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Equality in the Bosonic Quantum Entropy Power Inequality
Yonglong Li
TL;DR
The paper asks which independent finite-energy inputs make the bosonic quantum entropy power inequalities tight, a question not settled by prior proofs of the inequalities. A thermal-assisted doubling and noisy-channel argument links equality to product outputs, and the paper proves that both inequality forms have the same Gaussian equality class.
Problem
Which independent inputs make either qEPI bound exact, given that prior results established the inequalities but not their complete finite-energy equality characterization?
Source: §1 · p. 1Method
A thermal auxiliary and doubling turn one-output equality into preserved mutual information through a noisy channel, whose instrument argument forces product structure before the quantum Darmois–Skitovich theorem gives Gaussianity.
Source: §3.2 · p. 9, §3.5 · p. 14, §5 · p. 18Results
For either qEPI, equality holds exactly for Gaussian inputs with the same covariance matrix, allowing arbitrary displacements.
Source: §1.1 · p. 2Takeaways & Limitations
For independent finite-energy inputs, equality in either entropy inequality is equivalent to independence of the two beam-splitter outputs.
Source: §5 · p. 18The proof provides no stability estimate from exact injectivity alone; energy-dependent bounds remain an open question.
Source: §5 · p. 18
Abstract
from arXiv · showhide
The bosonic quantum entropy power inequality bounds the entropy of a beam-splitter output in terms of the input entropies. We determine its complete equality class among independent inputs of finite mean photon number: for any number of modes and any transmissivity strictly between zero and one, equality in either the linear or the exponential form holds if and only if the inputs are Gaussian with the same covariance matrix, allowing arbitrary displacements. The main step shows that equality for one output forces the two outputs to be independent: a thermal auxiliary converts equality into preservation of mutual information by a fixed noisy channel, and a two-outcome instrument shows that preservation of mutual information requires a product state with the reference. The quantum Darmois--Skitovich theorem then gives Gaussianity, while a central-limit argument reduces exponential equality to linear equality.
1 Introduction
The paper completes the finite-energy equality characterization for the bosonic quantum entropy power inequalities. It connects equality for one beam-splitter output to independence of both outputs, then to Gaussian inputs with matching covariances.
1 Introduction
- The missing result is a complete finite-energy equality characterization, beyond the previously established inequalities.The paper asks which independent inputs make either bound exact.Source: §1 · p. 1
1.1 Main results
- For either qEPI and any m ≥ 1 and 0 < η < 1, equality holds exactly for Gaussian inputs with the same covariance matrix, with arbitrary displacements.The two inequalities therefore have the same equality class; unequal input entropies make the bound strict for every finite-energy pair.Source: §1.1 · p. 2
1.2 A first calculation: the two output deficits
- For mixed inputs, equality in either output deficit forces the other deficit to vanish, while product outputs make both deficits vanish.The pure-input implication is immediate because a zero-entropy output is pure and hence uncorrelated with the other output.Source: §1.2 · p. 3
1.3 Related work and the proof idea
- The doubling construction turns a vanishing conditional mutual information into preservation of mutual information by a noisy channel, whose vacuum branch and injective filter force product structure.The proof uses identical thermal auxiliaries; the channel can prepare a fixed vacuum output with input-dependent probability.Source: §1.3 · p. 3
- The quantum Darmois–Skitovich characterization then gives Gaussian inputs with equal covariances, while a central-limit argument establishes exponential equality.The characterization permits unrestricted means; the exponential proof iterates at a fixed affine weight and uses the quantum central limit theorem and entropy continuity.Source: §1.3 · p. 3
2 Conventions and the conditional qEPI
This section sets conventions for finite-energy bosonic states and their entropy, introduces characteristic functions and Gaussian states, and defines the affine deficit used in the conditional qEPI.
