editors: nobody
Two invariants dominate the modern theory of free entropy. For a
bounded self-adjoint tuple X = (X_1, ..., X_n) in a von Neumann algebra
with a faithful normal tracial state tau, the microstates free entropy
chi(X) is the asymptotic exponential volume, at matrix size d after
dividing the logarithm by d^2, of self-adjoint d x d tuples whose
normalized trace moments of short noncommutative words approximate those
of X. The nonmicrostates free entropy chi*(X) integrates Voiculescu's
free Fisher information Phi*(X) along free convolution with a standard
semicircular family, the free analogue of integrating classical Fisher
information along heat flow.
Voiculescu introduced both quantities in a sequence of Inventiones
papers hitting in 1991, 1994, and 1998, and consolidated the framework
in a 2002 Bulletin of the London Mathematical Society survey that also
posed the finiteness question that motivates the construction. Guionnet's Probability
Surveys article collects the definitions, the standard comparison
results, and the original form of the conjecture.
Online (publisher): https://doi.org/10.1007/BF01245072
Online (publisher): https://doi.org/10.1007/BF01231539
Online (publisher): https://doi.org/10.1007/s002220050222
Online (publisher): https://doi.org/10.1112/S0024609301008992
Local PDF: pdf/p_guionnet-survey.pdf
Online: https://www.emis.mi.sanu.ac.rs/EMIS/journals/PS/images/getdoc1af6.pdf?id=48&article=18&mode=pdf
The inequality chi(X) <= chi*(X) holds for every bounded self-adjoint
tuple. Biane, Capitaine, and Guionnet proved it via large deviation
bounds for Hermitian matrix-valued Brownian motion; Jekel and Pi gave an
elementary proof via classical entropy and Fisher information, and
extended the inequality to conditional free entropy. Equality chi =
chi* holds in several regular settings: Dabrowski's theorem covers a
class of strictly convex potentials with Lipschitz derivatives and
noncommutative polynomial approximation; Jekel's theorem covers convex
matrix potentials with appropriate concentration and trace-polynomial
approximation. These hypotheses rule out the geometric obstruction
exploited below.
Online (publisher): https://doi.org/10.1007/s00222-002-0281-4
Local PDF: pdf/p_jekelpi.pdf
Online: https://arxiv.org/pdf/2305.02574
Local PDF: pdf/p_dabrowski.pdf
Online: https://arxiv.org/pdf/1604.06420
Local PDF: pdf/p_jekel.pdf
Online: https://arxiv.org/pdf/1805.08814
Finiteness of chi(X) does not force equality. Jekel and Pi recall
nonapproximable laws with chi = -infinity < chi*. Guionnet formulated
the remaining version of the question as Conjecture 7.1 of her survey:
whether chi(X) > -infinity implies chi(X) = chi*(X). The result proved
here answers it negatively:
-infinity < chi(X) <= chi*(X) - 1/2 < infinity
for some bounded self-adjoint tuple X in a von Neumann algebra with a
faithful normal tracial state. The tuple uses a large but fixed number
of generators; no claim is made for small n or for tuples of independent
interest.
The construction runs on two scales of semicircular noise around a rough
signal C built from p free semicirculars plus a commuting uniform
variable Y. Tensor independence gives C sparse higher cumulants.
Small noise X = C + sqrt(h) S keeps chi(X) finite. A larger noise
V_a = C + sqrt(a) S, at a comparable to t = p^{3/4}, gives a controlled
Fisher deficit at positive noise: the free Fisher information of V_a is
close to that of a semicircular family of the same covariance, so chi*(V_a)
sits just below the classical entropy L(a) of that family. Applying the
Jekel-Pi comparison along the same evolved matrix ensemble and matching
its terminal-time normalized entropy against the Gaussian reference of
the same covariance yields the strict gap of 1/2. The ingredients are:
(i) two matrix convergence lemmas - a variational formula for Phi* and
uniform trace convergence for bounded deterministic input;
(ii) a cumulant-Fisher estimate reducing the deficit of Phi* against
the semicircular baseline to squared higher-cumulant norms;
(iii) the specific signal C and its sparse cumulants;
(iv) a de Bruijn identity for the evolved ensemble, whose Fisher
information dominates Phi*(V_{h+s}) in the matrix limit;
(v) a block event - the persistence of a simultaneous p-block
decomposition of the last coordinate at cutoff R - whose
probability stays above 1/2 under the evolved law and below
exp(-4 d^2) under the matching Gaussian reference;
(vi) an entropy sandwich that closes the argument.
