---
title: Tensorial Free Additive Convolution
url: https://www.emergentmind.com/topics/tensorial-free-additive-convolution
type: topic
---

# Tensorial Free Additive Convolution

Tensorial free additive convolution is a family of related constructions that extend free additive convolution while retaining non-scalar structure. In operator-valued free probability, it is the free additive convolution with amalgamation over a fixed subalgebra \(B\), typically a matrix algebra such as \(M_N(\mathbb C)\), and it serves as the analytic mechanism behind the computation of asymptotic eigenvalue distributions for selfadjoint polynomials in independent random matrices [1303.3196]. In recent tensor-probabilistic frameworks, the same expression denotes an additive operation linearized by tensorial free cumulants indexed either by combinatorial maps or by colored permutation-tuples, producing higher-order analogues of semicircular and free Poisson laws and cumulant-based convolution rules for LU-invariant tensors [2412.02572, 2605.01887]. A third usage appears in finite free probability, where the “tensorial” or asymmetric additive convolution of polynomials is the operation \(p \boxplus_t q\), realized as an expected characteristic polynomial of \((A+RBQ)(A+RBQ)^*\) and related to rectangular free additive convolution after the lift \(S(p)(x)=p(x^2)\) [1504.00350].

## 1. Operator-valued additive convolution and analytic subordination

In the operator-valued formulation, the basic object is an operator-valued non-commutative probability space \((M,E,B)\), where \(M\) is a unital Banach algebra, \(B\subset M\) a unital Banach subalgebra, and \(E:M\to B\) a unit-preserving conditional expectation; throughout the main results one works in the \(C^*\)-setting, with \(B\subset M\) a \(C^*\)-subalgebra and \(E\) completely positive. Freeness is freeness with amalgamation over \(B\): subalgebras \(A_1,A_2\subset M\) containing \(B\) are free over \(B\) if \(E[x_1x_2\cdots x_n]=0\) whenever \(x_j\in A_{i_j}\), \(E[x_j]=0\) for all \(j\), and \(i_j\neq i_{j+1}\). The analytic domain is the operator upper half-plane
\[
H^+(B):=\{b\in B:\Im b := (b-b^*)/(2i)>0\}.
\]
For a selfadjoint \(B\)-valued random variable \(X\in M\), the Cauchy transform, reciprocal transform, and \(h\)-transform are
\[
G_X(b):=E[(b-X)^{-1}],\qquad F_X(b):=G_X(b)^{-1},\qquad h_X(b):=F_X(b)-b.
\]
On \(H^+(B)\), \(G_X\) is holomorphic, \(F_X\) maps \(H^+(B)\) into itself, and one has the positivity bound \(\Im F_X(b)\ge \Im b\), hence \(\Im h_X(b)\ge 0\) [1303.3196].

If \(X=X^*\) and \(Y=Y^*\) are free over \(B\), there exist unique Fréchet analytic self-maps \(\omega_1,\omega_2:H^+(B)\to H^+(B)\) such that for all \(b\in H^+(B)\),
\[
\Im \omega_j(b)\ge \Im b,
\]
\[
F_X(\omega_1(b))+b = F_Y(\omega_2(b))+b = \omega_1(b)+\omega_2(b),
\]
and
\[
G_X(\omega_1(b)) = G_Y(\omega_2(b)) = G_{X+Y}(b).
\]
Equivalently,
\[
\omega_1(b)+\omega_2(b)-b = F_{X+Y}(b).
\]
This is the operator-valued Bercovici–Voiculescu identity on the full operator upper half-plane, established globally as a Fréchet analytic subordination theorem and without any traciality assumption.