2.1 Bosonic systems and entropy
- Finite-energy bosonic states have finite covariance matrices and finite entropy, and photon-number-preserving beam splitters keep the entropy bound applicable throughout the circuits.Finite energy is equivalent to finite quadrature second moments.Source: §2.1 · p. 5
2.2 Characteristic functions and Gaussian states
- The Weyl characteristic function determines a state; Gaussian states are characterized through it, while displacement changes the mean but preserves covariance and entropy.A real symmetric matrix V is a valid covariance matrix exactly when V + iΩ/2 ≥ 0.Source: §2.2 · p. 5
2.3 The affine deficit
- Under conditional independence, the conditional qEPI makes the affine deficit nonnegative for finite-energy bosonic inputs and finite-entropy memory.The deficit combines conditional output entropy, weighted input conditional entropies, and a binary relative-entropy correction.Source: §2.3 · p. 6
3 Proof of linear equality
The proof shows that equality in the linear inequality forces independent outputs, then characterizes the resulting equality cases through Gaussian inputs with equal covariances and the converse implication. It also identifies the centered Gaussian pair as the unique minimizer of the deficit at fixed covariance and interior transmissivity.
3.1 Conditional doubling
- Doubling decomposes the two input-pair deficits into plus and conditional-minus deficits and a conditional mutual information, all nonnegative.This identity uses two possibly different independent input pairs and the conditional qEPI.Source: §3.1 · p. 7
3.2.1 Measure-and-prepare representation
- The fixed channel Φ is a transmissivity-1/2 thermal attenuator with environment mean photon number 2, realized by heterodyne measurement and preparation of displaced τ1/2 states.The prepared states contain a vacuum component, and this channel choice is independent of the physical beam-splitter transmissivity η.Source: §3.2 · p. 9, §3.2.1 · p. 9
3.2.2 The vacuum component
- Each prepared state πz decomposes as c(z)|0⟩⟨0| + (1 − c(z))κz, providing a vacuum branch for the channel instrument.That branch discards correlations with a reference system.Source: §3.2.2 · p. 10, §3.3 · p. 10
3.3 A two-outcome instrument and strict information loss
- For finite-energy X and a finite-entropy separable reference K, equality I(X; K) = I(Y; K) after Φ holds if and only if the input is a product state.A two-outcome instrument uses a positive-probability vacuum branch and a filter that cannot turn a correlated state into a product.Source: §3.3 · pp. 10–11
3.4 From equality to independent outputs
- A thermal auxiliary and doubling turn zero linear deficit into zero channel information loss, so noise removal and Weyl-characteristic uniqueness prove the original outputs are independent.The auxiliary outputs are independent, and the resulting zero conditional information allows Proposition 6 to force a product before the noise is removed.Source: §3.4 · pp. 12–13
3.5 Gaussian characterization and the converse
- Equation (5) gives the converse implication because its two deficits are nonnegative.This passage states only the converse implication and does not provide further Gaussian characterization details.Source: §3.5 · p. 14
- Independent finite-energy inputs with product outputs must be Gaussian with equal covariance matrices, while their means remain unrestricted.The proof applies the finite-energy quantum Darmois–Skitovich characterization.Source: §3.5 · p. 14
- Gaussian inputs with the same covariance produce product outputs and satisfy S(C) = S(A) = S(B), so Lη(A, B) = 0.The Gaussian entropy depends only on covariance; the means may differ.Source: §3.5 · p. 14
- Finite energy controls circuit entropies and scalar quadrature variances, while 0 < η < 1 ensures both mixing coefficients are nonzero; at either endpoint, arbitrary inputs have zero linear deficit.Source: §3.5 · p. 14
- For 0 < η < 1, the centered Gaussian pair uniquely minimizes the linear deficit among independent centered m-mode pairs with both covariance matrices fixed at V, attaining zero.The minimizer is (γV, γV), where γV is the centered Gaussian state of covariance V.Source: §3.5 · p. 14
4 Equality in the exponential qEPI
The exponential qEPI equality problem is reduced to linear equality through the affine-family formulation and a finite-energy Gaussian limit. Strictness within the Gaussian family then fixes the input covariances and affine weight.
