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# Quantum Scrambling of Observable Algebras

Paolo Zanardi  [ Department of Physics and Astronomy, and Center for Quantum
Information Science and Technology, University of Southern California, Los
Angeles, California 90089-0484, USA

###### Abstract

In this paper we describe an algebraic/geometrical approach to quantum
scrambling. Generalized quantum subsystems are described by an hermitian-
closed unital subalgebra $\cal A$ of operators evolving through a unitary
channel. Qualitatively, quantum scrambling is defined by how the associated
physical degrees of freedom get mixed up with others by the dynamics.
Quantitatively, this is accomplished by introducing a measure, the geometric
algebra anti-correlator (GAAC), of the self-orthogonalization of the commutant
of $\cal A$ induced by the dynamics. This approach extends and unifies
averaged bipartite OTOC, operator entanglement, coherence generating power and
Loschmidt echo. Each of these concepts is indeed recovered by a special choice
of $\cal A$. We compute typical values of GAAC for random unitaries, we prove
upper bounds and characterize their saturation. For generic energy spectrum we
find explicit expressions for the infinite-time average of the GAAC which
encode the relation between $\cal A$ and the full system of Hamiltonian
eigenstates. Finally, a notion of ${\cal A}$-chaoticity is suggested.

Introduction.—  Quantum dynamics can quickly spread information, which was
initially encoded in some physical degrees of freedom, into a larger set of
degrees of freedom, in this way quantum information gets _delocalized_ and
highly non-local correlations can be built. This so-called _quantum
scrambling_ , has over the last few years attracted a growing amount of
attention in the context of quantum chaos and also quantum computing. The Out
of Time Order Correlation functions (OTOCs) are among the most popular tools
to analyze scrambling from a quantitative point view [Larkin and
Ovchinnikov(1969), Kitaev(2015), Maldacena  _et al._(2016)Maldacena, Shenker,
and Stanford, Roberts and Stanford(2015), Polchinski and Rosenhaus(2016),
Mezei and Stanford(2017), Roberts and Yoshida(2017)].

The goal of this paper is to lay down a novel formalism for quantum
scrambling. Roughly speaking, we will characterize scrambling by how much a
_whole_ set of distinguished degrees of freedom gets far from itself by
unitary evolution. The underlying philosophy of this paper is an extension of
the observable-algebra approach to quantum subsystems originally advocated in
[Zanardi(2001a), Zanardi  _et al._(2004)Zanardi, Lidar, and Lloyd] (see also
recent developments in [Kabernik(2018), Kabernik  _et al._(2020)Kabernik,
Pollack, and Singh]). As such the strategy can be applied to situations in
which there is no an _a priori_ locality structure which gives a natural way
of defining subsystems e.g., see [Carroll and Singh(2021)].

We will show that specific instances of our construction allow one to recover
apparently different concepts including operator entanglement [Zanardi(2001b),
Prosen and Pižorn(2007)], averaged bipartite OTOCs [Yan  _et al._(2020)Yan,
Cincio, and Zurek, Styliaris  _et al._(2021)Styliaris, Anand, and Zanardi],
coherence generating power [Zanardi  _et al._(2017a)Zanardi, Styliaris, and
Campos Venuti, Zanardi  _et al._(2017b)Zanardi, Styliaris, and Campos Venuti,
Styliaris  _et al._(2018)Styliaris, Campos Venuti, and Zanardi] and Loschmidt
echo [Jalabert and Pastawski(2001a), Goussev  _et al._(2012a)Goussev,
Jalabert, Pastawski, and Wisniacki]. This conceptual unification provides one
of the main motivations for this work. Another one is to design candidate
tools for unveiling novel facets of quantum chaos.

For the sake of clarity, the main technical results of this paper are
organized in “Propositions” whose proofs are in 111 See supplemental material.