The same theory yields a fixed-point description suitable for computation. For each \(b\in H^+(B)\), define
\[
f_b(w):=h_Y(h_X(w)+b)+b.
\]
Then \(\omega_1(b)\) is the unique fixed point of \(f_b\), and for any \(w\in H^+(B)\),
\[
\omega_1(b)=\lim_{n\to\infty} f_b^{\circ n}(w).
\]
The analogous statement holds for \(\omega_2\) after swapping \(X\) and \(Y\). On tubes of the form \(\Im b\ge \epsilon\cdot 1\), the Earle–Hamilton theorem gives a strict Carathéodory contraction, and Lemma 2.3 provides the explicit bound
\[
\|h_Y(w)\|\le 4\|Y\|(1+2\epsilon^{-1}\|Y\|),
\]
which supplies boundedness and stability on invariant subsets.

## 2. The tensorial viewpoint in random matrix polynomial problems

The operator-valued theory becomes “tensorial” when the amalgamation algebra is itself a tensor or matrix algebra. For polynomials in independent random matrices, the natural choice is
\[
B=M_N(\mathbb C),\qquad M=M_N(\mathbb C)\otimes A,\qquad E=\mathrm{id}_{M_N(\mathbb C)}\otimes \phi.
\]
Here \(E\) collapses the second tensor factor to scalars while preserving the operator structure on the matrix factor. If \(L_P\in M_N(\mathbb C)\otimes A\) is a linearization of a polynomial \(P\), then \(L_P\) is a \(B\)-valued random variable, and its decomposition into summands \(b_j\otimes X_j\) turns the original polynomial problem into an operator-valued free additive convolution problem over \(B\) [1303.3196].

The key bridge is Anderson’s selfadjoint linearization trick. For any selfadjoint noncommutative polynomial \(p\), there exists a selfadjoint block matrix
\[
L_p=b_0\otimes 1+b_1\otimes X_1+\cdots+b_k\otimes X_k
\]
with \(b_j\in M_N(\mathbb C)_{sa}\), of the form
\[
L_p=
\begin{bmatrix}
0 & u\\
v & Q
\end{bmatrix},
\qquad Q\ \text{invertible},
\]
such that
\[
p=-uQ^{-1}v.
\]
Applying the Schur complement to \(\Lambda(z)-L_P\), where \(\Lambda(z)\) is the block-diagonal embedding of \(z\), yields
\[
[(\Lambda(z)-L_P)^{-1}]_{1,1}=(z-P)^{-1}.
\]
Thus the scalar resolvent of \(P\) is obtained as the \((1,1)\)-entry of an operator-valued resolvent.

This leads to a concrete algorithm. One computes \(G_{L_P}(b)=E[(b-L_P)^{-1}]\) on \(H^+(M_N(\mathbb C))\), typically by iterating the operator-valued subordination maps for the free summands of \(L_P\). The scalar Cauchy transform is then extracted by
\[
G_P(z)=\lim_{\epsilon\downarrow 0}[G_{L_P}(\Lambda_\epsilon(z))]_{1,1},
\qquad z\in \mathbb C^+,
\]
and the limiting spectral measure is recovered by Stieltjes inversion. In tensorial cases of the form \(X=B_1\otimes x_1\), the operator-valued Cauchy transform can be written as the Bochner integral
\[
G_X(b)=\int_{\mathbb R}(b-tB_1)^{-1}\,d\mu_{x_1}(t),
\]
so the nonlinear part of the computation is carried entirely by matrix inversions in \(B\).

The anticommutator example makes the mechanism explicit. For
\[
p(X_1,X_2)=X_1X_2+X_2X_1,
\]
a selfadjoint linearization is
\[
L_p=
\begin{bmatrix}
0 & X_1 & X_2\\
X_1 & 0 & -1\\
X_2 & -1 & 0
\end{bmatrix}.
\]
Evaluated on independent Wigner matrices, this becomes an element of \(M_3(\mathbb C)\otimes M_N(\mathbb C)\), amalgamated over \(B=M_3(\mathbb C)\) with \(E=\mathrm{id}\otimes \mathrm{tr}_N\). The limiting eigenvalue distribution is then obtained by the same subordination procedure, and the paper reports agreement with simulations.