A companion result, established elsewhere, resolves the free group
factor isomorphism problem affirmatively and shows each of the four free
entropy dimensions of L(F_2) attains every integer value n >= 2 on
suitable generating tuples. That statement varies the tuple inside a
fixed von Neumann algebra; the construction fixes a tuple and
compares chi and chi* on it, both finite.
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The quantitative definitions are as follows. Let m >= 1 be an integer,
epsilon > 0, R > 0 a cutoff, and d a matrix dimension. Write
Gamma_R(X; m, d, epsilon) for the set of self-adjoint d x d matrix
tuples (A_1, ..., A_n) with coordinate operator norms at most R and with
the normalized trace of every word of length at most m within epsilon of
its value on X. Volume is taken in the real Euclidean space of tuples
with inner product <A, B> = Re sum_i Tr(A_i^* B_i). Then
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),
with the convention that chi is -infinity when every chi_R is. The
additive (n/2) log d matches the classical entropy normalization below.
Conjugate variables live in L^2(M, tau). For a noncommutative polynomial
P in n formal indeterminates, the free difference quotient partial_i is
the linear map satisfying partial_i x_j = delta_{ij} (1 tensor 1) and
the Leibniz rule. Given a bounded self-adjoint tuple Z, the conjugate
variables xi_1, ..., xi_n are the unique elements of L^2 satisfying
tau(xi_i P(Z)) = (tau tensor tau)((partial_i P)(Z))
for every polynomial P. Put Phi*(Z) = sum_i ||xi_i||_2^2, possibly
infinite when conjugate variables fail to exist. That conjugate
variables are self-adjoint and unique in L^2 is standard.
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Lemma 1 (Variational formula for Phi*). If Z has conjugate variables
xi_i in L^2, then
Phi*(Z) = sup sum_i [ 2 (tau tensor tau)((partial_i P_i)(Z)) - tau(P_i(Z)^2) ],
the supremum over self-adjoint noncommutative polynomials P_1, ..., P_n.
The supremum may be restricted to polynomials with coefficients in Q + iQ.
Proof. Polynomial density in L^2. The closed span K of {Q(Z) : Q
polynomial} is invariant under each bounded self-adjoint Z_j, because
Z_j Q(Z) is again a polynomial in Z. The orthogonal projection onto K
commutes with Z_j, hence with W^*(Z). Since it fixes 1, it fixes every
bounded algebra vector; those are dense in L^2. So K = L^2.
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 L^2 density makes xi_i self-adjoint.
Variational identity. For self-adjoint P_i,
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 recovers sum_i ||xi_i||_2^2. Restricting the supremum
to rational coefficients and self-adjoint P_i preserves the value,
because Z has bounded moments. QED.
The 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.
For a semicircular family of variances w_i the bracket vanishes and
chi* reduces to (n/2) sum_i log(2 pi e w_i), matching the free entropy
of that family. For general Z the integral is finite on every compact
interval of positive noise times once Phi*(Z + sqrt(s) S) <= n/s, an
inequality proved below (Lemma 3).
Matrix conventions. Write tr_d = d^{-1} Tr and
||D||_{2,d}^2 = sum_i tr_d(D_i^* D_i) for a matrix tuple D. A standard
GUE tuple G^{(d)} consists of n independent centered Gaussian
self-adjoint matrices whose real Hilbert-Schmidt coordinates are
independent with variance 1/d. For a random matrix tuple with density f,
the 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), matching the microstate normalization.
Lemma 2 (Bounded matrix inputs). Let K_d be sets of self-adjoint matrix
tuples with a common coordinate operator-norm bound. Suppose the
normalized trace moments converge uniformly over K_d to those 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:
(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 n-tuple free from T;
(ii) products of finitely many such traces converge in expectation;
(iii) the dimensions may range over any sequence tending to infinity.
Proof. GUE norm bound. 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 because Tr((v v^*)^2) = 1. So
P(||G|| > L) <= 2 exp(d (2 log 9 - L^2/8)). (1)
In particular the coordinate norms of any fixed finite GUE tuple are at
most 8 with probability tending to one, and every fixed moment of
||G|| is uniformly bounded in d.
