==============================================================================
A FINITE-ENTROPY SEPARATION OF MICROSTATES AND
NONMICROSTATES FREE ENTROPY
==============================================================================
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 a faithful normal
tracial state such that
-infinity < chi(X) <= chi*(X) - 1/2 < infinity.
Here chi is the original microstates entropy with an operator-norm
cutoff and a limsup over matrix sizes. The counterexample uses a
large but fixed number of variables.
1. INTRODUCTION
Free entropy connects the joint moments of noncommuting variables with
the geometry of their matrix approximations. For a bounded self-adjoint
tuple X = (X_1, ..., X_n) in a tracial von Neumann algebra, Voiculescu's
microstates free entropy chi(X) measures the asymptotic volumes of
matrix tuples whose normalized trace moments approximate those of X.
His nonmicrostates free entropy chi*(X) instead integrates free Fisher
information along addition of freely independent semicircular noise.
Both definitions, with normalizations, appear in Section 2.
For a single self-adjoint variable these two quantities agree. Their
agreement for general tuples was one of the questions in Voiculescu's
unification program. The finite-entropy formulation asks whether
chi(X) > -infinity implies chi(X) = chi*(X).
Guionnet stated this explicitly (Conjecture 7.1). The finiteness
hypothesis is substantial: as discussed by Jekel and Pi, there exist
nonapproximable laws with chi = -infinity < chi*. Our construction
provides a strict gap with both entropies finite.
Main theorem. There exist an integer n >= 2 and a bounded self-adjoint
tuple X = (X_1, ..., X_n) in a von Neumann algebra with faithful normal
tracial state such that
-infinity < chi(X) <= chi*(X) - 1/2 < infinity,
where chi uses an operator-norm cutoff and a limsup over matrix sizes.
This answers the finite-microstates-entropy equality question
negatively. The number of variables is large but fixed; nothing is
claimed for smaller prescribed n.
Prior work. Biane, Capitaine, and Guionnet proved chi <= chi* through
large-deviation bounds for matrix Brownian motion. Jekel and Pi gave an
elementary proof using classical entropy and Fisher information, and
extended it to conditional free entropy. Equality was known in
strictly convex smooth-potential regimes (Dabrowski; Jekel). Those
results also identify conditions under which the upper and lower
matrix-size limits agree. Our example shows that finite entropy alone
does not force equality.
Idea of the proof. Two noise scales.
Step 1 (signal). Fix a large integer p and set n = p + 1. Let
B_1, ..., B_p be free variance-one semicirculars. In a separate tensor
factor let Y be uniformly distributed on p equally spaced points in
[-1, 1]. Then Y commutes with every B_i but is not free from them.
Write C = (B_1, ..., B_p, Y). For a standard semicircular n-tuple S
free from C, put X = C + sqrt(h) S with h small. The tuple X has finite
microstates entropy, because on the matrix side it is generated by
deterministic models of C plus small Gaussian noise. Its last
coordinate X_n = Y + sqrt(h) S_n partitions the spectrum into p blocks
e_1, ..., e_p, and those blocks still contain most of the energy of the
first p coordinates.
Step 2 (matrix side). Uniform laws on increasingly accurate
cutoff-microstate sets of X, evolved by independent GUE noise of
variance t, retain a detectable simultaneous block decomposition: the
probability is at least 1/2 along a suitable sequence of dimensions,
whereas the corresponding independent GUE law with the same coordinate
variances has probability at most exp(-4 d^2). Relative-entropy data
processing then produces a normalized classical-entropy deficit of at
least 1 in the limit.
Step 3 (analytic side). For V_a = C + sqrt(a) S, Speicher's
noncrossing-cumulant calculus and orthogonality of semicircular Wick
polynomials give
chi*(V_a) = L(a) - (1/2) integral_a^infinity Phi-deficit(u) du,
where
L(a) = (1/2) sum_{i=1}^n log(2 pi e (a + c_i)), c_i = tau(C_i^2),
is the free entropy of the semicircular family with the same coordinate
variances, and Phi-deficit(u) = Phi*(V_u) - sum_i 1 / (u + c_i). The
integrand is bounded by squared higher-cumulant norms. The signal's
cumulants are sparse, so at large a this deficit is small.