## 3. High-order tensor freeness and convolution of measures

A different line of work develops tensorial free additive convolution directly for random tensors of higher order. In this framework, the tensor space is \(E_p^N=(\mathbb C^N)^{\otimes p}\) or \(E_p^N=(\mathbb R^N)^{\otimes p}\), and trace invariants are indexed by combinatorial maps. If \(m\in M_q\) is a map with boundary sequence \(\partial\), then for \(q\ge 1\),
\[
m((x_v)_{v\in V(m)})_{i_\partial}=\sum_{i\in [N]^{E(m)}} \prod_{v\in V(m)} (x_v)_{i_{\partial v}},
\]
while for \(q=0\),
\[
m((x_v)_{v\in V(m)})=\frac{1}{N^\gamma}\sum_{i\in [N]^{E(m)}} \prod_{v\in V(m)} (x_v)_{i_{\partial v}},
\]
with \(\gamma\) the number of connected components. Freeness is defined by means of a non-crossing poset on maps generated by switches, together with chromatic conditions on components. Tensorial free cumulants \(\kappa_m\) are introduced by Möbius inversion on that poset:
\[
m(x)=\sum_{n:m_0\le n\le m}\kappa_n(x),\qquad
\kappa_m(x)=\sum_{m_0\le n\le m}\mu(n,m)\,n(x).
\]
For even families, tensorial freeness is characterized by vanishing mixed cumulants: for every connected non-monochromatic map \((m,x)\), one has \(\kappa_m(x)=0\). In the matrix case \(p=2\), connected maps are cycles, the poset becomes the non-crossing partitions lattice, and the construction reduces to classical free cumulants [2407.18881].

In this map-based setting, tensorial free additive convolution is defined for tensors of the same order \(p\). If \(X,Y\in E_p\) are centered and free, then for every connected \(p\)-regular map \(m\),
\[
\kappa_m(X+Y)=\kappa_m(X)+\kappa_m(Y).
\]
This is the exact tensorial analogue of cumulant additivity in ordinary free probability. The same framework also proves a free CLT for tensors: if \(\{a_i\}_{i\ge 1}\subset E_p\) are centered, free, and uniformly moment-bounded, and if the two-vertex melonic moments \(f_p^\sigma(a_i)=t_p^\sigma\) are independent of \(i\), then
\[
S_n=\frac{1}{\sqrt n}\sum_{i=1}^n a_i
\]
converges in distribution to a tensorial Gaussian law with cumulants supported only on the melons \(f_p^\sigma\) [2407.18881].

The compactly supported measure version refines this picture by aggregating cumulants over connected rooted \(p\)-regular trace maps. For a tensorial distribution \(a\), one sets
\[
m_n(a):=\sum_{b\in B_n^{(p)}} b(a),\qquad
\kappa_n(a):=\sum_{b\in B_n^{(p)}} \kappa_b(a),
\]
and defines generating series
\[
M_a(z):=\sum_{n\ge 0}m_n(a)z^n,\qquad
C_a(z):=\sum_{n\ge 0}\kappa_n(a)z^n.
\]
They satisfy the functional equation
\[
M(z)=C\!\big(z\,M(z)^{p/2}\big).
\]
For compactly supported measures \(\mu,\nu\), tensorial free additive convolution is denoted \(\mu\oplus_p \nu\) and is defined by the rule
\[
\kappa_n(\mu\oplus_p \nu)=\kappa_n(\mu)+\kappa_n(\nu).
\]
The associated tensorial \(R\)-transform is
\[
R_\mu(z):=\frac{C_\mu(z)-1}{z}=\sum_{n\ge 1}\kappa_n(\mu)z^{n-1},
\]
and it linearizes the convolution:
\[
R_{\mu\oplus_p \nu}(z)=R_\mu(z)+R_\nu(z).
\]
For \(p=2\), this recovers Voiculescu’s free additive convolution [2412.02572].

