==============================================================================
A FINITE-ENTROPY SEPARATION OF MICROSTATES AND
NONMICROSTATES FREE ENTROPY
==============================================================================
PREFACE: HOW TO READ THIS PAPER
Technical argument: high-school readers can follow the arc but not
verify every line. In one sentence, chi and chi* are two information
measures; chi counts matrix approximations, chi* integrates free
Fisher information along semicircular noise; they usually agree, but
we build X with both finite and chi(X) <= chi*(X) - 1/2.
Proof idea in one paragraph. Small noise keeps chi(X) finite. Larger
noise leaves a detectable block structure in the matrices, invisible
to chi*. Data processing turns that structure into the gap.
Guide. Section 1 is narrative; Sections 2-3 build tools; Sections 4-5
build the example; Section 6 assembles the proof. Notation appears
in-line; further symbols are introduced where first used.
ABSTRACT
We answer the finite-entropy equality question for microstates and
nonmicrostates free entropy negatively. We construct a bounded
self-adjoint tuple X in a von Neumann algebra with faithful normal
tracial state such that
-infinity < chi(X) <= chi*(X) - 1/2 < infinity.
The construction uses a large but fixed number of variables.
1. INTRODUCTION
1.1 The two invariants.
Fix a bounded self-adjoint tuple X = (X_1, ..., X_n) in a tracial von
Neumann algebra (M, tau). Voiculescu attached to X two numerical
invariants, both modelled on entropy.
Microstates free entropy chi(X). For matrix size d, take the set of
d x d self-adjoint n-tuples whose normalized trace of every short
noncommutative word agrees with X to tolerance epsilon, and whose
operator norms are bounded by a cutoff R. Its volume grows like
exp(chi(X) d^2) up to polynomial factors. The precise definition is in
Section 2.
Nonmicrostates free entropy chi*(X). The free Fisher information
Phi*(X) is the sum of squared L^2 norms of the conjugate variables of
X, the free analogue of classical scores. Then chi* is an integral of
Phi* along addition of standard semicircular noise. A standard
semicircular variable is the free analogue of a centered Gaussian; the
noise it generates plays the role of Brownian motion. The precise
definition is in Section 2.
For a single self-adjoint variable (n = 1), both invariants reduce to
the same classical entropy.
1.2 The question.
Two facts were commonly expected:
(a) chi(X) <= chi*(X) for every bounded self-adjoint X;
(b) chi(X) > -infinity forces chi(X) = chi*(X).
Fact (a) is a theorem. Biane-Capitaine-Guionnet proved it through
large deviations for matrix Brownian motion. Jekel-Pi gave an
elementary proof via classical entropy and Fisher information, and
extended it to conditional free entropy.
Fact (b) was formulated by Guionnet (Conjecture 7.1 in her survey).
The finiteness condition is a genuine hypothesis: Jekel-Pi recall
examples with chi = -infinity < chi*. The open question was whether
finiteness alone is enough.
Our main result answers it in the negative.
Theorem 1 (Main). There exist n >= 2 and a bounded self-adjoint tuple
X = (X_1, ..., X_n) in a von Neumann algebra with a faithful normal
tracial state such that
-infinity < chi(X) <= chi*(X) - 1/2 < infinity.
The number of variables n is large but fixed.
1.3 Road map of the proof.
Fix p large and put n = p + 1. Let B_1, ..., B_p be free variance-one
semicirculars. In a separate tensor factor let Y be uniform on the p
points y_l = -1 + 2(l-1)/(p-1), l = 1, ..., p. Then Y commutes with
each B_i but is not free from them. Set
C = (B_1, ..., B_p, Y),
V_a = C + sqrt(a) S (S a standard semicircular n-tuple free from C),
X = V_h (h > 0 small),
L(a) = (1/2) sum_i log(2 pi e (a + c_i)) (c_i = tau(C_i^2)).
Step A (finite chi; Lemma 4.2). chi(X) > -infinity: bounded matrix
models of C, noised by Gaussian matrices of variance h, supply a
positive-volume matching set.
Step B (block event; Lemma 6.1). Evolved microstate ensembles of X
retain a detectable simultaneous block structure in the last
coordinate: probability at least 1/2; at most exp(-4 d^2) for the
matching GUE reference.
Step C (data processing). Both bounds give KL >= 2 d^2 in the limit.
The GUE reference has explicit density, so
limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1.
Step D (analytic chi*). For a > 0,
chi*(V_a) = L(a) - (1/2) integral_a^infinity
[ Phi*(V_u) - sum_i 1/(u + c_i) ] du.
The Fisher deficit is bounded by squared higher-cumulant norms
(Proposition 3.2); the signal's high cumulants are sparse (Lemma 4.1),
so the deficit is small for large a.
Step E (entropy production; Lemma 5.1). Jekel-Pi along the SAME
ensemble:
liminf_d H_d(Z^{(d)}(t))
>= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds.
Step F (assembly). Combine the two bounds on the same terminal law,
using chi*(V_{h+t}) - chi*(V_h) = (1/2) int_0^t Phi*(V_{h+s}) ds:
chi_R(X) <= L(h+t) - 1 - (1/2) int_0^t Phi*(V_{h+s}) ds
<= chi*(V_h) - 1/2.
Supremum over cutoffs R gives Theorem 1.
1.4 Why two noise scales.
The two bounds make competing scale demands.
- Step D needs t >> sqrt(p), so the analytic deficit (order
K^6 p^{3/2} t^{-3}) is small.
- Step B needs t log(p) / p -> 0, so the unitary-net cost dominates
Gaussian concentration in the block event.