Step 4 (transfer). The classical-to-free Fisher comparison of Jekel
and Pi, applied to the exact terminal matrix ensemble, yields an
entropy-production lower bound for that same terminal law in terms of
the initial microstates entropy plus the free Fisher integral.
Step 5 (matching). The terminal law's entropy is squeezed between the
upper bound from Step 2 (deficit 1) and the lower bound from Step 4
(deficit 0). This produces the gap.
Two noise scales. The analytic deficit is of order t^{-3} p^{3/2},
which is small whenever t >> sqrt(p). The Gaussian block estimate
beats the unitary covering cost whenever t log(p) / p -> 0. The choice
t = p^{3/4} satisfies both. We then choose one sufficiently large
finite p, then a small finite h, and only then let matrix sizes tend to
infinity with everything else fixed.
Relation to companion work. A companion paper proves that 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
within a fixed algebra; here we compare chi and chi* on a single tuple
with both values finite.
Organization. Section 2 fixes conventions and the matrix convergence
facts. Section 3 proves the cumulant bound on the Fisher deficit.
Section 4 constructs the signal, fixes both noise scales, and proves
chi(X) > -infinity. Section 5 proves the specified-ensemble
entropy-production lemma. Section 6 establishes the block event and
combines everything into the proof of the main theorem.
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2. CONVENTIONS AND PRELIMINARY FACTS
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Setting. All noncommutative variables live in a von Neumann algebra
(M, tau) with a faithful normal tracial state tau. Write
||a||_2^2 = tau(a^* a)
for the Hilbert norm. A standard semicircular family has variance one
in each coordinate and is free. Every noise family introduced below is
free from the tuple it perturbs.
Microstates. Let X = (X_1, ..., X_n) be a bounded self-adjoint tuple.
For R > 0, an integer m >= 1, and epsilon > 0, let
Gamma_R(X; m, d, epsilon)
be the set of self-adjoint d x d matrix tuples whose coordinate operator
norms are at most R and whose normalized traces of every word of length
at most m differ 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 Tr(A_i^* B_i). Set
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 chi_R(X).
The additive (n/2) log d normalizes like the entropy below. No-cutoff
laws with chi = -infinity are allowed.
Conjugate variables. 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
partial_i x_j = delta_ij (1 tensor 1),
partial_i (PQ) = (partial_i P)(Q) + (P tensor 1)(partial_i Q),
evaluated by substituting Z. Define Phi*(Z) = sum_i ||xi_i||_2^2 if
they exist, and Phi*(Z) = infinity otherwise. Conjugate variables, when
they exist, are unique and self-adjoint.
Lemma 2.1 (Polynomial variational formula). If a bounded self-adjoint
tuple 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) ],
the supremum being over self-adjoint noncommutative polynomials
P_1, ..., P_n. It suffices to take P_i = (Q_i + Q_i^*)/2 with
coefficients of Q_i in Q + i Q. Consequently, along a family of tuples
with continuous moments and existing conjugate variables, Phi* is a
measurable function of the parameter.
Proof. 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 follows from polynomial L^2 density, so each xi_i
is self-adjoint. 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 ].
For the density: the closed span K of polynomial vectors is invariant
under each bounded self-adjoint left multiplication operator Z_j. Its
orthogonal projection commutes with Z_j and with W^*(Z); fixing 1, it
fixes every bounded algebra vector, and those are dense in L^2. So
K = L^2. Symmetrizing approximants gives the supremum over self-adjoint
P_i; approximating coefficients by rationals gives a countable
supremum. Each functional in that supremum is continuous in the
moments, hence measurability of Phi*. QED.
Nonmicrostates normalization. With S a standard semicircular family
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.
The n / (1 + s) is the free Fisher contribution of n free semicircular
coordinates; for a semicircular family the bracket vanishes identically.
Matrix trace 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=1}^n log(2 pi e w_i).
This matches the microstates normalization exactly.