The same paper introduces higher-order analogues of the semicircular and free Poisson laws. The higher-order semicircular law \(\mu_p\) has moments
\[
m_n(\mu_p)=
\begin{cases}
F_p(n/2), & n\ \text{even},\\
0, & n\ \text{odd},
\end{cases}
\]
where \(F_p(k)=\frac{1}{pk+1}\binom{pk+1}{k}\), and cumulants
\[
\kappa_n(\mu_p)=
\begin{cases}
1, & n=0\ \text{or}\ n=2,\\
0, & \text{otherwise}.
\end{cases}
\]
The higher-order free Poisson law \(\nu_{p,t}\) satisfies, for even \(p\),
\[
\kappa_n(\nu_{p,t})=
\begin{cases}
t, & n\ge 1,\\
1, & n=0,
\end{cases}
\qquad
R_{\nu_{p,t}}(z)=\frac{t}{1-z},
\]
and one has
\[
\nu_{p,t}\oplus_p \nu_{p,t'}=\nu_{p,t+t'}.
\]

## 4. LU-invariant tensors, permutation-indexed cumulants, and HCIZ/BGW asymptotics

A third tensorial framework organizes moments and cumulants by colored permutation data. A mixed tensor \(A\) has \(D\) inputs and \(D\) outputs,
\[
A=(A_{i_1,\ldots,i_D;\,j_1,\ldots,j_D}),
\]
while a pure tensor is a pair \((T,\bar T)\). Local unitary invariance means invariance in distribution under \(U=U_1\otimes\cdots\otimes U_D\) with \(U_c\in U(N)\). Trace-invariants are indexed by \(\boldsymbol{\sigma}=(\sigma_1,\ldots,\sigma_D)\in S_n^D\); for mixed tensors,
\[
\Tr_{\boldsymbol{\sigma}}(\boldsymbol A)
=
\sum_{\boldsymbol j:[n]\to [N]^D}
\boldsymbol A_{\boldsymbol j\circ \boldsymbol \sigma;\,\boldsymbol j},
\]
and analogous formulas hold in the pure case. The connectedness of \(\boldsymbol{\sigma}\) is encoded by the join \(\Pi(\boldsymbol{\sigma})\) of the cycle-partitions of \(\sigma_1,\ldots,\sigma_D\), with \(K(\boldsymbol{\sigma})=\#\Pi(\boldsymbol{\sigma})\) [2605.01887].

Finite-\(N\) tensorial cumulant precursors \(K_{\boldsymbol{\sigma}}\) are defined as classical cumulants of the \(G\)-observables. In the mixed case,
\[
K_{\boldsymbol{\sigma}}[A]
=
\sum_{\substack{\pi\in P(n)\\ \Pi(\boldsymbol{\sigma})\le \pi}}
\mu_\pi
\prod_{S\in \pi}
G_{\boldsymbol{\sigma}\vert_S}[A],
\]
and there are analogous formulas for pure tensors. The asymptotic tensorial free cumulants are the rescaled limits
\[
K_{\boldsymbol{\sigma}}[\cdot]
=
N^{r(\boldsymbol{\sigma})}\bigl(\kappa_{\boldsymbol{\sigma}}+o(1)\bigr),
\]
with scaling exponent \(r\) determined by the asymptotic class. For independent LU-invariant tensors \(A_1,A_2\), additivity holds already at finite \(N\):
\[
K_{\boldsymbol{\sigma}}[A_1+A_2]
=
K_{\boldsymbol{\sigma}}[A_1]+K_{\boldsymbol{\sigma}}[A_2],
\]
and asymptotically this becomes
\[
\kappa_{\boldsymbol{\sigma}}(X+Y)
=
\kappa_{\boldsymbol{\sigma}}(X)+\kappa_{\boldsymbol{\sigma}}(Y).
\]
Accordingly, tensorial free additive convolution is defined by
\[
\kappa_{\boldsymbol{\sigma}}(X\boxplus_{\mathrm{tensor}} Y)
=
\kappa_{\boldsymbol{\sigma}}(X)+\kappa_{\boldsymbol{\sigma}}(Y)
\]
for all \(n\) and all \(\boldsymbol{\sigma}\in S_n^D\).