Gradient bound. Fix a deterministic tuple a of coordinate norms at
most R and put F = tr_d P(a, G) for a fixed polynomial P. Choose 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
g_{j alpha} are independent centered 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 variance bound. A polynomial in standard real Gaussian
coordinates has variance at most the expected squared gradient:
differentiation weights each nonconstant Hermite coefficient's squared
modulus by its positive total degree. After 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}).
Convergence. So F converges in probability on every deterministic
moment-convergent sequence a_d, with expectation converging to the value
at (T, S) by Voiculescu's trace-convergence theorem. Uniformity over
K_d follows: a failure would produce a subsequence d_k and points
a_{d_k} in K_{d_k} whose error probabilities are bounded away from
zero, forming a bounded deterministic moment-convergent sequence that
contradicts the previous sentence. 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. The
squared estimate is uniformly integrable by the norm-moment bound, so
convergence in probability gives convergence of expectations. QED.
Free cumulants are the standard bookkeeping device. For a centered
bounded self-adjoint n-tuple C, let kappa_m(C) denote the order-m free
cumulant array, with
||kappa_m(C)||_2^2 = sum_{i_1, ..., i_m} |kappa_m(C_{i_1}, ..., C_{i_m})|^2.
Three facts enter: the moment-cumulant formula and its Mobius inversion
on noncrossing partitions; vanishing of mixed cumulants between free
algebras; and that a standard semicircular family has only second
cumulants, given by its covariance. All three are in Nica-Speicher.
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Lemma 3 (Wick testing). Let C be a centered bounded self-adjoint
n-tuple, S a standard semicircular n-tuple free from C, and
V_u = C + sqrt(u) S for u > 0. For a multi-index j = (j_1, ..., j_r)
with r >= 1 and a monomial P in x_1, ..., x_n, set
D_j(P) = sum_{P = P_0 x_{j_1} P_1 ... x_{j_r} P_r} prod_{a=0}^r tau(P_a(V_u)),
the sum over increasing factorization positions with empty gaps
contributing one. Extend D_j linearly to all 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(V_u)). (2)
In particular the conjugate variables of V_u exist and equal
xi_i = u^{-1/2} E[S_i | W^*(V_u)].
Proof. Construction. From the word S_{j_1} ... S_{j_r}, sum 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}}.
An empty subset yields the original word. The definition gives
U_j^* = U_{(j_r, ..., j_1)}.
Multilinear expansion. Expand tau(U_j^* P(V_u)) via noncrossing
cumulants after multilinear expansion V_u = C + sqrt(u) S. Freeness
and semicircularity force every block touching an initial S position
to be a pair, with covariance delta_{jk} against another initial S and
sqrt(u) delta_{jk} against a selected S in P.
Cancellation of deletions. Fix a choice of C or S at every P-position
and a noncrossing partition pi on all positions. Let A(pi) be the
initial-initial pair blocks whose endpoints were previously adjacent.
For fixed choices, contributions correspond exactly to subsets
F subset A(pi): delete pairs in F, expand the remaining word, restore.
Restoring adjacent pairs preserves noncrossingness and covariance
factors, so the total coefficient is
sum_{F subset A(pi)} (-1)^{|F|} = (1 - 1)^{|A(pi)|}.
If pi has any initial-initial pair, its innermost one has adjacent
endpoints - intervening positions are all initial, and each must pair
inside to avoid crossing - so A(pi) is nonempty and pi cancels. This
argument covers nested internal pairs and deletes nothing newly
adjacent after each step.
Surviving partitions. In each surviving partition, all initial
positions pair externally with positions of P, in reverse order. The
initial indices of U_j^* are j_r, ..., j_1, so the selected indices in
P occur as j_1, ..., j_r. External arch pairs prevent any remaining
block from connecting distinct gaps. Summing the unrestricted
partitions and the C/S choices within each gap gives the trace of that
gap, even when entries of C fail to be free. Each external pair
contributes sqrt(u), proving (2).
Orthonormality. With C = 0 and u = 1, a Wick word tests to zero against
shorter words and to a Kronecker delta against a word of its own
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)(V_u)),
so (2) together with the definition of conditional expectation gives
xi_i = u^{-1/2} E[S_i | W^*(V_u)]. QED.