Preliminaries.—  In this section we introduce the main formal ingredients
utilized in this paper and set the notation. Let ${\cal
H}=\mathrm{span}\\{|m\rangle\\}_{m=1}^{d}$ be a $d$-dimensional Hilbert space
and $L({\cal H})$ its the full operator algebra (see 222 $L({\cal H})$ has a
Hilbert space structure via the Hilbert-Schmidt scalar product: $\langle
X,\,Y\rangle:=\mathrm{Tr}\left(X^{\dagger}Y\right)$ and norm
$\|X\|_{2}^{2}:=\langle X,\,X\rangle.$ This equips the space of superoperators
i.e., $L(L({\cal H}))$ with the scalar product $\langle{\cal T},\,{\cal
F}\rangle:=\mathrm{Tr}_{HS}\left({\cal T}^{\dagger}{\cal
F}\right)=\sum_{l,m}\langle m|{\cal T}^{\dagger}{\cal F}(|m\rangle\langle
l|)|l\rangle,$ and the norm $\|{\cal T}\|_{HS}^{2}=\langle{\cal T},\,{\cal
T}\rangle=\sum_{l,m}\|{\cal T}(|m\rangle\langle l|)\|_{2}^{2}.$ If ${\cal
T}(X)=\sum_{i}A_{i}XA_{i}^{\dagger},$ then $\|{\cal
T}\|_{HS}^{2}=\sum_{i,j}|\mathrm{Tr}(A_{i}^{\dagger}A_{j})|^{2}$ for further
notation). In the following by the notation ${\mathbf{C}}\\{X\\}$ we will
denotes the vector space spanned by the $X^{\prime}$s.

The key formal ingredients of this investigation are _hermitian-closed unital
subalgebras_ ${\cal A}\subset L({\cal H})$ and their commutants ${\cal
A}^{\prime}:=\\{X\in L({\cal H})\,/\,[X,\,Y]=0,\,\forall Y\in{\cal A}\\}.$ The
intersection ${\cal A}\cap{\cal A}^{\prime}=:{\cal Z}({\cal A})$ is the
_center_ of the algebra ${\cal A}.$ The fundamental structure theorem of these
objects states that the Hilbert space breaks into a direct sum of
$d_{Z}:=\mathrm{dim}\,{\cal Z}({\cal A})$ orthogonal blocks and each of them
has a tensor product bi-partite structure: ${\cal H}=\oplus_{J}{\cal
H}_{J},\,{\cal H}_{J},\cong\mathbf{C}^{n_{J}}\otimes\mathbf{C}^{d_{J}}.$
Moreover,

$\displaystyle{\cal A}\cong\oplus_{J}\mathbbm{1}_{n_{J}}\otimes
L(\mathbf{C}^{d_{J}}),\qquad{\cal
A}^{\prime}\cong\oplus_{J}L(\mathbf{C}^{n_{J}})\otimes\mathbbm{1}_{d_{J}}.$
(1)

Whence, $d=\sum_{J}n_{J}d_{J},$ $\mathrm{dim}\,{\cal
A}=\sum_{J}d_{J}^{2}=:d({\cal A})$ and $\mathrm{dim}\,{\cal
A}^{\prime}=\sum_{J}n_{J}^{2}=:d({\cal A}^{\prime}).$ Also, ${\cal Z}({\cal
A})=\mathbf{C}\\{\Pi_{J}:=\mathbbm{1}_{n_{J}}\otimes\mathbbm{1}_{d_{J}}\\},$
namely the center of $\cal A$ is spanned by the projections over the ${\cal
H}_{J}$ blocks.

Associated to any algebra $\cal A$ we have an orthogonal (super) projection
CP-map: $\mathbb{P}_{\cal A}^{\dagger}=\mathbb{P}_{\cal A},\,\mathbb{P}_{\cal
A}^{2}=\mathbb{P}_{\cal A}$ and $\mathrm{Im}\,\mathbb{P}_{\cal A}={\cal A}.$
Such maps can be written in the Kraus form $\mathbb{P}_{\cal
A}(X)=\sum_{\alpha=1}^{d({\cal A}^{\prime})}e_{\alpha}Xe_{\alpha}^{\dagger},$
where the $e_{\alpha}$ are a suitable orthogonal  basis of ${\cal
A}^{\prime}.$ Notice that $\mathrm{Tr}_{HS}\mathbb{P}_{\cal
A}=\sum_{\alpha=1}^{d({\cal A}^{\prime})}|\mathrm{Tr}\,e_{\alpha}|^{2}=d({\cal
A}).$ In terms of the decomposition (1) one has $\mathbb{P}_{{\cal
A}}(X)=\sum_{J}\frac{\mathbbm{1}_{n_{J}}}{n_{J}}\otimes\mathrm{tr}_{n_{J}}(X),$
and $\mathbb{P}_{{\cal
A}^{\prime}}(X)=\sum_{J}\mathrm{tr}_{d_{J}}(X)\otimes\frac{\mathbbm{1}_{d_{J}}}{d_{J}}.$