Lemma 2.2 (Bounded matrix inputs). Let K_d be sets of self-adjoint
matrix tuples with a common coordinate operator-norm bound. Suppose
normalized trace moments converge uniformly over K_d to the moments of
a bounded tuple T. Let A^{(d)} be a random tuple supported on K_d,
independent of a fixed finite standard GUE tuple G^{(d)}. Then 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. Products of finitely many such traces converge in
expectation. The dimensions may run over any sequence tending to
infinity.
Proof. Uniform norm bounds on GUE. A 1/4-net of the unit sphere of
C^d, viewed 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 centered real Gaussian of variance 1/d, since
Tr((v v^*)^2) = 1. Hence
P( ||G|| > L ) <= 2 exp( d (2 log 9 - L^2 / 8) ). (2.1)
In particular the coordinate norms of any fixed finite GUE tuple are at
most 8 with probability tending to one; integrating the tail yields
uniformly bounded moments of ||G||.
Gradient bound. Let a be deterministic with coordinate norms at most R
and put F = tr_d P(a, G) for a fixed polynomial P. Fix a Hermitian
basis E_alpha orthonormal for the unnormalized Hilbert-Schmidt inner
product and write G_j = sum_alpha g_{j,alpha} E_alpha. The coordinates
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. For a polynomial in
standard real Gaussian coordinates, expansion in orthonormal Hermite
polynomials shows that variance is at most the expected squared
gradient (each differentiation weights the squared modulus of a
nonconstant coefficient by its positive total degree). Rescaling to
variance 1/d, and applying separately 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}).
Since each B_j has fixed degree and a is uniformly bounded, the cited
trace-convergence theorem (Voiculescu, arbitrary deterministic-tuple
form) yields convergence in probability for every deterministic input
sequence with the required moments.
Uniformity. Suppose convergence over K_d failed: then for some
polynomial and some positive threshold, a subsequence of dimensions and
points a_d in K_d with error probability bounded below. These points
form a bounded deterministic moment-convergent sequence, contradicting
the previous conclusion. Conditioning on A^{(d)} gives convergence in
probability for the random input.
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
==============================================================================
We compare Phi*(V_u) to the value sum_i 1 / v_i(u) that a semicircular
family with the same covariance would have.
Free cumulants. For a centered bounded self-adjoint n-tuple C, let
kappa_m(C) be the 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 the moment-cumulant formula and its Mobius inversion on
noncrossing partitions, the vanishing of mixed cumulants between free
algebras, and the fact that a standard semicircular family has only
second cumulants equal to its covariance.
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) satisfying
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. Define 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 yields the original word. This
definition gives U_j^* = U_{(j_r, ..., j_1)}.
Expansion. Expand tau(U_j^* P(Z)) by noncrossing cumulants after
multilinear expansion Z = C + sqrt(u) S. Freeness and semicircularity
show that every block touching an initial S position is a pair; its
covariance is delta_{ij} against another initial S and sqrt(u) delta_{ij}
against a selected S in 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 its initial-initial pair blocks with originally
adjacent endpoints. For these fixed choices, contributions correspond
exactly to subsets F of A(pi): delete pairs in F, expand the remaining
word, then restore. Restoring adjacent pairs preserves noncrossingness
and the covariance factors, so their total coefficient is
sum_{F subset A(pi)} (-1)^{|F|} = (1 - 1)^{|A(pi)|}.
If pi has an initial-initial pair, an innermost one has adjacent
endpoints: its intervening positions are all initial and must each pair
inside that interval to avoid crossing. Thus A(pi) is nonempty and the
partition cancels. This includes nested internal pairs and does not use
deletion of newly adjacent positions.
Surviving partitions. In each surviving partition all initial positions
pair externally with positions of P, in reverse order. The indices in
U_j^* are j_r, ..., j_1, so the selected indices in P occur in order
j_1, ..., j_r. External arches prevent any remaining block from
connecting distinct gaps. Summing unrestricted partitions and choices
of C or S in each gap gives the trace of the gap, even though entries of
C need not be free. 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 a shorter word and produces a Kronecker delta
against a word of its own degree. Every other term of a Wick polynomial
has smaller degree, so U_j is orthonormal against Wick words of equal or
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.