In this approach, tensorial \(R\)-transform data arise from free-energy derivatives of tensor HCIZ and BGW integrals. In the mixed matrix-product scaling regime,
\[
\lim_{N\to\infty}\frac{1}{N}\frac{\partial^n}{\partial z^n}
E\Bigl[\log \mathcal I_{D,N}(A;zB)\Bigr]\Big|_{z=0}
=
n!\sum_{\boldsymbol{\sigma}\in \mathbb M_n^D/\sim}
\frac{t_{\boldsymbol{\sigma}}}{|\mathrm{Aut}(\boldsymbol{\sigma})|}
\,\kappa_{\boldsymbol{\sigma}}(a),
\]
and in the pure Gaussian or \(\epsilon\)-scaling regime,
\[
\lim_{N\to\infty}\frac{1}{N}\frac{\partial^n}{\partial z^n}\frac{\partial^n}{\partial \bar z^n}
E\Bigl[\log \mathcal J_{D,N}(T,\bar T;zJ,\bar z \bar J)\Bigr]\Big|_{z=\bar z=0}
=
(n!)^2\sum_{\boldsymbol{\sigma}\in \mathbb M_n^D/\sim}
\frac{t_{\boldsymbol{\sigma}}}{|\mathrm{Aut}(\boldsymbol{\sigma})|}
\,\kappa_{\boldsymbol{\sigma}}(t,\bar t).
\]
In the melonic first-order sector, addition of these generating series mirrors addition of tensorial cumulants.

This formalism also distinguishes between invariance classes. Under a coarser LU grouping \(\xi:[D]\to [D']\), cumulants collapse via the canonical embedding \(g_\xi\). In the global unitary invariant case \(\xi=(1_D)\), tensorial free cumulants coincide with matricial free cumulants if and only if all \(\sigma_c\) are equal; otherwise they vanish. The paper emphasizes that Gaussian pure tensors with non-trivial covariance provide concrete examples with genuinely non-trivial tensorial cumulants, including explicit formulas when the covariance is a tensor product of matrices.

## 5. The finite polynomial operation \(p\boxplus_t q\)

In finite free probability, tensorial free additive convolution appears as an operation on monic polynomials of degree at most \(d\). If
\[
p(x)=\sum_{i=0}^d x^{d-i}(-1)^i a_i,
\qquad
q(x)=\sum_{i=0}^d x^{d-i}(-1)^i b_i,
\]
their asymmetric or tensorial additive convolution is
\[
(p\boxplus_t q)(x)
=
\sum_{k=0}^d x^{d-k}(-1)^k
\sum_{i+j=k}
\Bigg(\frac{(d-i)!(d-j)!}{d!(d-k)!}\Bigg)^2
a_i b_j.
\]
Equivalently, if \(p(x)=P(DxD)x^d\) and \(q(x)=Q(DxD)x^d\), then
\[
(p\boxplus_t q)(x)=P(DxD)\,Q(DxD)\,x^d.
\]
The random-matrix model is
\[
(p\boxplus_t q)(x)
=
\mathbb E_{R,Q\in U(d)}
\chi\!\left((A+RBQ)(A+RBQ)^*;x\right),
\]
for square \(d\times d\) matrices \(A,B\) such that \(\chi(AA^*;x)=p(x)\) and \(\chi(BB^*;x)=q(x)\). The paper states explicitly that this “tensorial” convolution is not a Kronecker-sum [1504.00350].