The single choice t = p^{3/4} satisfies both as p -> infinity. In
explicit numbers, p = 2^64 and t = 2^48: then 2 K^2 sqrt(p) = 2^47 < t
and 32 t / p = 2^{-11}. Choose p and t first; then take h small; then
let matrix sizes d -> infinity.
1.5 Relation to companion work.
A companion paper proves each of the four free entropy dimensions of
L(F_2) attains every integer value n >= 2 on some generating tuple.
That statement varies the tuple inside a fixed algebra. Here we
compare chi and chi* on a single fixed tuple, with both values finite.
1.6 Organization.
Section 2: conventions; Lemma 2.1 (variational formula for Phi*),
Lemma 2.2 (bounded matrix inputs).
Section 3: cumulants; Lemma 3.1 (Wick testing), Proposition 3.2
(Fisher deficit bound).
Section 4: the signal; Lemma 4.1 (cumulant bound), Lemma 4.2
(chi(X) > -infinity).
Section 5: entropy production; Lemma 5.1.
Section 6: block event, Lemma 6.1, and proof of Theorem 1.
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2. CONVENTIONS AND PRELIMINARY FACTS
==============================================================================
We set up notation once and prove the two matrix-limit facts used in
later sections. Section 2.1 fixes the algebra. Section 2.2 defines
the microstates entropy chi. Section 2.3 defines conjugate variables
and the free Fisher information Phi*. Section 2.4 proves a variational
formula for Phi* (Lemma 2.1). Section 2.5 defines the nonmicrostates
entropy chi*. Section 2.6 fixes matrix and GUE notation. Section 2.7
proves the matrix convergence lemma (Lemma 2.2).
2.1 Setting.
Noncommutative variables live in a von Neumann algebra (M, tau) with a
faithful normal tracial state tau. Write
||a||_2^2 = tau(a^* a).
A standard semicircular family has variance one in each coordinate and
is free. Every semicircular noise introduced below is free from the
tuple it perturbs.
2.2 Microstates entropy.
Let X = (X_1, ..., X_n) be a bounded self-adjoint tuple. For R > 0,
m >= 1, epsilon > 0, let Gamma_R(X; m, d, epsilon) be the set of
self-adjoint d x d matrix tuples with coordinate operator norms at most
R whose normalized trace of every word of length at most m differs from
the value on X by less than epsilon. Take volume in the real Euclidean
space of matrix tuples with inner product
<A, B> = Re sum_{i=1}^n Tr(A_i^* B_i).
Define
chi_R(X) = inf_{m, epsilon} limsup_{d -> infinity}
[ d^{-2} log Vol Gamma_R(X; m, d, epsilon)
+ (n/2) log d ],
chi(X) = sup_{R > 0} chi_R(X).
The additive (n/2) log d matches the classical entropy normalization
below. If no matrices ever match X this way for a given R, the value
is -infinity, which is allowed.
2.3 Conjugate variables and free Fisher information.
A bounded self-adjoint tuple Z has conjugate variables xi_1, ..., xi_n
in L^2(M, tau) if for every noncommutative polynomial P in the
indeterminates x_1, ..., x_n,
tau(xi_i P(Z)) = (tau tensor tau)((partial_i P)(Z)),
where the free difference quotient partial_i is linear and satisfies the
Leibniz rule
partial_i x_j = delta_ij (1 tensor 1),
partial_i (P Q) = (partial_i P)(Q) + (P tensor 1)(partial_i Q),
applied by substituting Z. Define the free Fisher information
Phi*(Z) = sum_{i=1}^n ||xi_i||_2^2
when the xi_i exist, and Phi*(Z) = infinity otherwise. Conjugate
variables are unique and self-adjoint when they exist.
2.4 A variational formula for Phi*.
Lemma 2.1 (Polynomial variational formula). If a bounded self-adjoint
Z has conjugate variables, then
Phi*(Z) = sup sum_{i=1}^n ( 2 (tau tensor tau)((partial_i P_i)(Z))
- tau(P_i(Z)^2) ), (2.1)
the supremum being over self-adjoint noncommutative polynomials
P_1, ..., P_n. It suffices to take P_i = (Q_i + Q_i^*)/2 with Q_i
having coefficients in Q + i Q. Consequently, along any family of
tuples with continuous moments and existing conjugate variables, Phi*
is a measurable function of the parameter.
Proof. Self-adjointness. The identity
partial_i(P^*) = flip((partial_i P)^*) together with symmetry of
tau tensor tau shows xi_i^* is also a conjugate variable. Uniqueness
from polynomial L^2 density (below) makes xi_i = xi_i^*.
Variational identity. The expression under the supremum equals
sum_i [ 2 tau(xi_i P_i(Z)) - ||P_i(Z)||_2^2 ]
= sum_i [ ||xi_i||_2^2 - ||xi_i - P_i(Z)||_2^2 ],
so the supremum over all self-adjoint polynomials recovers
sum_i ||xi_i||_2^2.
Polynomial density. Let K be the closed span in L^2 of 1 and the
polynomial vectors. Each bounded self-adjoint Z_j preserves K, so the
orthogonal projection onto K commutes with Z_j and with W^*(Z). Since
it fixes 1, it fixes every bounded algebra vector, and those are dense
in L^2. Hence K = L^2.
Countable and measurable versions. Symmetrizing P_i via
(Q_i + Q_i^*)/2 gives the self-adjoint supremum. Rational
coefficients give a countable dense subset. Each functional in the
countable supremum is continuous in the moments of Z, so Phi* is
measurable in the parameter. QED.