Proposition 3.2 (Squared-cumulant Fisher bound). Let C be a centered
bounded self-adjoint n-tuple with diagonal covariance
tau(C_i C_j) = delta_ij c_i, c_i >= 0. Let S be a standard semicircular
family 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. Put Z = V_u. Conjugate variables xi_i come
from Lemma 3.1, 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. Start from 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 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,
and diagonal covariance makes the order-two term v_i(u) D_i(P). Adding
free semicircular noise leaves all higher cumulants unchanged.
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 the summability assumption
gives L^2 convergence with
sum_i ||R_i||_2^2 = sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2.
For a fixed P, terms with r > deg P have zero trace pairing with P(Z)
by (3.1). Since A -> tau(A P(Z)) is continuous on L^2, the finite
cumulant expansion gives
tau((Z_i - v_i(u) xi_i) P(Z)) = tau(R_i P(Z)).
The bilinear trace pairing needs no complex conjugation. Testing with
adjoint polynomials converts this to Hilbert orthogonality, and
polynomial L^2 density identifies
Z_i - v_i(u) xi_i = E[R_i | W^*(Z)].
Assembly. Since xi_i are self-adjoint by their conditional-expectation
formula 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 division by v_i(u)^2, using projection contraction and
v_i(u) >= u, gives (3.3). QED.
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4. THE SIGNAL AND THE TWO NOISE SCALES
==============================================================================
Competing requirements. The signal's higher cumulants must be sparse
enough for the analytic bound, while many coordinates must share a
block decomposition. Tensor independence supplies the latter without
spoiling the former.
Signal. Fix an integer p >= 2 and set n = p + 1. Let B_1, ..., B_p be
free variance-one semicirculars. In a separate tensor factor take Y
uniform on the p equally spaced points
y_l = -1 + 2 (l - 1) / (p - 1), l = 1, ..., p.
Thus 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)),
where S is a standard semicircular n-tuple free from C and
c_i = tau(C_i^2). Then c_i = 1 for i <= p, c_{p+1} = (1/p) sum_l y_l^2,
and C is centered with diagonal covariance. The benchmark 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.
Faithful realization and bounded matrix models. Choose uniformly
bounded deterministic matrix models B^{(q_k)} of the free semicircular
p-tuple (asymptotic freeness together with norm bounds from Section 2).
Tensor with I_p and take I_{q_k} tensor diag(y_1, ..., y_p) for Y. In
dimensions d_k = p q_k these give models of the required tensor law.
Adjoin an independent GUE n-tuple and pass to a subsequence satisfying
successively more moment tests and one fixed norm bound. The joint
limiting moments define a positive tracial functional in (C, S) with
the asserted freeness. In its GNS representation, left and right
coordinate multiplication are bounded, because the matrix inequality
tr_d(Q^* D_i^2 Q) <= R_i^2 tr_d(Q^* Q)
passes to the limit. The vacuum is cyclic for right polynomial
multiplication, which commutes with the weak closure of the left
algebra, so it is separating for that closure. Its vector state is
therefore faithful and normal; the trace identity extends by separate
weak continuity. This supplies the required tracial von Neumann
algebra. Further independent noise can be included. Faithfulness and
the semicircular distribution give ||B_i|| = ||S_i|| = 2.
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 and their Mobius
coefficients have magnitude at most 4^m (Catalan product formula,
interval factorization of the noncrossing lattice). Each product of
block moments has absolute value at most 2^m since ||B_i|| = 2 and
||Y|| <= 1. Thus each individual cumulant is at most 32^m.
Suppose b of the m positions carry B-indices. By tensor independence,
each block moment factors into its B-word moment times its Y-moment.
A nonzero Mobius sum requires, after deleting Y positions in each block,
a semicircular word admitting a noncrossing pairing with equal indices.
The union of these pairings across blocks is still noncrossing, because
crossings between distinct blocks 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. Therefore
||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.
