We study lower bounds for the quantum relative entropy between a density matrix and its block-diagonal part. For a block matrix with diagonal blocks A,C>0 and off-diagonal coherence block B, we prove a lower bound expressed through the associated Bogoliubov--Kubo--Mori (BKM) kernel.
The proof uses a midpoint estimate for the BKM Hessian along the affine interpolation between the matrix and its block-diagonal projection. The resulting estimate is genuinely noncommutative and retains information about the joint spectral structure of the diagonal blocks and the coherence term.
As a consequence, under a spectral gap condition on A relative to C, we obtain an explicit logarithmic lower bound proportional to the squared Frobenius norm of the coherence block. The appearance of the BKM metric is natural in this setting because it coincides with the Hessian of quantum relative entropy.
We study lower bounds for the quantum relative entropy between a density matrix and its block-diagonal part. For a block matrix with diagonal blocks A,C>0 and off-diagonal coherence block B, we prove a lower bound expressed through the associated Bogoliubov--Kubo--Mori (BKM) kernel.
The proof uses a midpoint estimate for the BKM Hessian along the affine interpolation between the matrix and its block-diagonal projection. The resulting estimate is genuinely noncommutative and retains information about the joint spectral structure of the diagonal blocks and the coherence term.
As a consequence, under a spectral gap condition on A relative to C, we obtain an explicit logarithmic lower bound proportional to the squared Frobenius norm of the coherence block. The appearance of the BKM metric is natural in this setting because it coincides with the Hessian of quantum relative entropy.
For block-diagonal quantum states with prescribed block weights and dimensions, we determine the minimum von Neumann entropy excess at each trace-norm distance from the blockwise-pure states. The one-block extremal is the classical reverse-Fano profile; the new problem is its constrained assembly when the block weights and dimensions cannot be redistributed. The resulting value function Φq,n(D) is attained on the full admissible interval and is the lower envelope of finitely many rank-lattice edges. Every minimizing distance allocation has at most one block between consecutive integer ranks, and the active edge can switch nonmonotonically with D. We also determine the lower convex envelope of Φq,n explicitly: it is the polygonal curve through synchronized rank vertices and admits a one-parameter dual representation. The exact nonconvex envelope differs from this convex surrogate by at most q⋄log(5/4), with a sharper stage-dependent bound, and the constant is optimal. For binomial charge-sector weights the uniform gap is O(L−1/2). The framework yields a symmetry-sector purity certificate; in a three-qubit phase-damping example the fixed-weight certificate remains nontrivial where the universal classical entropy-error envelope is already vacuous.
How much high-photon quantum coherence is controlled by a nonnegative phase-space distribution? This article gives quantitative guarantees for approximating an optical state by a finite photon-number block, and constructs physical states showing why the guarantee cannot be inferred from mean energy or Wigner positivity alone at the strength available for a Gaussian-mixture source. Source classes and scope. The comparison concerns one optical mode with [Q,P]=i, mean photon number at most E>0, and a known parity centre. The full class WE consists of normal Wigner-positive states at the exact vacuum min-entropy endpoint, equivalently states supported on the even photon sector in that frame. The subclass GE consists of actual probability mixtures of centred pure squeezed vacua. Mixture labels are unavailable, component energies may be arbitrarily large, and their true average energy is charged. Infinite photon tails are retained. A hierarchy of coherent-tail guarantees. Let PL project onto photon numbers at most L, and define the cutoff error by eL(ρ)=21∥ρ−PLρPL∥1. This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard OE(L−1/2) guarantee. At the exact Wigner endpoint, the article proves supρ∈WEeL(ρ)≤CE(L+1)−3/4,supρ∈GEeL(ρ)≤L+14E,L≥2. The stronger endpoint bound combines operator positivity and Wigner positivity on the same density kernel, followed by a controlled conversion to trace norm. It requires only mean energy, without a uniform second photon moment. Physical coherence beyond the Gaussian-mixture bound. A family τd∈WE has exact energy E, covariance (E+1/2)I, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs Ld=Θ(d3), eLd(τd)≥cEd−5/2,(Ld+1)eLd(τd)⟶∞. Thus the Gaussian-mixture L−1 law fails uniformly on the full endpoint class. The construction preserves coherence through an actual optical partial trace and pays its physical energy cost. Positive phase mixing further yields distinct states with identical complete photon statistics and covariance but nearly indistinguishable bounded optical responses. The rare bright branches obey the prescribed mean-energy budget; no efficient experimental preparation protocol or uniform branch-energy bound is assumed. Consequences for a specified measurement system. Each independent copy is measured by bounded centred squeezing followed by a vacuum/nonvacuum binary test. At fixed energy, squeezing range and confidence, let nX(δ) be the minimum worst-case number of copies for trace-distance accuracy δ over source class X. As δ↓0, the bounds are lognGE(δ)=Θ(δ−1),cδ−6/5≤lognWE(δ)≤Cδ−4/3. These are measurement-restricted reconstruction costs. The lower bounds allow adaptive randomized settings. The upper bounds use finite designs and estimates in the original source class, with the energy budget respected for every record; they establish statistical existence rather than an efficient catalogue-generation algorithm. The full-class exponents remain distinct lower and upper bounds. The article also characterizes infinite-order Wigner entropy equality by displaced even-parity support and corrects the pure-Gaussian-only equality clause at infinite order in Dias and Prata (2023). The 27-page integrated article includes the complete proofs, figures and references.