2.5 Nonmicrostates entropy.
Our nonmicrostates normalization is, for S a standard semicircular
n-tuple free from Z,
chi*(Z) = (n/2) log(2 pi e)
+ (1/2) integral_0^infinity
[ n / (1 + s) - Phi*(Z + sqrt(s) S) ] ds. (2.2)
The n / (1 + s) is the free Fisher integrand of n free semicircular
coordinates; for a semicircular family the bracket vanishes identically
and chi* equals the free entropy of that family, namely
(n/2) log(2 pi e sigma^2).
2.6 Matrix notation and GUE.
Write tr_d = d^{-1} Tr and, for a matrix tuple D,
||D||_{2,d}^2 = sum_{i=1}^n tr_d(D_i^* D_i).
A standard GUE tuple G^{(d)} consists of independent centered Gaussian
matrices whose real Hilbert-Schmidt coordinates are independent with
variance 1/d. For a random matrix tuple with density f, its normalized
classical entropy is
H_d = -(1/d^2) integral f log f + (n/2) log d.
Independent GUE coordinates of variances w_i > 0 have
H_d = (1/2) sum_i log(2 pi e w_i).
This matches the microstates normalization exactly.
2.7 Matrix convergence under bounded input.
The classical input to Lemma 2.2 is Voiculescu's asymptotic freeness:
if deterministic matrix tuples converge in normalized trace moments,
adjoining independent GUE matrices makes them converge to free products
with semicirculars. Lemma 2.2 supplies the extra probability and
expectation statements we need, including a uniform version over
microstate sets.
Lemma 2.2 (Bounded matrix inputs). Let K_d be sets of self-adjoint
matrix tuples with a common coordinate operator-norm bound. Suppose
their normalized trace moments converge uniformly over K_d to the
moments of a bounded tuple T. Let A^{(d)} be any random tuple supported
on K_d, independent of a fixed finite standard GUE tuple G^{(d)}. Then
(i) every normalized polynomial trace in (A^{(d)}, G^{(d)})
converges in probability to its value at (T, S), where S is a
standard semicircular family free from T;
(ii) products of finitely many such traces converge in expectation;
(iii) the dimensions may run over any sequence tending to infinity.
Proof.
GUE norm bound. A 1/4-net of the unit sphere of C^d in real dimension
2d has cardinality at most 9^{2d}. For Hermitian G,
||G|| <= 2 max_{v in net} |v^* G v|.
For standard GUE, each v^* G v is a real centered Gaussian of variance
1/d because Tr((v v^*)^2) = 1. Therefore
P( ||G|| > L ) <= 2 exp( d (2 log 9 - L^2 / 8) ). (2.3)
In particular, the coordinate norms of any fixed finite GUE tuple are
at most 8 with probability tending to one; integrating the tail gives
uniformly bounded moments of ||G||.
Gradient bound. Let a be a deterministic tuple with coordinate norms
at most R and put F = tr_d P(a, G) for a fixed polynomial P. Let
(E_alpha) be a Hermitian basis orthonormal for the unnormalized
Hilbert-Schmidt inner product and write G_j = sum_alpha g_{j,alpha}
E_alpha. The g_{j,alpha} are independent Gaussians of variance 1/d.
Cyclicity of trace gives
partial_{g_{j,alpha}} F = d^{-1} Tr(E_alpha B_j),
sum_{j,alpha} |partial_{g_{j,alpha}} F|^2
= d^{-1} sum_j tr_d(B_j^* B_j),
where B_j is the sum of cyclically ordered products obtained from P by
deleting each occurrence of the j-th GUE letter.
Hermite expansion. A polynomial in standard real Gaussian coordinates
has variance at most the expected squared gradient: each differentiation
weights the squared modulus of a nonconstant Hermite coefficient by its
positive total degree. Rescaling to variance 1/d and applying the same
to real and imaginary parts:
E |F - E F|^2 <= d^{-2} sum_j E tr_d(B_j^* B_j)
= O_{P,R}(d^{-2}).
So for every deterministic input sequence a_d converging in moments,
F converges in probability to its value at (T, S). Voiculescu's
trace-convergence theorem upgrades the expectation convergence to
convergence in probability for the limit.
Uniformity over K_d. Suppose uniformity failed: then for some
polynomial and some threshold eta > 0, there existed a subsequence of
dimensions and points a_d in K_d with error probability at least eta.
Those a_d form a bounded deterministic moment-convergent sequence,
contradicting the previous paragraph. Conditioning on A^{(d)} gives
convergence in probability for the random input A^{(d)}.
Expectations. For a fixed product Q of normalized polynomial traces,
|Q(A^{(d)}, G^{(d)})| <= C (1 + max_j ||G_j^{(d)}||)^D
with C, D depending only on the polynomials and the common cutoff.
Thus E |Q|^2 is uniformly bounded, Q is uniformly integrable, and
convergence in probability implies convergence of expectations. QED.
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3. A CUMULANT BOUND FOR THE FISHER DEFICIT
==============================================================================
Goal of this section: bound how far Phi*(V_u) is from sum_i 1 / (u + c_i),
the value for a semicircular family of the same covariance. The bound
is in terms of squared higher-cumulant norms of the signal C. Section
3.1 recalls free cumulants. Section 3.2 tests higher-cumulant terms
against semicircular Wick polynomials. Section 3.3 assembles the bound
(Proposition 3.2).
3.1 Free cumulants.
For a centered bounded self-adjoint n-tuple C, let kappa_m(C) be its
order-m free cumulant array with
||kappa_m(C)||_2^2
= sum_{i_1, ..., i_m = 1}^n |kappa_m(C_{i_1}, ..., C_{i_m})|^2.