4.1 The fixed affine weight
- The complete affine family is equivalent to the exponential qEPI, while its p=η case is the linear qEPI.Exponential equality is Fp*=0; linear equality requires Lη=0, equivalently δ=0 and S(A)=S(B).Source: §4.1 · p. 15
4.2 Strict Gaussian entropy-power concavity
- For Gaussian inputs, Fp=0 holds exactly when the covariance matrices match and p=η.When the covariances differ, strict Gaussian entropy-power concavity rules out zero affine deficit.Source: §4.2 · pp. 15–16
4.3 Quantum convolution and the finite-energy limit
- Centered finite-energy quantum convolutions preserve covariance and energy, and converge in trace norm and entropy to the Gaussian state with the same covariance.The convolution combines independent copies with a passive unitary, retains the normalized sum output, and discards the others.Source: §4.3 · pp. 16–17
4.4 Proof of Theorem 2
- Exponential equality forces equal input covariances, p*=η, linear equality, and Gaussianity; matched-covariance Gaussian inputs also attain exponential equality.The proof propagates zero affine deficit through convolution iterates and passes to their Gaussian limit.Source: §4.4 · pp. 17–18
5 Discussion
The discussion recasts equality as independence of the beam-splitter outputs and distinguishes the thermal auxiliary's role from the Gaussian limit. It identifies quantitative stability as a natural extension.
5 Discussion
- Either qEPI equality is equivalent to independence of the two outputs, after which the quantum Darmois–Skitovich theorem gives the Gaussian characterization.The thermal auxiliary connects an entropy condition on one output to the structural condition on both outputs.Source: §5 · p. 18
- The thermal states create strict information loss, whereas the Gaussian limit determines the affine weight and input covariances; linear equality is proved first.The linear equality proof is complete before the limit argument is used.Source: §5 · p. 18
- Quantitative stability is a natural extension, but exact injectivity alone gives no stability estimate; such bounds must control vacuum-branch information loss and thermal-noise removal.The inverse characteristic multiplier is unbounded, and energy-dependent estimates are identified as a next question.Source: §5 · p. 18
6 Declarations of AI use
The declaration describes the use of two ChatGPT models during the work.
6 Declarations of AI use
- ChatGPT 5.6 Sol explored circuit design and proof strategy; ChatGPT 6 Astra helped draft the paper and refine mathematical arguments.The author states that the proofs were independently checked and approved and takes responsibility for the scientific content.Source: §6 · p. 18
A.1 Moments and characteristic functions
The section establishes the covariance and characteristic-function tools used in the main text. Finite second moments ensure the relevant quadrature quantities are well-defined, and Weyl expectations determine a trace-class operator uniquely.
A.1 Moments and characteristic functions
- Each covariance entry is a quadratic-form expectation, equivalently the real part of a Hilbert–Schmidt inner product.The vectors in that inner product are centered quadrature operators applied to the square root of the state.Source: §A.1 · p. 18
- The second-moment assumption makes the Hilbert–Schmidt norms finite and justifies the second moment of every scalar quadrature.Source: §A.1 · p. 18
- Weyl expectations uniquely determine a trace-class operator because the Weyl representation has ultraweakly dense linear span.Thus a trace-class operator with zero expectation against every Weyl operator must be zero.Source: §A.1 · p. 18
A.2 Oscillator entropy bound
The section derives an oscillator entropy bound from relative entropy to a Gibbs state. Optimizing over the Gibbs parameter yields (9), with the vacuum as the zero-energy case.
A.2 Oscillator entropy bound
- Nonnegativity of relative entropy to a Gibbs state is the starting point for the oscillator entropy bound.The argument considers β > 0.Source: §A.2 · p. 19
- Minimizing over β yields the entropy bound in (9).Source: §A.2 · p. 19
- At zero energy the state is the vacuum, and nonnegative mode number operators bound each subsystem energy by the total energy.Source: §A.2 · p. 19
A.3 The heterodyne Fourier identity
These appendices justify the heterodyne Fourier identity and the integral-map arguments used in the paper. They also establish the finiteness of the entropies needed for the instrument chain rules.