Proposition 1 (Squared-cumulant Fisher bound). Let C be centered,
bounded, self-adjoint, with diagonal covariance tau(C_i C_j) = delta_ij c_i
for c_i >= 0. Let S be a standard semicircular n-tuple free from C,
and put V_u = C + sqrt(u) S with v_i(u) = u + c_i for u > 0. Then
sum_i 1 / v_i(u) <= Phi*(V_u) <= n / u. (3)
If additionally 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. (4)
Proof. Bounds. Put Z = V_u. Lemma 3 supplies xi_i, and conditional
expectation 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
left half of (3).
Cumulant expansion. Extract the block containing the first position in
tau(Z_i P(Z)). Its other positions select letters in D_j(P); 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.
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. 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)^*
converges by orthonormality of the adjoint Wick family, 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 vanish in trace against P(V_u) by
Lemma 3. Continuity of A -> tau(A P(V_u)) on L^2 and finite truncation
of the cumulant expansion give
tau((Z_i - v_i(u) xi_i) P(V_u)) = tau(R_i P(V_u)).
Trace is bilinear, so no complex conjugation is needed here. Testing
with adjoints turns this into Hilbert-space orthogonality; polynomial
L^2 density identifies
Z_i - v_i(u) xi_i = E[R_i | W^*(V_u)].
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) ).
Divide by v_i(u)^2, sum, and use projection contraction plus v_i(u) >= u
to obtain (4). QED.
Two consequences enter later. Equation (3) makes the Fisher deficit
integrable on any compact interval of positive noise times, which is
what makes the free-flow identity (8) legitimate; equation (4) reduces
the large-noise entropy estimate to control of the squared higher-
cumulant norms of the signal C.
The witness 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 take Y uniform on p equally spaced points y_l = -1 + 2(l-1)/(p-1)
for l = 1, ..., p, so Y commutes with every B_i but is not free from
them. Put
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),
where S is a standard semicircular n-tuple free from C. Then c_i = 1
for 1 <= i <= p and c_n = (1/p) sum_l y_l^2, so C is centered with
diagonal covariance. L(a) is at once the nonmicrostates entropy of the
semicircular family with variances a + c_i and the normalized classical
entropy of the corresponding independent GUE law.
Faithful realization. Choose uniformly bounded deterministic matrix
models B^{(q_k)} for the free semicircular p-tuple (Voiculescu's trace-
convergence and norm-boundedness theorem), 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 give models of the required tensor law. Adjoin an independent
GUE n-tuple and select a subsequence satisfying successively more
moment tests with a single fixed operator-norm bound. The joint
limiting moments define a positive tracial functional on the variables
(C, S) with the asserted freeness. In its GNS representation, left
multiplication by C_i is bounded because the matrix inequality
tr_d(Q^* D_i^2 Q) <= R_i^2 tr_d(Q^* Q) passes to the limit. The GNS
vacuum is cyclic for right polynomial multiplication, which commutes
with the weak closure of the left algebra, so it is separating for that
closure and its vector state is faithful and normal. Faithfulness and
the semicircular distribution give ||B_i|| = ||S_i|| = 2.
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Lemma 4 (Cumulant bound). With K = 128 and C as above, for every
m >= 1,
||kappa_m(C)||_2 <= K^m p^{m/4}. (5)
Proof. There are at most 4^m noncrossing partitions, with Mobius
coefficients bounded by 4^m; each block moment has modulus at most 2^m
because ||B_i|| = 2 and ||Y|| <= 1. So 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 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. 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.
Fisher deficit for the signal. Combine Proposition 1 with Lemma 4:
the geometric series sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 has
ratio K^2 sqrt(p) / u, so for u > 2 K^2 sqrt(p) its value is at most
8 K^6 p^{3/2} u^{-4} and
0 <= Phi*(V_u) - sum_i 1 / (u + c_i) <= 2 K^6 p^{3/2} u^{-4}. (6)
Entropy deficit. Adding free semicircular noise of variance s to V_a
yields V_{a+s}. By the definition of chi*,
chi*(V_a) = L(a) - (1/2) integral_a^infinity
[ Phi*(V_u) - sum_i 1 / (u + c_i) ] du. (7)
All terms are finite for a > 0: the integrand is measurable by Lemma 1,
nonnegative by Proposition 1, 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_{h+t}) - chi*(V_h)
= (1/2) integral_0^t Phi*(V_{h+s}) ds, (8)
chi*(V_a) >= L(a) - K^6 p^{3/2} / (3 a^3) > L(a) - 1/2
whenever a > 2 K^2 sqrt(p). (9)
Parameters. Fix p large and 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. (10)
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 2 + t < 2^49 gives
2^64 / (256 (2 + t)) > 2^64 / 2^57 = 128, while
128 sqrt(2 + t) < 2^32 makes the right-hand side of the third
inequality at most 67 log 2 + 8 < 55. From this point p, h, t are
fixed; h will be chosen small next; only matrix dimensions tend to
infinity.