We use three standard facts: the moment-cumulant formula and its Mobius
inversion on noncrossing partitions; vanishing of mixed cumulants
between free algebras; semicirculars have only second cumulants.
3.2 Wick testing.
Lemma 3.1 (Wick testing). Let C be a centered bounded self-adjoint
n-tuple, S a standard semicircular family free from C, and Z = C +
sqrt(u) S with u > 0. For a multi-index j = (j_1, ..., j_r) with
r >= 1 and a monomial P in x_1, ..., x_n, define
D_j(P) = sum over factorizations P = P_0 x_{j_1} P_1 ... x_{j_r} P_r
of prod_{a=0}^r tau(P_a(Z)),
selecting increasing positions and letting empty gaps contribute one.
Extend D_j linearly to polynomials. Then there is an orthonormal family
of semicircular Wick polynomials U_j(S) with
D_j(P) = u^{-r/2} tau(U_j(S)^* P(Z)). (3.1)
In particular, the conjugate variables of Z exist and equal
xi_i = u^{-1/2} E[ S_i | W^*(Z) ].
Proof. Definition. Build U_j from the word S_{j_1} ... S_{j_r} by
summing over subsets of pairwise disjoint adjacent pairs in the original
index list, deleting each chosen pair (j_a, j_a + 1) with weight
-delta_{j_a, j_{a+1}}. The empty subset gives the original word. This
gives U_j^* = U_{(j_r, ..., j_1)}.
Expansion. Expand tau(U_j^* P(Z)) by noncrossing cumulants of
Z = C + sqrt(u) S. Freeness and semicircularity force every block
touching an initial S position to be a pair: covariance delta_{ij}
against another initial S, sqrt(u) delta_{ij} against an S coming from P.
Cancellation of deletions. Fix a choice of C or S at every P-position
in the multilinear expansion, and a noncrossing partition pi on all
positions. Let A(pi) be the set of initial-initial pair blocks with
originally adjacent endpoints. Contributions correspond exactly to
subsets F of A(pi) (delete pairs in F, expand the rest, restore), and
restoring adjacent pairs preserves both noncrossingness and the
covariance factors. Their net coefficient is
sum_{F subset A(pi)} (-1)^{|F|} = (1 - 1)^{|A(pi)|}.
If pi contains any initial-initial pair, an innermost one has adjacent
endpoints (its gaps are all initial and must pair inside), so A(pi) is
nonempty and pi cancels. This covers nested internal pairs and never
requires deleting newly adjacent positions.
Surviving partitions. All initial positions pair externally with
positions of P, in reverse order. Since U_j^* has initial indices
j_r, ..., j_1, the selected indices in P occur as j_1, ..., j_r.
External pair arches prevent any remaining block from joining distinct
gaps, so each gap's trace factors out independently of the choices of
C or S within it. Each external pair contributes sqrt(u), proving
(3.1).
Orthonormality. Apply the identity with C = 0 and u = 1. A Wick word
tests to zero against shorter words and to a Kronecker delta against a
word of matching degree; every other term of a Wick polynomial has
smaller degree. Conjugate symmetry covers the remaining cases.
Conjugate variables. For r = 1, D_i(P) = (tau tensor tau)((partial_i P)(Z)),
so (3.1) and the definition of conditional expectation give
xi_i = u^{-1/2} E[S_i | W^*(Z)]. QED.
3.3 Fisher deficit bound.
Proposition 3.2 (Squared-cumulant Fisher bound). Let C be centered
bounded self-adjoint with diagonal covariance tau(C_i C_j) = delta_ij c_i,
c_i >= 0. Let S be standard semicircular free from C. Put V_u = C +
sqrt(u) S and v_i(u) = u + c_i for u > 0. Then for every u > 0,
sum_i 1 / v_i(u) <= Phi*(V_u) <= n / u. (3.2)
If, in addition, sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 < infinity,
then
0 <= Phi*(V_u) - sum_i 1 / v_i(u)
<= u^{-2} sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2. (3.3)
Proof. General bounds. Set Z = V_u. Lemma 3.1 supplies xi_i, so
projection contraction gives Phi*(Z) <= n / u. Testing the conjugate
identity with Z_i gives tau(Z_i xi_i) = 1. Since ||Z_i||_2^2 = v_i(u),
Cauchy-Schwarz gives ||xi_i||_2^2 >= 1 / v_i(u), and summing proves the
lower bound in (3.2).
Deficit bound. Extract the block containing the first position in
tau(Z_i P(Z)). Its other positions select letters in D_j(P), and its
gaps contribute their moments:
tau(Z_i P(Z)) = sum_{r >= 1, j}
kappa_{r+1}(Z_i, Z_{j_1}, ..., Z_{j_r}) D_j(P).
Only r <= deg P contributes. Centering removes the order-one cumulant;
diagonal covariance makes the order-two term v_i(u) D_i(P). Adding free
semicircular noise leaves all higher cumulants equal to those of C.
Residual. Define the ambient L^2 series
R_i = sum_{r >= 2, j}
u^{-r/2} kappa_{r+1}(C_i, C_{j_1}, ..., C_{j_r}) U_j(S)^*.
The adjoint Wick family is orthonormal and summability gives L^2
convergence with
sum_i ||R_i||_2^2 = sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2.
For fixed P, terms with r > deg P have zero trace pairing against P(Z)
by (3.1). Since A -> tau(A P(Z)) is L^2 continuous, the truncation gives
tau( (Z_i - v_i(u) xi_i) P(Z) ) = tau(R_i P(Z)).
