gamma(\mathbb C^n)$8

where $\Gamma(\mathbb C^n)$9 is the relative Shannon entropy of Bernoulli laws with success probabilities $N$0. The quantum contribution is

$N$1

The paper’s formulation is therefore bosonic through Gaussian mode decomposition, thermal occupation statistics, and Weyl/Gaussian-unitary structure, while the entropy notion remains Umegaki (Parthasarathy, 2021).

In the bosonic mean-field problem, the same Umegaki quantity controls the distance between symmetric $N$2-body dynamics and the tensorized Hartree state. The setting uses

$N$3

$N$4

and symmetric density matrices $N$5 with

$N$6

The $N$7-body Hamiltonian is

$N$8

and the reference state is the tensor product $N$9, where $S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$0 solves the Hartree equation (Guo et al., 9 May 2026).

The main entropy is

$S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$1

whose derivative obeys the exact entropy production identity

$S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$2

with

$S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$3

Under the stated assumptions, the paper proves

$S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$4

followed by block subadditivity and quantum Pinsker estimates for marginals (Guo et al., 9 May 2026). Here the bosonic interpretation comes from permutation symmetry and comparison with a condensate-like product state, while the entropy formula itself is unchanged.

4. Araki-Uhlmann relative entropy in bosonic quantum field theory

In algebraic QFT, local von Neumann algebras are type III, so von Neumann entropy is not the natural local information measure. “Quantum Relative Entropy implies the Semiclassical Einstein Equations” uses the relative (Araki-Uhlmann) entropy for a free scalar bosonic field on a bifurcate Killing horizon (Dorau et al., 28 Oct 2025). The operational formula is

$S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$5

where $S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$6 is a quasifree $S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$7-invariant Hadamard/KMS state, $S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$8 its coherent excitation, $S(\Gamma,\Gamma')= \begin{cases} \operatorname{Tr}_{H}\!\left(\Gamma(\log \Gamma -\log \Gamma')\right),&\text{if }\Ker(\Gamma')\subset \Ker(\Gamma),\[1mm] +\infty,&\text{otherwise}. \end{cases}$9 the horizon von Neumann algebra, and $\gamma_t^{\otimes N}$0 the modular operator (Dorau et al., 28 Oct 2025).

The bosonic structure is explicit. The field is a real, minimally coupled scalar satisfying

$\gamma_t^{\otimes N}$1

On the horizon, the symplectic form is

$\gamma_t^{\otimes N}$2

the Weyl algebra is generated by

$\gamma_t^{\otimes N}$3

and coherent excitations are implemented by

$\gamma_t^{\otimes N}$4

Using the geometric modular action

$\gamma_t^{\otimes N}$5

the paper derives

$\gamma_t^{\otimes N}$6

and then

$\gamma_t^{\otimes N}$7

since

$\gamma_t^{\otimes N}$8

The entropy is therefore identified with the energy flux along the Killing flow through the horizon (Dorau et al., 28 Oct 2025).

A closely related result in the Rindler wedge proves that the Araki relative entropy between the vacuum and a coherent state equals the entropy of the associated classical Klein-Gordon wave (Ciolli et al., 2019). For the translated wedge $\gamma_t^{\otimes N}$9,

$\gamma(f)=f\log f-(1+f)\log(1+f),$0

and

$\gamma(f)=f\log f-(1+f)\log(1+f),$1

This gives a bosonic coherent-state realization of local relative entropy controlled by null stress-energy (Ciolli et al., 2019).

These works are directly relevant to bosonic quantum relative entropy, but they are not about Bose-Einstein condensation or grand-canonical bosonic gases. Their entropy is modular-theoretic and local, attached to CCR/Weyl algebras and coherent excitations (Dorau et al., 28 Oct 2025, Ciolli et al., 2019).

5. Bose-Einstein relative entropy in kinetic theory

In the 1-D Bose-Einstein Kac grazing limit model, the entropy framework is kinetic rather than operator-theoretic. The equation is

$\gamma(f)=f\log f-(1+f)\log(1+f),$2

with

$\gamma(f)=f\log f-(1+f)\log(1+f),$3

and the bosonic enhancement factor $\gamma(f)=f\log f-(1+f)\log(1+f),$4 is built into both diffusion and drift (Alexandre et al., 2013).

The Bose-Einstein entropy density is

$\gamma(f)=f\log f-(1+f)\log(1+f),$5

with

$\gamma(f)=f\log f-(1+f)\log(1+f),$6

The entropy functional is

$\gamma(f)=f\log f-(1+f)\log(1+f),$7

Its equilibrium minimizer under fixed mass and energy is the Bose distribution

$\gamma(f)=f\log f-(1+f)\log(1+f),$8

and the relative entropy is

$\gamma(f)=f\log f-(1+f)\log(1+f),$9

Because $f$0 is convex, this functional is nonnegative (Alexandre et al., 2013).

The exact entropy dissipation law is

$f$1

where

$f$2

Under the assumptions $f$3 and $f$4 suitably small, the paper proves exponential decay of entropy production, relative entropy, and $f$5 distance to equilibrium (Alexandre et al., 2013).

This is “quantum” only in the sense of Bose-Einstein statistics for indistinguishable bosons. It is not the same object as the operator-relative-entropy constructions used in quantum information or algebraic QFT.

6. Conceptual distinctions, standard quantum relative entropy, and current usage

The general axiomatic literature characterizes the standard quantum relative entropy as

$f$6

and shows that under monotonicity, weak additivity, and lower asymptotic continuity, an admissible quantum relative entropy must equal a constant multiple of $f$7 (Matsumoto, 2010). This is relevant because several bosonic works—Gaussian states and mean-field bosons in particular—use exactly this Umegaki quantity (Parthasarathy, 2021, Guo et al., 9 May 2026).

The resulting landscape is therefore sharply stratified. In bosonic Gaussian and bosonic many-body theories, the entropy is standard Umegaki relative entropy on density operators (Parthasarathy, 2021, Guo et al., 9 May 2026). In local algebraic QFT, the entropy is Araki-Uhlmann relative entropy on type III von Neumann algebras, because local von Neumann entropy is not well-defined (Dorau et al., 28 Oct 2025). In kinetic Bose-Einstein models, the relevant object is the Bregman divergence generated by the scalar Bose entropy density $f$8 (Alexandre et al., 2013). In optimization over unnormalized positive operators, the proposed canonical divergence is the Bose-Einstein quantum relative entropy $f$9, designed to remain natural on the unbounded positive semidefinite cone where the unnormalized Umegaki-type expression can become negative (Minervini et al., 26 May 2026).

A common misconception is to identify all bosonic relative entropies with relative entropy between Bose-Einstein thermal distributions. The recent literature does not support that identification. Free-field coherent-state entropy on horizons, Gaussian-state Umegaki formulas, bosonic Hartree stability estimates, and Bose-Einstein Bregman divergences on PSD operators belong to distinct mathematical frameworks and solve different problems (Dorau et al., 28 Oct 2025, Parthasarathy, 2021, Guo et al., 9 May 2026, Minervini et al., 26 May 2026).

A plausible implication is that “Bose-Einstein quantum relative entropy” should be read contextually. When the ambient variables are unnormalized occupation operators $S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$00, $S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]$01 is the specifically Bose-Einstein divergence (Minervini et al., 26 May 2026). When the states are bosonic density operators on Fock space, recent work usually retains Umegaki or Araki-Uhlmann relative entropy and lets the bosonic structure enter through the state class, the algebra, or the dynamics (Parthasarathy, 2021, Guo et al., 9 May 2026, Dorau et al., 28 Oct 2025).