These structural results provide the mathematical underpinnings of the theory
of decoherence-free subspaces [Zanardi and Rasetti(1997), Lidar  _et
al._(1998)Lidar, Chuang, and Whaley], noiseless subsystems [Knill  _et
al._(2000)Knill, Laflamme, and Viola, Zanardi(2000)] and in general to all
quantum-information stabilizing techniques [Zanardi(2000)]. From the physical
point of view two special cases are worth emphasizing:

_Factors_ : ${\cal Z}({\cal A})=\mathbf{C}\mathbbm{1},$ in this case ${\cal
H}\cong\mathbf{C}^{n_{1}}\otimes\mathbf{C}^{d_{1}}$ namely the algebra $\cal
A$ endows $\cal H$ with a bipartition into _virtual_ subsystems
[Zanardi(2001a), Zanardi  _et al._(2004)Zanardi, Lidar, and Lloyd]. The case
in which ${\cal H}={\cal H}_{A}\otimes{\cal H}_{B}$ with ${\cal A}=L({\cal
H}_{A})\otimes\mathbbm{1}_{B}$ and ${\cal A}^{\prime}=\mathbbm{1}_{A}\otimes
L({\cal H}_{B})$ clearly falls in this category.

_Super-Selection_ : ${\cal A}^{\prime}\subset{\cal A}$ this is when the
commutant is an Abelian algebra. This implies $n_{J}=1,\,(\forall J)$ and
therefore the Hilbert space breaks into $d_{J}$-dimensional super-selection
sectors i.e., ${\cal H}\cong\oplus_{J=1}^{d({\cal A}^{\prime})}{\cal
H}_{J},\,{\cal H}_{J}\cong\mathbf{C}^{d_{J}}$ and ${\cal
A}\cong\oplus_{J=1}^{d({\cal A}^{\prime})}L(\mathbf{C}^{d_{J}}).$ If $\cal A$
is a maximal abelian subalgebra one has ${\cal A}={\cal A}^{\prime}$ and
$n_{J}=d_{J}=1,\,(\forall J).$ This is the case that is relevant to the study
of quantum coherence [Streltsov  _et al._(2017)Streltsov, Adesso, and Plenio]
and its dynamical generation [Zanardi  _et al._(2017a)Zanardi, Styliaris, and
Campos Venuti, Zanardi and Campos Venuti(2018)].

When the $d_{Z}$-dimensional (integer-valued) vectors ${\bf{d}}:=(d_{J})_{J}$,
and ${\bf{n}}:=(n_{J})_{J}$ are proportional to each other i.e.,
${\bf{d}}=\lambda{\bf{n}}$ one has that $d^{2}=d({\cal A})d({\cal
A}^{\prime}).$ If this is the case we shall say that the pair $({\cal
A},\,{\cal A}^{\prime})$ is _collinear_. Note that both factors and maximal
abelian subalgebras are of this type.

General results.—  We are now in the position to define the central
mathematical object of this paper: the _geometric algebra anti-correlator_
(GAAC) by

$\displaystyle G_{\cal A}(U):=1-\frac{\langle\mathbb{P}_{{\cal
A}^{\prime}},\,\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}\rangle}{\|\mathbb{P}_{{\cal A}^{\prime}}\|_{HS}^{2}}.$ (2)

The geometrical meaning of GAAC should be evident from Eq. (2): the larger
$G_{\cal A}(U)$ the smaller is the intersection between ${\cal A}^{\prime}$
and its unitarily evolved image ${\cal U}({\cal A}^{\prime}):=\\{{\cal
U}(X)\,/\,X\in{\cal A}^{\prime}\\}$ 333This can be seen from the fact that
given two projectors $P$, and $Q$ of rank $d$ one has: $\mathrm{dim}(V_{P}\cap
V_{Q})\leq\mathrm{tr}(PQ)\leq d.$ Where $V_{P/Q}$ are the images of $P/Q.$ The
lower (upper) bound is achieved when $P$ and $Q$ commute (coincide).

_Remark.–_ In the RHS of Eq. (2) we use the ${\cal A}^{\prime}$ (and not $\cal
A$) as the dynamics $\cal U$ is in the Heisenberg picture. Symmetries mapped
out of ${\cal A}^{\prime}$ by ${\cal U}$ is equivalent to states mapped out of
$\cal A$ by ${\cal U}^{\dagger}.$ This choice is somewhat arbitrary (See Prop.
1).