The trace pairing is bilinear and needs no complex conjugation. Testing
with adjoints makes this Hilbert orthogonality; polynomial L^2 density
identifies
Z_i - v_i(u) xi_i = E[R_i | W^*(Z)].
Assembly. Since xi_i is self-adjoint and tau(Z_i xi_i) = 1,
||Z_i - v_i(u) xi_i||_2^2 = v_i(u)^2 ( ||xi_i||_2^2 - 1/v_i(u) ).
Summing after dividing by v_i(u)^2, and using projection contraction
plus v_i(u) >= u, proves (3.3). QED.
Remark. Two consequences are used later. The upper bound Phi*(V_u) <=
n/u gives integrability of the Fisher-deficit integrand on every
compact interval of positive noise times, so the free-flow identity
(4.5) is legitimate. The deficit bound (3.3) reduces the large-noise
entropy computation to control of ||kappa_{r+1}(C)||_2^2 for the signal
C.
==============================================================================
4. THE SIGNAL AND THE TWO NOISE SCALES
==============================================================================
Requirements. Higher cumulants of the signal must be sparse (so the
analytic bound holds), and many coordinates must share a block
decomposition (so the matrix bound applies). Tensor independence
supplies both at once.
4.1 The signal.
Fix p >= 2 and n = p + 1. Let B_1, ..., B_p be free variance-one
semicirculars. In a separate tensor factor take Y uniform on p equally
spaced points y_l = -1 + 2(l-1)/(p-1), l = 1, ..., p. Then Y commutes
with every B_i but is not free from them. Set
C = (B_1, ..., B_p, Y),
V_a = C + sqrt(a) S,
L(a) = (1/2) sum_{i=1}^n log(2 pi e (a + c_i)), c_i = tau(C_i^2),
with S a standard semicircular n-tuple free from C. So c_i = 1 for
i <= p, c_{p+1} = (1/p) sum_l y_l^2, and C is centered with diagonal
covariance. L(a) is both the nonmicrostates entropy of the semicircular
family with variances a + c_i and the classical entropy of the
corresponding independent GUE law.
4.2 Faithful realization.
Choose uniformly bounded deterministic matrix models B^{(q_k)} of the
free semicircular p-tuple, tensor with I_p, and use
I_{q_k} tensor diag(y_1, ..., y_p) for Y. In dimensions d_k = p q_k
these have the required tensor law. Adjoin independent GUE and take a
subsequence satisfying moment tests with one fixed norm bound: the
limiting moments give a positive tracial functional with asserted
freeness, bounded on C_i by passing tr_d(Q^* D_i^2 Q) <= R_i^2
tr_d(Q^* Q) to the limit. The GNS vacuum is cyclic for right
multiplication (commuting with the weak closure of the left algebra)
and separating there, so the state is faithful and normal. Faithfulness
and semicircularity give ||B_i|| = ||S_i|| = 2.
4.3 Cumulant bound.
Lemma 4.1 (Cumulant bound). With K = 128 and C as above, for every
m >= 1,
||kappa_m(C)||_2 <= K^m p^{m/4}. (4.1)
Proof. There are at most 4^m noncrossing partitions, with Mobius
coefficients bounded by 4^m. Each product of block moments has modulus
at most 2^m since ||B_i|| = 2 and ||Y|| <= 1. So each cumulant is at
most 32^m.
Suppose b of the m positions carry B-indices. Tensor independence
factors each block moment into its B-word moment times its Y-moment. A
nonzero Mobius sum requires, after deleting Y positions within each
block, a semicircular word with a noncrossing pairing matching equal
indices. Pairings from distinct blocks stay noncrossing (crossing
them would cross the containing partition). So for fixed b positions at
most 4^b p^{b/2} labelings contribute. Summing over positions bounds
the potentially nonzero entries by (1 + 4 sqrt p)^m. Then
||kappa_m(C)||_2 <= 32^m (1 + 4 sqrt p)^{m/2}
<= (32 sqrt 5)^m p^{m/4}
<= K^m p^{m/4}. QED.
4.4 Fisher deficit for the signal.
Apply Proposition 3.2 with the cumulant bound (4.1). The series
sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 has ratio K^2 sqrt(p)/u;
for u > 2 K^2 sqrt(p) it is < 1/2 and the geometric sum is bounded by
2 K^6 p^{3/2} u^{-2}. Therefore
0 <= Phi*(V_u) - sum_i 1/(u + c_i) <= 2 K^6 p^{3/2} u^{-4}. (4.2)
4.5 Entropy deficit formula.
Adding free semicircular noise of variance s to V_a gives V_{a+s}.
Equation (2.2) then yields
chi*(V_a) = L(a) - (1/2) integral_a^infinity
[ Phi*(V_u) - sum_i 1/(u + c_i) ] du. (4.3)
All terms are finite for a > 0: the integrand is measurable (Lemma 2.1),
nonnegative (Prop 3.2), and bounded by sum_i c_i / (u (u + c_i)) <=
sum_i c_i / u^2, which is integrable on [a, infinity). The covariance
correction is integral_0^infinity [(1+s)^{-1} - (a+c_i+s)^{-1}] ds =
log(a + c_i), so no divergent integral is subtracted. In particular,
chi*(V_a) >= L(a) - K^6 p^{3/2}/(3 a^3) > L(a) - 1/2
whenever a > 2 K^2 sqrt(p), (4.4)
chi*(V_{h+t}) - chi*(V_h)
= (1/2) integral_0^t Phi*(V_{h+s}) ds. (4.5)
4.6 Parameters.
Fix p large and set t = p^{3/4}, so that
t > 2 K^2 sqrt(p), 32 t / p < 1/8,
p / (256 (2 + t)) > 2 log(1 + 128 sqrt(2 + t)) + log 2 + 8. (4.6)
These are compatible because p/t = p^{1/4} grows faster than log p.