Consequence. Proposition 3.2 combined with a geometric series of ratio
K^2 sqrt(p) / u < 1/2 for u > 2 K^2 sqrt(p) gives
0 <= Phi*(V_u) - sum_i 1 / (u + c_i)
<= 2 K^6 p^{3/2} u^{-4}. (4.2)
Entropy deficit. Adding free semicircular noise of variance s to V_a
produces V_{a+s}. By chi*'s definition,
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, and bounded above by
sum_i c_i / ( u (u + c_i) ) <= sum_i c_i / u^2,
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 subtraction of divergent integrals is used. In particular,
chi*(V_a) >= L(a) - K^6 p^{3/2} / (3 a^3) > L(a) - 1/2
for 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)
Parameters. Fix p large enough 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.
An explicit choice is p = 2^64 and t = 2^48. Then 2 K^2 sqrt(p) = 2^47
is less than t; 32 t / p = 2^{-11}; and for the last inequality, since
2 + t < 2^49 we have 2^64 / (256 (2 + t)) > 2^64 / 2^57 = 2^7 = 128,
while 128 sqrt(2 + t) < 2^32 makes the right side less than
67 log 2 + 8 < 55. From now on p, h, t are fixed and only matrix sizes
tend to infinity.
Spectral blocks. For small h > 0, partition the spectrum of
X_n = Y + sqrt(h) S_n by midpoints between the atoms y_l; write
e_l(h) for the resulting spectral projections and Delta = 2/(p - 1)
for the atom spacing. Since ||sqrt(h) S_n|| = 2 sqrt(h), the condition
h < Delta^2 / 64 places the spectrum inside disjoint Delta/4-neighborhoods
of the atoms, with every threshold lying in a spectral gap. Choose
continuous functions agreeing with the interval indicators on those
common neighborhoods. Continuous functional calculus, or uniform
polynomial approximation on those neighborhoods followed by norm
continuity, 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.
We may 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)
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 and
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 (2.1) show
that these noised models lie inside every fixed moment neighborhood of
X at cutoff R_0, with probability tending to one. Their density is
bounded above by (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
==============================================================================
We transfer a later entropy bound for one evolved matrix law back to
the initial microstates entropy. This uses the classical-to-free Fisher
comparison of Jekel and Pi, keeping the terminal ensemble.
Lemma 5.1 (Entropy production). Fix X = V_h. Let R > 0 be a cutoff
with chi_R(X) > -infinity. There are then 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 exactly this ensemble.
Proof. Upper entropy 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 its normalized
entropy above by (n/2) log(2 pi e R^2). Thus chi_R(X) is a finite real
number.
Choosing the dimension sequence. Take an increasing sequence d_k with
the normalized log-volume at parameters (m, epsilon) = (k, 1/k) at
least chi_R(X) - 1/k. Such arbitrarily large dimensions exist by the
defining limsup, and the corresponding microstate sets have positive
volume. The pairs (k, 1/k) are cofinal among all moment accuracies.
The uniform laws satisfy
liminf_d H_d(A^{(d)}) >= chi_R(X). (5.2)
Here and below, d ranges over the selected d_k. Every tuple in the
support of A^{(d)} has the same fixed cutoff and increasingly accurate
moments of X, and the selection does not depend on the subsequent noise
time or polynomial tests.
De Bruijn identity. Put Z^{(d)}(s) = A^{(d)} + sqrt(s) G^{(d)} and let
f_s be its density for s > 0. In the n d^2 real Hilbert-Schmidt
coordinates, the added Gaussian has covariance (s/d) I. Thus
partial_s f_s = (1/(2 d)) Laplacian f_s. Differentiating entropy and
integrating by parts gives
d/ds H_d(Z^{(d)}(s)) = (1/2) I_d(s), I_d(s) = sum_i E tr_d(F_i^2),
F_i = -(1/d) grad_i log f_s evaluated at Z^{(d)}(s),
I_d(s) = d^{-3} integral ||grad log f_s||^2 f_s. (5.3)
Gaussian convolution of a compactly supported distribution justifies
differentiation and integration by parts 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. In particular the entropy and Fisher
information are finite at positive times.