Algebraically, (2) measures how much the symmetries of the generalized quantum
subsystem associated to $\cal A$ are dynamically broken by the channel ${\cal
U}.$ Let us now start by further unveiling the geometrical nature of GAACs.
First notice that, using the algebra super-projections, one can define a
distance between two algebras $\cal A$ and $\cal B$: $D({\cal A},\,{\cal
B}):=\|\mathbb{P}_{\cal A}-\mathbb{P}_{\cal B}\|_{HS}.$ This metric structure
allows one to draw a quite simple geometrical picture of algebra scrambling.

######  Proposition 1.

i) The GAAC is the (squared and normalized) distance between the algebra
${\cal A}^{\prime}$ and its image ${\cal U}({\cal A}^{\prime}).$

$\displaystyle G_{\cal A}(U)=\frac{1}{2}\frac{D^{2}\left({\cal
A}^{\prime},\,{\cal U}({\cal A}^{\prime})\right)}{d({\cal A}^{\prime})}$ (3)

ii) $G_{\cal A}(U)=0\Leftrightarrow{\cal U}({\cal A}^{\prime})={\cal
A}^{\prime}\Leftrightarrow{\cal U}({\cal A})={\cal A}.$ In words: the GAAC Eq.
(2) vanishes if and only if both algebras $\cal A$ and ${\cal A}^{\prime}$ are
invariant under ${\cal U}$ i.e., there is no algebra scrambling.

The definition of GAAC given by Eq. (2) has the drawback of relying of
superoperator projections and therefore may seem somewhat abstract and removed
from practical calculations. Hence, before moving on to physical examples and
applications of our formalism, we would like to re-express the GAAC at the
more familiar operator level.

######  Proposition 2.

i) One can find an orthogonal basis of $\cal A$
$\\{e_{\alpha}\\}_{\alpha=1}^{d({\cal A})}$ and an orthonormal basis of ${\cal
A}^{\prime}$ $\\{f_{\gamma}\\}_{\gamma=1}^{d({\cal A}^{\prime})}$ such that

$\displaystyle 1-G_{\cal A}(U)=\frac{\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega_{\cal A})\rangle}{\|\Omega_{\cal
A}\|_{2}^{2}}=\frac{\langle\tilde{\Omega}_{{\cal A}},\,{\cal
U}^{\otimes\,2}(\tilde{\Omega}_{{\cal A}})\rangle}{\|\tilde{\Omega}_{{\cal
A}}\|_{2}^{2}}$ (4)

where $\Omega_{\cal A}:=\sum_{\alpha=1}^{d({\cal A})}e_{\alpha}\otimes
e^{\dagger}_{\alpha},$ and $\tilde{\Omega}_{{\cal A}}=\sum_{\gamma=1}^{d({\cal
A}^{\prime})}f_{\gamma}\otimes f^{\dagger}_{\gamma}.$ Also,
$\tilde{\Omega}_{{\cal A}}=S\Omega_{\cal A},$ where $S$ is the swap on ${\cal
H}^{\otimes\,2},$ and $\|\Omega_{{\cal A}}\|_{2}^{2}=\|\tilde{\Omega}_{{\cal
A}}\|_{2}^{2}=d({\cal A}^{\prime}).$

ii) If $({\cal A},\,{\cal A}^{\prime})$ is collinear then $G_{\cal
A}(U)=G_{{\cal A}^{\prime}}(U),(\forall U)$.

In the above proposition, all the (Hilbert-Schmidt) scalar products and norms
are ordinary operators ones. Moreover, the $\Omega$’s operator can be
expressed in the same way if the bases $e_{\alpha}$’s and $f_{\alpha}$’s are
replaced by unitarily equivalent ones. The connection between Eqs (2) and (4)
is given by

$\displaystyle\mathbb{P}_{{\cal
A}^{\prime}}(X)=\mathrm{Tr}_{1}\left(S\Omega_{\cal
A}(X\otimes\mathbbm{1})\right)=\mathrm{Tr}_{1}\left(\tilde{\Omega}_{\cal
A}(X\otimes\mathbbm{1})\right).$ (5)

Interestingly, the no-scrambling condition $G_{\cal A}(U)=0$ using Prop. 2 can
be expressed by the operator fixed-point equations ${\cal
U}^{\otimes\,2}(\Omega_{\cal A})=\Omega_{\cal A}.$ The (unsurprising) price to
pay is that now the Hilbert space is doubled.