Explicit choice: p = 2^64, t = 2^48. Then 2 K^2 sqrt(p) = 2^47 < t;
32 t / p = 2^{-11}; and 2 + t < 2^49 makes the left side exceed
2^64/2^57 = 2^7 = 128, beating 67 log 2 + 8 < 55 since
128 sqrt(2 + t) < 2^32. From here on p, h, t are fixed; only matrix
sizes d -> infinity.
4.7 Spectral blocks.
For small h > 0, partition the spectrum of X_n = Y + sqrt(h) S_n at
the midpoints between the y_l. Let e_l(h) be the resulting spectral
projections and Delta = 2/(p-1) the atom spacing. Since
||sqrt(h) S_n|| = 2 sqrt(h), the condition h < Delta^2 / 64 puts the
spectrum in disjoint Delta/4-neighborhoods of the atoms, with every
threshold in a spectral gap. Choose continuous functions agreeing
with the interval indicators on those neighborhoods; continuous
functional calculus gives e_l(h) -> e_l(0) in norm. At h = 0 the
e_l(0) commute with every B_i, have trace 1/p, and satisfy
sum_{i=1}^p sum_{l=1}^p ||e_l(0) B_i e_l(0)||_2^2
= sum_{i=1}^p ||B_i||_2^2 = p.
Therefore fix 0 < h < min{1, Delta^2/64} and write e_l = e_l(h). For
X = V_h,
tau(e_l) < 3/(2 p) for each l,
sum_{i=1}^p sum_{l=1}^p ||e_l X_i e_l||_2^2 > 3 p / 4. (4.7)
4.8 Finite microstates entropy.
Lemma 4.2 (Finite chi). The tuple X = V_h has chi(X) > -infinity.
Proof. Use the deterministic models for C in dimensions d_k = p q_k,
then add independent GUE noise of variance h. If M bounds the
coordinate norms of the models, take R_0 = M + 8 sqrt(h). Lemma 2.2
and the GUE norm bound (2.3) show that these noised models lie in every
fixed moment neighborhood of X at cutoff R_0 with probability tending
to one. Their density is at most (d / (2 pi h))^{n d^2 / 2}. If q_d
denotes that probability,
Vol Gamma_{R_0}(X; m, d, epsilon)
>= q_d (2 pi h / d)^{n d^2 / 2}.
Since q_d -> 1 along the model dimensions, the normalized limsup
logarithm is at least (n/2) log(2 pi h). Take infimum over
(m, epsilon), then supremum over cutoffs. The limsup definition allows
restricting to this subsequence of dimensions. QED.
==============================================================================
5. ENTROPY PRODUCTION FOR SPECIFIED MATRIX ENSEMBLES
==============================================================================
Goal: transfer a terminal-time entropy bound back to the initial
microstates entropy, via Jekel-Pi's classical-to-free Fisher
comparison along the exact evolved ensemble.
Lemma 5.1 (Entropy production). Fix X = V_h. Let R > 0 be a cutoff
with chi_R(X) > -infinity. There exist dimensions d_k -> infinity and
random tuples A^{(d)}, uniform on increasingly accurate positive-volume
microstate sets of X at cutoff R, such that for independent standard
GUE tuples G^{(d)},
limsup_d H_d( A^{(d)} + sqrt(t) G^{(d)} )
>= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds. (5.1)
The limsup is over this particular ensemble.
Proof.
Entropy upper bound. Every uniform law on a positive-volume microstate
set at cutoff R has sum_i E tr_d(A_i^2) <= n R^2. Comparison with
independent GUE of variance R^2 bounds H_d(A) <= (n/2)log(2 pi e R^2).
Thus chi_R(X) is a finite real number.
Dimension sequence. Take d_k -> infinity with normalized log-volume at
parameters (k, 1/k) at least chi_R(X) - 1/k. The pairs (k, 1/k) are
cofinal among (m, epsilon); the sets have positive volume. So
liminf_d H_d(A^{(d)}) >= chi_R(X). (5.2)
Here and below d ranges over d_k. Every tuple in the support of A^{(d)}
has fixed cutoff and successively accurate moments of X; the choice
does not depend on the later noise time or polynomial tests.
De Bruijn identity. Put Z^{(d)}(s) = A^{(d)} + sqrt(s) G^{(d)} with
density f_s for s > 0. In the n d^2 real Hilbert-Schmidt coordinates,
the added Gaussian has covariance (s/d) I, so
partial_s f_s = (1/(2 d)) Delta f_s. Differentiating entropy and
integrating by parts:
d/ds H_d(Z^{(d)}(s)) = (1/2) I_d(s),
I_d(s) = sum_i E tr_d(F_i^2) = d^{-3} integral |grad log f_s|^2 f_s,
F_i = -(1/d) grad_i log f_s evaluated at Z^{(d)}(s). (5.3)
Gaussian convolution of a compactly supported distribution justifies
both steps on compact positive-time intervals: derivatives have
Gaussian decay, a Gaussian lower bound on f_s bounds |log f_s(x)| by
C(1 + ||x||^2), and grad log f_s(x) = -(d/s)(x - E[A^{(d)} | Z^{(d)}(s)
= x]) has at most linear growth. So entropy and Fisher information are
finite at positive times.