Classical gain. Entropy concavity under mixtures of translations gives
H_d(Z^{(d)}(epsilon)) >= H_d(A^{(d)}). Integrating from epsilon to t and
letting epsilon -> 0 gives
H_d(Z^{(d)}(t)) >= H_d(A^{(d)}) + (1/2) integral_0^t I_d(s) ds.
(5.4)
No continuity of entropy at zero is needed.
Matrix-to-free Fisher. For fixed s > 0, integration by parts against
any complex polynomial P gives
E tr_d(F_i P(Z^{(d)}(s)))
= E (tr_d tensor tr_d)((partial_i P)(Z^{(d)}(s))). (5.5)
To verify the normalization, take 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 for every complex D. Scalar
integration by parts on real and imaginary parts turns the left side of
(5.5) into
d^{-2} sum_alpha E Tr( E_alpha D_i P(Z^{(d)}(s))[E_alpha] ),
where D_i P(Z)[E] is the directional derivative in the ith matrix
coordinate along E. 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) with the prefix and suffix in the
same order as the free difference quotient.
Completing squares. For self-adjoint polynomial tests 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 become products of normalized polynomial
traces. Lemma 2.2 then gives the limit of the right side as
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 new semicircular noise is free
from X = V_h, and conjugate variables exist by Lemma 3.1. Taking
supremum over tests via Lemma 2.1 gives
liminf_d I_d(s) >= Phi*(V_{h+s}). (5.6)
This holds for every fixed s > 0 along the same selected dimensions.
The family s -> V_{h+s} has continuous moments, so Lemma 2.1 also gives
measurability of Phi*(V_{h+s}). Fatou's lemma on the nonnegative
functions I_d, then (5.2) and (5.4), yield
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 integral is
finite since Phi*(V_{h+s}) <= n/(h+s) and h > 0. Throughout, the
terminal law is that of the same A^{(d)} + sqrt(t) G^{(d)}; no
entropy-maximizing law at the terminal distribution has been chosen.
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, dimension sequence, uniform microstate 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,
D_{[p]} = ( D_1, ..., D_p ).
Spectral witness. Use the midpoint spectral partition of the last
coordinate of A^{(d)} to obtain P. Then for every tuple in the support
of A^{(d)} and all sufficiently large dimensions,
rank(P_l) <= 2 d / p, || D_P A^{(d)}_{[p]} ||_{2,d}^2 > p / 2.
(6.1)
Uniformity of the witness. Suppose a deterministic sequence of tuples
at cutoff R has joint moments converging to those of X. Let I_l be the
midpoint intervals (with any fixed endpoint convention) and nu_d the
empirical spectral measure of A_n^{(d)}. Since nu_d converges weakly to
the law of X_n, choose a union N of small closed neighborhoods of the
midpoint thresholds, disjoint from sigma(X_n). On a common compact
interval containing [-R, R] and sigma(X_n), choose continuous functions
0 <= f_l <= 1 agreeing with 1_{I_l} outside N. Then f_l(X_n) = e_l,
and with F_l^{(d)} = f_l(A_n^{(d)}),
|| P_l - F_l^{(d)} ||_{2,d}^2 <= nu_d(N) -> 0.
Here weak convergence gives nu_d(N) -> 0 because N is closed with zero
limiting mass. Small spectral outliers and eigenvalues exactly at
thresholds are permitted.
The cutoff bounds multiplication: for each i,
|| 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}.
Since the compressed norms are at most R, squared norms also differ by
o(1). Uniform polynomial approximation of f_l, plus joint moment
convergence, gives
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.
The block summands are orthogonal in Hilbert-Schmidt, so their squared
norms add to || D_P A^{(d)}_{[p]} ||_{2,d}^2. The strict margins in
(4.7) give (6.1). Uniformity over the selected supports holds; if it
failed, a violating tuple in each dimension of a subsequence would form
a bounded moment-convergent sequence, contradicting the argument.
Closure points of the microstate sets satisfy the same moment bounds
with weak inequalities whose errors tend to zero.