Another advantage of the formulation (4) is that it makes clear that the GAAC
can be computed in terms of $2$-point correlation functions. In fact, from Eq.
(4) one finds (see appendix)

$\displaystyle 1-G_{\cal A}(U)=\frac{1}{d({\cal
A}^{\prime})}\sum_{\alpha,\beta=1}^{d({\cal A})}|\langle e_{\alpha},\,{\cal
U}(e_{\beta})\rangle|^{2},$ (6)

(a similar expression hold for the $f_{\alpha}$’s). This expression suggests
how one could measure the GAAC by resorting to process tomography for $\cal
U$. Notice also that operational protocols to measure the GAAC were already
discussed, for the cases 1) and 2) here below, in [Styliaris  _et
al._(2021)Styliaris, Anand, and Zanardi] and [Zanardi  _et al._(2017a)Zanardi,
Styliaris, and Campos Venuti] respectively.

Physical Cases.—  To concretely illustrate the formalism let us now consider
several physically motivated examples in which the GAAC can be fully computed
analytically.

The first two examples show how the GAAC formalism allows one to understand
two ostensibly unrelated physical problems, operator entanglement
[Zanardi(2001b)] and coherence generating power (CGP) [Zanardi  _et
al._(2017a)Zanardi, Styliaris, and Campos Venuti, Zanardi  _et
al._(2017c)Zanardi, Styliaris, and Campos Venuti], from a single vantage
point. The first (second) concept is obtained when $\cal A$ is a factor
(maximal abelian). This means that one can also think of the GAAC either as an
extension of operator entanglement to algebras that are not factors, or as an
extension of coherence generating power to algebras that are not maximal
abelian subalgebras.

The third and fourth examples are "dual" to each other and show that, in
general, $G_{\cal A}(U)\neq G_{{\cal A}^{\prime}}(U).$ Finally, the fifth
illustrates in which sense even the concept of _Loschmidt echo_ , a valuable
tool in the study of quantum chaos [Peres(1984), Jalabert and
Pastawski(2001b), Goussev  _et al._(2012b)Goussev, Jalabert, Pastawski, and
Wisniacki, Gorin  _et al._(2006)Gorin, Prosen, Seligman, and Žnidarič], is
comprised by the GAAC. This last connection is perhaps unsurprising as the
Loschmidt echo is indeed a measure of auto-correlation of a dynamicaly
evolving state which is precisely what $1-G_{\cal A}(U)$ does at the more
general algebra level.

The special results 1–5 reported here below can be obtained by Eqs. (4) and
(6) by rather straightforward manipulations.

1) Now we consider a bipartite quantum system with ${\cal H}={\cal
H}_{A}\otimes{\cal H}_{B}$ and ${\cal A}=L({\cal
H}_{A})\otimes\mathbbm{1}_{B}$ and, therefore, ${\cal
A}^{\prime}=\mathbbm{1}_{A}\otimes L({\cal H}_{B}).$ In this case one finds
that $\mathbb{P}_{{\cal
A}^{\prime}}(X)=\frac{\mathbbm{1}}{d_{A}}\otimes\mathrm{Tr}_{A}(X),\,$
$\Omega_{\cal A}=\frac{S_{AA^{\prime}}}{d_{A}},$ where $S_{AA^{\prime}}$ is
the swap between the $A$ factors in ${\cal H}^{\otimes\,2}$ and
$d_{X}=\mathrm{dim}\,{\cal H}_{X}\,(X=A,B).$ One gets

$\displaystyle G_{\cal A}(U)=1-\frac{1}{d^{2}}\langle S_{AA^{\prime}},\,{\cal
U}^{\otimes\,2}(S_{AA^{\prime}})\rangle,$ (7)

where $d=d_{B}d_{A}=\mathrm{dim}\,{\cal H}.$ The same relation is true with
$S_{AA^{\prime}}\rightarrow
S_{BB^{\prime}}=SS_{AA^{\prime}}=d_{A}\tilde{\Omega}_{\cal A}.$