When does finite phase-space entropy stop guaranteeing proximity to coherent light? This article determines sharp thresholds for two broad optical source classes when the number of modes may grow. It separates the roles of entropy, mode number and photon energy in approximating an optical state by a single pure coherent state. For an m-mode state ρ, write Qρ(α)=⟨α∣ρ∣α⟩, with dμm=π−md2mα, and define M(ρ)=2m∫Qρ(α)2dμm(α),s=−lnM(ρ),D(ρ)=infα∥ρ−∣α⟩⟨α∣∥1. Here s is the second-order Wehrl entropy excess and D is the full trace-norm distance, whose maximum is 2. The collision quantity M is also the output purity after half loss and can be accessed using interference of two independent, identically prepared copies. Exact source-class thresholds. The two classes are states diagonal in a common passive-mode number basis, and arbitrary states supported on the vacuum together with configurations whose photons occupy only one mode at a time in a common passive basis. The latter class retains arbitrary complex coherences, including vacuum coherences. For either class X, with FX(s)=supρ∈X,−lnM(ρ)=sD(ρ)2 and the supremum over all finite mode numbers, FX(s)=4⟺s≥ln4. The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and M→1/4 force the one-photon probability to approach one, without an energy-bound assumption. This probability rigidity does not assert convergence of the mean energy. Sharp boundary geometry and energy. For the full number-diagonal class D, the approach from below has the exact corner lims↑ln4ln4−s4−FD(s)=e4. A small vacuum component changes the optimal coherent peak and produces the sharp slope. Separate all-state energy bounds show that vanishing coherent overlap requires at least one photon asymptotically, or two photons for pure states. Explicit sources attain these limiting energy costs, and exact-entropy constructions include their additional photon costs. The proofs retain arbitrary photon tails and constants uniform in the mode number. The earliest-threshold and corner results apply to the source classes stated above. The integrated article contains the complete proofs and two figure groups.
In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.
We establish upper and lower bounds for relative entropy contraction of generalized quantum depolarizing channels and semigroups. Our bound provides tight first order asymptotic of the contraction rate in terms of the dimension constant. One side estimate are based on sharp reverse ratio and convexity of relative entropy of two states, which can be derived from the recently introduced Hockey-Stick quantum f-divergence, and also independently, Bogoliubov--Kubo--Mori quantum Fisher information metric. The other side follows from the existence of index achieving pure state with respect to a general conditional expectation. Our results extend to the complete entropy contraction rate, tensor stable estimates for product dynamics. Examples include quantum depolarization, dephasing, and compact group symmetrization, with consequences for the decay rate of coherence and asymmetry.
Normal forms and completely positive gluing theory describe representation freedom in quantumdynamics. We extract dimension laws and sharp coherence constraints when every sector projectoris exactly conserved and the dynamics inside each sector is prescribed. Let ℋ = ⨁𝑞ℋ𝑞, let𝑑𝑞 = dimℋ𝑞, and let 𝑘𝑞 be the rank of the canonical dissipative matrix of the prescribed Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator on block 𝑞. We prove that the compatibleextensions have real affine dimension 2 ∑𝑞<𝑠(𝑘𝑞 +1)(𝑘𝑠 +1), recovering the linear restriction-kerneldimension at full local dissipative rank. For a scalar sector paired with an 𝑛-dimensional sector,the instantaneous cross-sector amplitude damping is bounded below by half the sum of local jumpoperator variances. The bound is attainable for every specified instantaneous direction and equalsone quarter of the initial purity-loss rate of the associated pure sector state. An isotropic qubitdissipator with Bloch decay rate 2𝜅 forces cross-sector damping of at least 𝜅/2; rank-one dephasingcan instead preserve a selected coherence. A joint observable with a reference accesses this differencewhile preserving total charge. We also give full-matrix and superselected-algebra dimensions, exactrefinement losses, and extremal multiplicity profiles. These results constrain a nonunitary chargesector model after its projectors and local dynamics are supplied; they do not determine theirphysical origin or select the realized extension