--- title: Bose-Einstein Quantum Relative Entropy url: https://www.emergentmind.com/topics/bose-einstein-quantum-relative-entropy type: topic --- # Bose-Einstein Quantum Relative Entropy “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 [2605.27228] [2102.06708] [2605.08652] [2510.24491]. 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 [2605.27228]. 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 [2102.06708]. 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 [2605.08652]. 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 [2510.24491]. 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 [1906.01707]. 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 [1302.6377]. | Setting | Relative-entropy notion | Bosonic content | |---|---|---| | Unbounded PSD cone | \(D_{\mathrm{BE}}(X\|Y)\) | Bose-Einstein entropy and occupation numbers | | Boson Fock Gaussian states | Umegaki \(S(\rho\|\sigma)\) | Gaussian bosonic modes | | Symmetric many-body mean field | Umegaki \(S(\Gamma^N,\gamma^{\otimes N})\) | Permutation-symmetric bosons | | Algebraic QFT | Araki-Uhlmann relative entropy | CCR/Weyl algebra, coherent states | | Bose kinetic theory | Bregman divergence of \(\gamma(f)\) | 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” [2605.27228]. There the scalar bosonic entropy is \[ g(x)\coloneqq(x+1)\ln(x+1)-x\ln x,\qquad x\ge 0, \] and its operator extension is \[ S_{\mathrm{BE}(X)\coloneqq\Tr\!\left[(X+I)\ln(X+I)-X\ln X\right]. \] The generator is the convex trace functional \[ F(X)=-S_{\mathrm{BE}(X) =\Tr\!\big[X\ln X-(X+I)\ln(X+I)\big], \] with \[ \nabla F(Y)=\ln Y-\ln(Y+I). \] The associated Bregman divergence is \[ D_{\mathrm{BE}(X\|Y) = F(X)-F(Y)-\Tr[\nabla F(Y)(X-Y)], \] equivalently \[ D_{\mathrm{BE}(X\|Y)=D(X\|Y)-D(X+I\,\|\,Y+I), \] where \[ D(A\|B)\coloneqq \Tr[A\ln A-A\ln B]. \] The intended domain is the **unbounded positive semidefinite cone** \(X,Y\ge 0\). The divergence is finite iff \[ \operatorname{supp}(X)\subseteq \operatorname{supp}(Y), \] since otherwise the \(-X\ln Y\) term diverges [2605.27228]. In scalar form, \[ d_{\mathrm{BE}(x\|y) \coloneqq x\ln\!\left(\frac{x}{y}\right) +(x+1)\ln\!\left(\frac{y+1}{x+1}\right), \] and the operator spectral expansion is \[ D_{\mathrm{BE}(X\|Y) = \sum_{i,j} d_{\mathrm{BE}(x_i\|y_j)\, \left|\langle \psi_i|\phi_j\rangle\right|^2. \] Several structural properties are proved. The divergence is nonnegative, with equality iff \(X=Y\); it is unitarily invariant; it is strictly convex in the first argument; it is additive under direct sums; and it is not jointly convex [2605.27228]. 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 \(a,b\ge 0\) satisfy \[ 2b+1\ge a, \] then \[ D_{\mathrm{BE}(X\|Y)\ge D_{\mathrm{BE}(aX+bI\,\|\,aY+bI). \] This applies to affine occupation-number maps modeling attenuator, amplifier, and additive-noise bosonic Gaussian channels [2605.27228]. The thermodynamic interpretation is central. The regularized primal problem \[ F_T(\mathcal{Q}, q) \coloneqq \min_{X \geq 0} \left\{ \begin{array}{c} \Tr[HX] - T \cdot S_{\mathrm{BE}(X) : \ \Tr[Q_i X] = q_i \ \forall i \in [c] \end{array} \right\} \] has optimal primal variable \[ X_T(\mu)\coloneqq \left(e^{\frac{1}{T}(H-\mu\cdot Q)}-I\right)^{-1}, \] the **Bose-Einstein thermal operator** [2605.27228]. The derivation uses \[ \Tr[AX]-T S_{\mathrm{BE}(X) = T D_{\mathrm{BE}(X\|Y)-T\Tr[\ln(Y+I)], \qquad Y=(e^{A/T}-I)^{-1}, \] so the minimizer is obtained by the positivity of \(D_{\mathrm{BE}}\). 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 \(n\)-mode Gaussian states \(\rho,\sigma\) on \(\Gamma(\mathbb C^n)\), \[ S(\rho\|\sigma)=\operatorname{Tr}\rho(\log \rho-\log \sigma) \] is computed by first reducing \(\sigma\) to a product thermal normal form using a Gaussian unitary [2102.06708]. If \[ \sigma = W(\ell)\,\Gamma(L)\,\rho(t_1)\otimes\cdots\otimes\rho(t_n)\,\Gamma(L)^{-1}W(\ell)^{-1}, \] then the transformed state \[ \rho'=(W(\ell)\Gamma(L))^{-1}\rho\,(W(\ell)\Gamma(L)) \] has transformed annihilation mean \(m'\) and covariance \(C'\), and one obtains a modewise formula for \(S(\rho\|\sigma)\) involving the thermal parameters of \(\rho\) and \(\sigma\), the transformed one-mode covariance blocks \(T_k\), and the transformed mean components \((m_\rho)_k\) [2102.06708]. The paper emphasizes a decomposition into a “classical” thermal part and a “quantum” correction. The classical contribution is \[ \sum_{k=1}^n \frac{H(e^{-s_k}:e^{-t_k})}{1-e^{-s_k}, \] where \(H(p_1:p_2)\) is the relative Shannon entropy of Bernoulli laws with success probabilities \(p_1,p_2\). The quantum contribution is \[ \sum_{k=1}^n \frac{t_k}{2} \left(\operatorname{Tr}T_k-\coth\frac{s_k}{2}+2|(m_\rho)_k|^2\right). \] 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 [2102.06708]. In the bosonic mean-field problem, the same Umegaki quantity controls the distance between symmetric \(N\)-body dynamics and the tensorized Hartree state. The setting uses \[ H=L^2(\Omega),\qquad \Omega\in\{\mathbb T^d,\mathbb R^d\}, \] \[ H_N=H^{\otimes N}\simeq L^2(\Omega^N), \] and symmetric density matrices \(\Gamma^N\in D_s(H_N)\) with \[ U_\pi \Gamma^N U_\pi^*=\Gamma^N\quad\forall \pi\in\mathfrak S_N. \] The \(N\)-body Hamiltonian is \[ H_N:=\sum_{j=1}^{N}h_j+\frac{1}{N-1}\sum_{1\le j