Classical gain. Entropy is concave under mixtures of translations, so
H_d(Z^{(d)}(epsilon)) >= H_d(A^{(d)}). Integrate epsilon -> t then let
epsilon -> 0:
H_d(Z^{(d)}(t)) >= H_d(A^{(d)}) + (1/2) integral_0^t I_d(s) ds. (5.4)
No entropy continuity at zero is needed.
Matrix-to-free Fisher. For fixed s > 0 and any complex polynomial P,
E tr_d(F_i P(Z^{(d)}(s)))
= E (tr_d tensor tr_d)((partial_i P)(Z^{(d)}(s))). (5.5)
Normalization check. Fix a Hermitian basis E_alpha orthonormal for the
unnormalized Hilbert-Schmidt inner product; it satisfies
sum_alpha E_alpha D E_alpha = Tr(D) 1. Scalar integration by parts
(real and imaginary parts separately) turns the left side into
d^{-2} sum_alpha E Tr(E_alpha D_i P(Z)[E_alpha]). At an occurrence
P = L Z_i R in a monomial, its contribution is
d^{-2} sum_alpha Tr(E_alpha L E_alpha R) = d^{-2} Tr(L) Tr(R) =
tr_d(L) tr_d(R). Summing over occurrences proves (5.5) in the correct
order.
Complete squares. For self-adjoint P_i,
I_d(s) >= sum_i E tr_d( 2 F_i P_i(Z^{(d)}(s)) - P_i(Z^{(d)}(s))^2 ).
By (5.5) the score terms reduce to products of normalized polynomial
traces. Lemma 2.2 then gives
lim_d RHS = sum_i [ 2 (tau tensor tau)((partial_i P_i)(V_{h+s}))
- tau(P_i(V_{h+s})^2) ].
The limiting law is V_{h+s} because the fresh semicircular noise is free
from X = V_h; conjugate variables exist by Lemma 3.1. Taking supremum
over tests via Lemma 2.1:
liminf_d I_d(s) >= Phi*(V_{h+s}). (5.6)
Along the same d_k, for every fixed s > 0. Since s -> V_{h+s} has
continuous moments, Phi*(V_{h+s}) is measurable (Lemma 2.1). Fatou on
the nonnegative I_d, together with (5.2) and (5.4):
liminf_d H_d(Z^{(d)}(t))
>= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds.
This proves (5.1), with liminf in place of limsup. The free Fisher
integral is finite since Phi*(V_u) <= n/u. Throughout the terminal law
is the same A^{(d)} + sqrt(t) G^{(d)}; no entropy-maximizing law at the
terminal distribution has been used. QED.
==============================================================================
6. A PERSISTENT BLOCK-STRUCTURE ENTROPY DEFICIT
==============================================================================
We bound the entropy of the terminal ensembles retained in Lemma 5.1.
Fix its cutoff R, sequence d_k, ensembles A^{(d)}, and parameters p, h, t.
For a decomposition of the identity into p orthogonal projections
P = (P_1, ..., P_p), write D_P D = (sum_l P_l D_i P_l)_i and
D_[p] = (D_1, ..., D_p).
6.1 Spectral witness.
Take P as the midpoint partition of the last coordinate of A^{(d)}.
Then for every tuple in the support of A^{(d)} and all large d,
rank(P_l) <= 2 d / p, ||D_P A^{(d)}_[p]||_{2,d}^2 > p / 2. (6.1)
Proof. Take a_d at cutoff R with moments converging to those of X.
Let I_l be the midpoint intervals (any endpoint convention) and nu_d
the empirical spectral measure of A_n^{(d)}. Weak convergence plus a
union N of small closed neighborhoods of thresholds disjoint from
sigma(X_n) give nu_d(N) -> 0. Pick continuous 0 <= f_l <= 1 agreeing
with 1_{I_l} outside N. Then f_l(X_n) = e_l and F_l^{(d)} = f_l(A_n^{(d)})
satisfies
||P_l - F_l^{(d)}||_{2,d}^2 <= nu_d(N) -> 0.
The cutoff gives ||P_l A_i^{(d)} P_l - F_l^{(d)} A_i^{(d)} F_l^{(d)}||_{2,d}
<= 2 R ||P_l - F_l^{(d)}||_{2,d}; compressed norms are at most R, so
squared norms also differ by o(1). Polynomial approximation and joint
moment convergence yield
rank(P_l)/d -> tau(e_l),
||P_l A_i^{(d)} P_l||_{2,d}^2 -> ||e_l X_i e_l||_2^2.
Block summands are Hilbert-Schmidt orthogonal, so their norms add to
||D_P A^{(d)}_[p]||^2. Strict margins in (4.7) then force (6.1).
Uniformity over d follows by the same subsequence contradiction as in
Lemma 2.2; closure points of microstate sets satisfy the same bounds.
6.2 The event.
Put v = 1 + h + t. For a matrix tuple D let E_d be the set of D with
||D_[p]||_{2,d} <= 4 sqrt(v p), (6.2a)
some P with rank(P_l) <= 2d/p
and ||D_P D_[p]||_{2,d} >= sqrt(p)/4. (6.2b)
E_d is a deterministic Borel event: for each rank list that respects
the rank bound, the space of decompositions is a compact unitary orbit;
the max of ||D_P D_[p]|| over that orbit is Lipschitz in D because
compressions are contractive. E_d does not depend on A^{(d)}; the
spectral decomposition of A_n^{(d)} only witnesses its probability under
the evolved law.