Spectral blocks. For h > 0 small, partition the spectrum of X_n = Y +
sqrt(h) S_n at the midpoints of the y_l, and write e_l(h) for the
resulting spectral projections. Let Delta = 2/(p - 1) be the spacing.
Since ||sqrt(h) S_n|| = 2 sqrt(h), the condition h < Delta^2 / 64 places
the spectrum inside the disjoint Delta/4-neighborhoods of the y_l,
with every threshold 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.
There is therefore a fixed 0 < h < min{1, Delta^2/64} for which, with
e_l = e_l(h) and 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. (11)
Lemma 5 (Finite microstates entropy). The tuple X = V_h has
chi(X) > -infinity.
Proof. Use deterministic models of C in dimensions d_k = p q_k and
adjoin independent GUE noise of variance h. If M bounds the coordinate
norms of the models, take R_0 = M + 8 sqrt(h). Lemma 2 and (1) place
the noised models inside 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, then
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). Infimum over (m, epsilon) and
supremum over R conclude. QED.
Lemma 6 (Entropy production). Fix X = V_h and a cutoff R > 0 with
chi_R(X) > -infinity. There exist dimensions d_k -> infinity and
random tuples A^{(d)}, uniform on increasingly accurate positive-volume
microstate sets 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. (12)
The limsup is over this specific 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) from
above by (n/2) log(2 pi e R^2); hence 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. Such arbitrarily large
dimensions exist by the defining limsup and the sets have positive
volume. The pairs (k, 1/k) are cofinal among (m, epsilon). Their
uniform laws satisfy
liminf_d H_d(A^{(d)}) >= chi_R(X). (13)
Here and below d ranges over d_k. Every tuple in the support of A^{(d)}
has the same fixed cutoff and increasingly accurate moments of X; the
choice is independent of the later 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, so
partial_s f_s = (1/(2 d)) Delta f_s. Differentiating the 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) = d^{-3} integral |grad log f_s|^2 f_s,
F_i = -(1/d) grad_i log f_s evaluated at Z^{(d)}(s). (14)
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 controls |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. Entropy and Fisher information are
therefore finite at positive times.
Classical gain. Entropy is concave under mixtures of translations,
so H_d(Z^{(d)}(epsilon)) >= H_d(A^{(d)}) for each epsilon > 0.
Integrating from 0 to t and letting epsilon -> 0,
H_d(Z^{(d)}(t)) >= H_d(A^{(d)}) + (1/2) integral_0^t I_d(s) ds. (15)
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))). (16)
Normalization. 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 for every complex D. Scalar
integration by parts on real and imaginary parts turns the left side
into
d^{-2} sum_alpha E Tr(E_alpha D_i P(Z)[E_alpha]),
where D_i P(Z)[E] is the directional derivative in the i-th 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 (16) with the prefix and suffix in the
same order as the free difference quotient.
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 (16) the score terms become products of normalized polynomial
traces. Lemma 2 then gives the limit of the right-hand 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 fresh semicircular noise is
free from X = V_h, and conjugate variables exist by Lemma 3. Taking
the supremum over these tests via Lemma 1,
liminf_d I_d(s) >= Phi*(V_{h+s}). (17)
This holds along the same d_k for every fixed s > 0. Continuous
moments of s -> V_{h+s} make Phi*(V_{h+s}) measurable (Lemma 1).
Fatou's lemma on the nonnegative I_d, then (13) and (15), yield
liminf_d H_d(Z^{(d)}(t))
>= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds.
That proves (12), even with liminf in place of limsup. QED.
Lemma 7 (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. For all
sufficiently large d,
mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (20)
Proof. Witness for mu_d. By (11) and Lemma 2, the empirical spectral
decomposition of A_n^{(d)} supplies a decomposition P with
rank(P_l) <= 2d/p and
||D_P A^{(d)}_[p]||_{2,d}^2 > p/2. (18)
Fix A^{(d)} together with its witness P; both are independent of
G^{(d)}. Put r_l = rank(P_l). GUE covariance 1/d and normalized
trace give
E[ ||D_P sqrt(t) G^{(d)}_[p]||_{2,d}^2 | A^{(d)} ]
= t p sum_l (r_l/d)^2 <= t p * p * (2/p)^2 = 4 t.