A.3 The heterodyne Fourier identity
- The heterodyne Fourier identity follows by representing the bounded operator as a weak integral and evaluating its matrix elements on coherent vectors.The coherent vectors span a dense subspace, so agreement there identifies the bounded operators; taking the expectation in ρ gives (42).Source: §A.3 · p. 19
- The integral in (41) exists in trace norm because its measurable trace-class integrand has norm Qρ(z), an integrable probability density.Source: §A.3 · p. 19
A.4 Integral maps and finite instrument entropies
- The integral-map construction yields bounded completely positive maps, allowing the data-processing steps to use ordinary quantum channels on separable Hilbert spaces [25, Theorem 1].The residual map is formed from Bz = πz − c(z)|0⟩⟨0| ≥ 0; the same argument applies to Λ1 and their sum.Source: §A.4 · p. 19
- The instrument chain rules apply to finite-entropy states: each output branch has finite energy and entropy, and subadditivity gives finite entropy for the joint states, including those with the finite flag.Source: §A.4 · p. 19
A.5 Minkowski’s determinant inequality
The section states and proves the positive-definite Hermitian case of Minkowski’s determinant inequality for matrices of size 2m. Equality holds exactly when the matrices are positive scalar multiples of one another.
A.5 Minkowski’s determinant inequality
- The inequality is treated for positive-definite Hermitian matrices H and K of size d = 2m and 0 < t < 1.The proof introduces the eigenvalues of H^-1/2 K H^-1/2 and their elementary symmetric sums.Source: §A.5 · p. 20
- Arithmetic–geometric mean inequalities applied to the eigenvalue products yield the determinant inequality.The proof expands the product using the elementary symmetric sums.Source: §A.5 · p. 20
- Equality holds if and only if K = cH for a positive scalar c.Equality in the k = 1 arithmetic–geometric mean step forces all eigenvalues of H^-1/2 K H^-1/2 to be equal.Source: §A.5 · p. 20
B Gaussian characterization from independent outputs
Independent outputs of a beam splitter characterize the inputs: finite-energy inputs must be Gaussian with equal covariance matrices, while allowing arbitrary displacements. The proof applies the scalar Darmois–Skitovich result to quadrature laws and extends the resulting Gaussian characterization globally.
Scalar Darmois–Skitovich lemma
- Independent finite-variance variables whose orthogonal linear forms are independent are Gaussian with equal variance, while their means may differ.The lemma permits zero variance, corresponding to a constant random variable.Source: B Gaussian characterization from… · p. 20
Scalar proof
- Finite second moments permit differentiating the characteristic-function identity; its local logarithmic form yields the Gaussian structure and equal variances.The proof takes logarithms near zero, differentiates in both variables, and shows the second derivatives are constant.Source: B Gaussian characterization from… · pp. 20–21
- A contraction recursion extends the local Gaussian characteristic-function formulas to the whole real line.Since q = max{a², b², ab} < 1, each recursion enlarges the interval where the formulas hold until the intervals exhaust the real line.Source: B Gaussian characterization from… · p. 21
Gaussian characterization
- For 0 < η < 1, independent finite-energy inputs with independent beam-splitter outputs are Gaussian with equal covariance matrices and arbitrary displacements.The characterization holds for any number of modes.Source: B Gaussian characterization from… · p. 21
Quantum characterization
- For each quadrature, the beam splitter maps independent input laws to its output laws, so independent outputs let the scalar lemma establish Gaussian inputs with equal covariance matrices.Applying the result for every ξ establishes Gaussianity; equality of the quadratic forms implies V_A = V_B.Source: B Gaussian characterization from… · p. 21