The event. Put v = 1 + h + t. For a matrix tuple D, let E_d be the
set of D satisfying
||D_{[p]}||_{2,d} <= 4 sqrt(v p),
there is a decomposition P with rank(P_l) <= 2 d / p
and || D_P D_{[p]} ||_{2,d} >= sqrt(p) / 4. (6.2)
This is a deterministic Borel event. For each admissible rank list the
space of projection decompositions is a compact unitary orbit, and the
maximum of || D_P D_{[p]} ||_{2,d} on that orbit is continuous in D (in
fact Lipschitz in the tuple 2-norm, since every compression is
contractive). E_d is a finite union of closed threshold sets
intersected with a closed norm ball. Its definition 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
sufficiently large d,
mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (6.3)
Proof.
Lower bound for mu_d. Use the witness of (6.1) before noise.
Conditional on A^{(d)}, the projections P are fixed and independent of
G^{(d)}. Put r_l = rank(P_l). The block-diagonal Hermitian subspace
has real dimension sum_l r_l^2, so the 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's inequality at threshold p/16 bounds failure by
2 t / (p / 16) = 32 t / p < 1/8. On success, the projected total norm
exceeds sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
Independence and centering also give
E || A^{(d)}_{[p]} + sqrt(t) G^{(d)}_{[p]} ||_{2,d}^2
= E || A^{(d)}_{[p]} ||_{2,d}^2 + t p
-> v p.
Markov's inequality at 16 v p bounds the failure of the norm condition
in (6.2) by 1/16 + o(1) < 1/8. The union bound proves
mu_d(E_d) >= 3/4, stronger than claimed.
Upper bound for gamma_d. Enumerate at most (d + 1)^p rank lists.
Choose an operator-norm delta-net of the unitary group with
delta = 1 / (64 sqrt(v)). A maximal delta-separated subset of U(d) 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 of
the real 2 d^2-dimensional space M_d(C); volume comparison bounds the
cardinality by (1 + 2/delta)^{2 d^2}.
Every decomposition of a fixed rank list is a unitary conjugate of the
standard coordinate blocks P^0 (including lists with zero ranks). For
unitaries with || U - V || <= delta,
|| U^* D_{[p]} U - V^* D_{[p]} V ||_{2,d}
<= 2 delta || D_{[p]} ||_{2,d}.
Fixed-block compression is contractive, so replacing a witnessing
unitary by a net point loses at most 2 delta * 4 sqrt(v p) = sqrt(p)/8
in (6.2). Thus some admissible 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 on 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) ) ).
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) because p is fixed. This proves gamma_d(E_d) <= exp(-4 d^2)
for large d. QED.
Proof of the main theorem.
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 and the bound on binary entropy give
KL(mu_d || gamma_d) >= alpha log(alpha / beta)
+ (1 - alpha) log((1 - alpha) / (1 - beta))
>= alpha log(1/beta) - log 2.
By Lemma 6.1,
liminf_d d^{-2} KL(mu_d || gamma_d) >= 2.
We retain one unit of this normalized deficit in the comparison.
At positive time the bounded-input Gaussian convolution has finite
entropy and finite second moments, and its relative entropy to gamma_d
is finite. The Gaussian reference has density
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 - integral f log f = - integral f log g_d - KL(f || g_d) together
with the normalization of H_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 second moments equal those of A_i^{(d)} plus t, and
converge to v_i(h+t). Hence
limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (6.4)
This concerns the exact terminal law Z^{(d)}(t) = A^{(d)} + sqrt(t) G^{(d)}
of Lemma 5.1. Combining the two bounds on that entropy with (4.4) and
(4.5):
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*(V_{h+s}) ds
= chi*(V_h) - 1/2.
This holds for every cutoff with chi_R(X) > -infinity; for the
remaining cutoffs it is automatic. Taking the supremum over cutoffs
preserves the inequality. Lemma 4.2 gives chi(X) > -infinity, and
(4.3) with a = h gives chi*(X) finite. The bounded matrix realization
in Section 4 supplies the required faithful normal tracial state and
bounded self-adjoint tuple. This completes the proof. QED.
==============================================================================
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==============================================================================
END OF PAPER
==============================================================================