Markov at squared threshold p/16 bounds failure by 64 t/p < 1/4. On
success, the triangle inequality and (18) yield
||D_P (A + sqrt(t) G)_[p]||_{2,d}
>= sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
Centeredness and independence also give
E ||(A^{(d)} + sqrt(t) G^{(d)})_[p]||_{2,d}^2 = v p (1 + o(1)),
which is at most 2 v p for large d. Markov at squared norm 16 v p
bounds failure by 1/8 + o(1). Union bound: both conditions hold with
probability at least 3/4 + 7/8 - 1 = 5/8 > 1/2. That proves
mu_d(E_d) >= 1/2.
Unitary net. Enumerate at most (d+1)^p admissible rank lists. For
each, take 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
real (2 d^2)-dimensional matrix space, giving cardinality at most
(1 + 2/delta)^{2 d^2}.
Local PDF: pdf/p_szarek.pdf
Online: https://arxiv.org/pdf/math/9701213
Compression and reduction. Every decomposition of a fixed rank list
(including zero ranks) is a unitary conjugate of the standard
coordinate blocks P^0. For unitaries U, V 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(vp) = sqrt(p)/8
in the event condition (19). Thus some admissible rank list and some
net unitary U satisfy
||D_{P^0}( U^* D_[p] U )||_{2,d} >= sqrt(p)/8,
equivalently squared norm at least p/64.
Chi-squared tail. 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 real
coordinates of variance v/d, where q <= 2 d^2 because the ranks sum to
d and each is at most 2d/p. 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 Q reaches p d^2/(64 v) - equivalently that the squared
norm reaches p/64 - is at most
exp( d^2 ( log 2 - p / (256 v) ) ).
Union bound. The log of the full union bound is at most
p log(d+1) + d^2 ( 2 log(1 + 128 sqrt(v)) + log 2 - p/(256v) ).
By v < 2 + t, (10) makes the bracket less than -8, and the first term
is o(d^2) because p is fixed. So gamma_d(E_d) <= exp(-4 d^2) for
large d. QED.
Proof of the strict gap. Retain the same cutoff, dimension sequence,
and ensembles throughout. Put alpha = mu_d(E_d) and beta = gamma_d(E_d).
Binary data processing for relative entropy and 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 7 yields liminf_d d^{-2} KL(mu_d || gamma_d) >= 2. Only one
unit of this normalized deficit is retained in the comparison below.
Upper bound on H_d. At positive time the bounded-input Gaussian
convolution has finite entropy and finite second moments, and its
relative entropy with respect to gamma_d is finite. The Gaussian
reference has density
g_d(D) = prod_i ( (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) and 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 squared coordinates equal those of A_i^{(d)} plus t,
converging to v_i(h+t). Hence
limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (21)
Assembly. The same terminal ensemble appears in Lemma 6. Match the
lower bound (12) against the upper bound (21):
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 and applying (8) and (9) at a = h+t > 2 K^2 sqrt(p) gives
chi_R(X) <= L(h+t) - 1 - (1/2) integral_0^t Phi* ds
<= chi*(V_{h+t}) - 1/2 - (1/2) integral_0^t Phi* ds
= chi*(V_h) - 1/2,
for each cutoff with chi_R(X) > -infinity. For the remaining cutoffs
the bound is automatic. Supremum over R gives
chi(X) <= chi*(V_h) - 1/2.
Lemma 5 shows chi(X) > -infinity. Equation (7) at a = h shows
chi*(V_h) finite.
Related context. The companion preprint proves that L(F_2) and L(F_3)
are isomorphic and shows each of the four free entropy dimensions of
L(F_2) attains every integer n >= 2 on suitable generating tuples;
Ge's prime factor analysis addresses the separate multiplicative
structure of II_1 factors. Both are independent of the strict gap
obtained above.
Local PDF: pdf/p_freegroup.pdf
Local PDF: pdf/p_ge.pdf
Online: https://europepmc.org/api/getPdf?pmcid=PMC23993
The bounded self-adjoint tuple X = V_h from the witness construction
supplies the required state and tuple. QED.