Lemma 6.1 (Event probability). Let mu_d be the law of
A^{(d)} + sqrt(t) G^{(d)} and gamma_d the law of independent centered
GUE coordinates of variances v_i(h+t) = h + t + c_i. Then for all
large d,
mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (6.3)
Proof.
Lower bound for mu_d. Fix A^{(d)} and its witness P; they are
independent of G^{(d)}. Put r_l = rank(P_l). The block-diagonal
Hermitian subspace has real dimension sum_l r_l^2. GUE covariance 1/d
and the normalized trace give
E[ ||sqrt(t) D_P G^{(d)}_[p]||_{2,d}^2 | A^{(d)} ]
= t p sum_l (r_l / d)^2 <= 2 t.
Markov at threshold p/16 bounds failure by 32 t / p < 1/8. On success,
||D_P (A + sqrt(t) G)_[p]||_{2,d}
>= ||D_P A_[p]||_{2,d} - ||D_P sqrt(t) G_[p]||_{2,d}
> sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
Independence and centering give
E||A_[p] + sqrt(t) G_[p]||^2 = E||A_[p]||^2 + t p -> v p.
Markov at 16 v p bounds failure of (6.2a) by 1/16 + o(1) < 1/8.
Union bound: mu_d(E_d) >= 13/16 > 1/2.
Upper bound for gamma_d. Enumerate at most (d+1)^p rank lists.
Choose an operator-norm delta-net of U(d) with delta = 1/(64 sqrt(v)).
A maximal delta-separated subset has unitary centers and is a delta-net.
Its ambient operator-norm balls of radius delta/2 are disjoint inside
the ball of radius 1 + delta/2 in real 2 d^2-dimensional M_d(C); volume
bounds the cardinality by (1 + 2/delta)^{2 d^2}.
Each decomposition of a fixed rank list is a unitary conjugate of the
standard coordinate blocks P^0 (including zero ranks). For
||U - V|| <= delta,
||U^* D_[p] U - V^* D_[p] V||_{2,d} <= 2 delta ||D_[p]||_{2,d}.
Fixed-block compression is contractive; replacing a witness unitary by a
net point loses at most 2 delta * 4 sqrt(vp) = sqrt(p)/8 in (6.2b). So
some rank list and some net unitary U satisfy
||D_{P^0}(U^* D_[p] U)||_{2,d} >= sqrt(p)/8.
For fixed list and unitary, unitary invariance preserves the independent
variance-v GUE law of the first p coordinates. The projected
Hilbert-Schmidt vector has q = p sum_l rank(P_l^0)^2 <= 2 d^2 real
coordinates of variance v/d. The normalized squared norm is (v/d^2) Q
with Q chi-squared of degree q. Since E e^{Q/4} = 2^{q/2}, the
probability that the squared norm reaches p/64, equivalently
Q >= p d^2/(64 v), is at most
exp( d^2 (log 2 - p/(256 v)) ). (6.4)
The union bound has logarithm at most
p log(d+1) + d^2 ( 2 log(1 + 128 sqrt(v)) + log 2 - p/(256 v) ).
Since v < 2 + t, (4.6) makes the bracket less than -8; the first term
is o(d^2). So gamma_d(E_d) <= exp(-4 d^2) for large d. QED.
6.3 Proof of Theorem 1.
Retain the same cutoff, dimension sequence, and ensembles. Put
alpha = mu_d(E_d), beta = gamma_d(E_d). Binary data processing for
relative entropy plus the binary-entropy bound give
KL(mu_d || gamma_d) >= alpha log(alpha/beta)
+ (1-alpha) log((1-alpha)/(1-beta))
>= alpha log(1/beta) - log 2.
Lemma 6.1 gives liminf_d d^{-2} KL(mu_d || gamma_d) >= 2.
Upper bound on H_d. At positive time the input is bounded, so the
Gaussian convolution has finite entropy and finite second moments. The
Gaussian reference density is
g_d(D) = prod_{i=1}^n ( (d / (2 pi v_i(h+t)))^{d^2/2}
exp(-d^2 tr_d(D_i^2) / (2 v_i(h+t))) ).
Using -int f log f = -int f log g_d - KL(f || g_d):
H_d(Z^{(d)}(t)) = (1/2) sum_i [ log(2 pi v_i(h+t))
+ E tr_d(Z_i^{(d)}(t)^2)/v_i(h+t) ]
- d^{-2} KL(mu_d || gamma_d).
The expected squared coordinates equal those of A_i^{(d)} plus t, and
converge to v_i(h+t). So
limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (6.5)
Assembly. The same terminal law is in Lemma 5.1. Match the lower
bound (Lemma 5.1) against the upper bound (6.5):
chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds
<= limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1.
Rearranging,
chi_R(X) <= L(h+t) - 1 - (1/2) integral_0^t Phi*(V_{h+s}) ds
<= chi*(V_{h+t}) - 1/2 - (1/2) integral_0^t Phi* ds
= chi*(V_h) - 1/2.
The middle inequality uses (4.4) at a = h+t > 2 K^2 sqrt(p), which
gives chi*(V_{h+t}) > L(h+t) - 1/2, hence L(h+t) - 1 < chi*(V_{h+t})
- 1/2. The last equality is (4.5). The bound holds for every cutoff
with chi_R(X) > -infinity; remaining cutoffs are automatic. Taking
supremum over R gives chi(X) <= chi*(V_h) - 1/2. Lemma 4.2 shows
chi(X) > -infinity; (4.3) at a = h shows chi*(V_h) finite. The
bounded matrix realization in Section 4 supplies the required faithful
normal tracial state and bounded self-adjoint tuple. QED.
==============================================================================
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END OF PAPER
==============================================================================