The basic spectral property is real-rootedness. If \(p\) and \(q\) have nonnegative real roots, then \(p\boxplus_t q\) also has nonnegative real roots. The transform inequality is formulated after the lift \(S(p)(x)=p(x^2)\):
\[
R_{S(p\boxplus_t q)}(w)\le R_{Sp}(w)+R_{Sq}(w),
\qquad w>0.
\]
Equivalently,
\[
\max\operatorname{root}\big((1-\alpha D)\,S(p\boxplus_t q)\big)
\le
\max\operatorname{root}\big((1-\alpha D)\,Sp\big)
+
\max\operatorname{root}\big((1-\alpha D)\,Sq\big)
-
2\alpha d.
\]
Equality holds if and only if \(p\) or \(q\) is \(x^d\).

The operation has characteristic differential identities. Degree reduction is controlled by the Laguerre derivative:
\[
p\boxplus_t q
=
\frac{1}{d^2}(DxD\,p)\boxplus_t q
\quad\text{at grade }(d-1),
\]
and if \(q(x)=x^{d-1}\), then
\[
p\boxplus_t q=DxD\,p.
\]
There is also a Laguerre identity:
\[
a^dL_d(x/a)\boxplus_t b^dL_d(x/b)=c^dL_d(x/c),
\qquad c=a+b>0.
\]
The paper interprets this finite convolution as a finite-dimensional analogue of rectangular free additive convolution, with the \(S\)-lift supplying the connection to the corresponding finite \(R\)-transform bound.

## 6. Relations, distinctions, and open directions

The literature does not use the expression “tensorial free additive convolution” in a single uniform sense. In one usage, it means operator-valued free additive convolution over a matrix or tensor algebra \(B\), with analytic subordination on \(H^+(B)\) and direct application to selfadjoint random matrix polynomials. In another, it means the cumulant-additive convolution of higher-order tensors indexed by combinatorial maps or by colored permutation-tuples. In a third, it denotes the finite polynomial operation \(p\boxplus_t q\). A plausible implication is that results about transforms, limits, or combinatorics cannot be transferred between these settings without additional structure.

A related but distinct algebraic viewpoint is due to Cebron, who enlarges polynomial calculus to a tensorial algebra \( \mathbb C\{X_i:i\in I\} \) with trace slots and a trace-product \( \cdot_{\mathrm{tr}} \). In that calculus, additive free convolution is realized by a derivation \(\Delta_A\) built from free cumulants, and the central formula is
\[
\tau(P(A+B)\mid B)=(e^{\Delta_A}P)(B).
\]
For semicircular noise \(S_t\), this gives the semigroup
\[
T_t^{\boxplus}:=e^{\Delta_{S_t}}=e^{t\Delta_{S_1}},
\]
which realizes the transition \(X\mapsto X+S_t\). This is a tensorial realization of scalar free convolution rather than a theory of high-order random tensors, but it shows that tensorial calculi also arise on the operator side of free probability [1304.1713].

The principal limitations are framework-specific. In the operator-valued theory, the global analytic subordination results are set up for selfadjoint variables in a \(C^*\)-operator-valued probability space; for non-selfadjoint polynomials one must pass to Brown measure. In the map-based tensor theory, analytic Cauchy or resolvent transforms and subordination are not developed, and extending such tools is explicitly left open. In the compactly supported high-order theory, even \(p\) simplifies the poset structure through uniqueness of the minimal map, while odd \(p\) requires additional care. In the LU-invariant permutation framework, one open direction is the full equivalence between different proposals for tensorial free cumulants, especially beyond connected cases and for broader invariance classes [1303.3196, 2407.18881, 2412.02572, 2605.01887].

These distinctions also clarify a common misconception. Tensorial free additive convolution is not a single universal object that has merely been expressed in different notation. The operator-valued theory is analytic and resolvent-based; the high-order tensor theories are combinatorial and cumulant-based; the finite polynomial theory is an expected-characteristic-polynomial construction. They agree in certain reductions—most notably at \(p=2\), where the tensor frameworks collapse to classical free probability—but the surrounding structures, admissible observables, and computational techniques remain different.

Source: https://www.emergentmind.com/topics/tensorial-free-additive-convolution