Eq. (7) _coincides exactly_ with the averaged OTOC discussed in [Styliaris
_et al._(2021)Styliaris, Anand, and Zanardi] i.e.,
$d^{-1}\mathbb{E}_{X\in{\cal A},Y\in{\cal A}^{\prime}}\left[\|[X,\,{\cal
U}(Y)]\|_{2}^{2}\right]$ (here $\mathbb{E}$ denotes the Haar average over the
unitary groups of $\cal A$ and ${\cal A}^{\prime}.$). Remarkably, this
quantity was shown to be equal to the _operator entanglement_ [Zanardi(2001b),
Wang and Zanardi(2002)] of the unitary $U.$

The latter concept has found important applications to a variety of quantum
information-theoretic problems [Prosen and Pižorn(2007), Chen and Zhou(2018),
Alba  _et al._(2019)Alba, Dubail, and Medenjak, Bertini  _et
al._(2020a)Bertini, Kos, and Prosen, Bertini  _et al._(2020b)Bertini, Kos, and
Prosen]. More recently, it has been shown that operator entanglement requires
_exponentially scaled_ computational resources to simulate [et al.(2021)].

Remark.– The bi-partite OTOC Eq. (7), because of the averages over the two
full sub-algebras, does not satisfy Lieb-Robinson type of bounds with
associated effective ‘light-cone" structures. Indeed the regions $A$ and $B$
are complementary and therefore contiguous (zero distance). The same is, in
general true, for the GAAC which does not even require a locality (tensor
product) structure to begin with.

2) Let ${\cal A}_{B}$ the algebra of operators which are diagonal with respect
to an orthonormal basis $B:=\\{|i\rangle\\}_{i=1}^{d}$ i.e., ${\cal
A}_{B}=\mathbf{C}\,\\{\Pi_{i}:=|i\rangle\langle i|\\}_{i=1}^{d}.$ This is a
$d$-dimensional maximal abelian subalgebra of $L({\cal H})$ such that ${\cal
A}={\cal A}^{\prime}.$ In this case $\mathbb{P}_{{\cal
A}^{\prime}}(X)=\sum_{i=1}^{d}\Pi_{i}X\Pi_{i},\,\Omega_{\cal
A}=\sum_{i=1}^{d}\Pi_{i}^{\otimes\,2},$ and

$\displaystyle G_{{\cal A}_{B}}(U)=1-\frac{1}{d}\sum_{i,j=1}^{d}|\langle
i|U|j\rangle|^{4},$ (8)

This expression coincides with the _coherence generating power_ (CGP) of $U$
introduced in [Zanardi  _et al._(2017a)Zanardi, Styliaris, and Campos Venuti,
Styliaris  _et al._(2019)Styliaris, Anand, Venuti, and Zanardi]. CGP is there
defined as the average coherence (measured by the the sum of the square of
off-diagonal elements, with respect $B$) generated by $U$ starting from any of
the pure incoherent states $\Pi$ i.e., $G_{{\cal
A}_{B}}(U)=\frac{1}{d}\sum_{i=1}^{d}\|\mathbb{Q}_{B}{\cal
U}(\Pi_{i})\|_{2}^{2},$ where $\mathbb{Q}=1-\mathbb{P}_{{\cal A}_{B}}$
projects onto the orthogonal complement of ${\cal A}_{B}.$ [Zanardi  _et
al._(2017a)Zanardi, Styliaris, and Campos Venuti, Zanardi  _et
al._(2017c)Zanardi, Styliaris, and Campos Venuti]. The fact that the CGP is
related to the distance between maximal abelian subalgebras was already
established in [Zanardi and Campos Venuti(2018)]. CGP has been applied to the
detection of the localization transitions in many-body systems [Styliaris  _et
al._(2019)Styliaris, Anand, Venuti, and Zanardi], detection of quantum chaos
in closed and open systems [Anand  _et al._(2021)Anand, Styliaris, Kumari, and
Zanardi].

3) ${\cal H}=\mathbf{C}^{d}\otimes\mathbf{C}^{d},$ ${\cal A}=L({\cal H})_{s},$
${\cal
A}^{\prime}=\mathbf{C}\\{\mathbbm{1},\,S\\}\cong\mathbf{C}\mathbf{Z}_{2}.$
Here, $L({\cal H})_{s}$ denotes the algebra of symmetric operators i.e.,
commuting with the swap $S.$ One can readily check that $\tilde{\Omega}_{\cal
A}=\frac{1}{2}\sum_{J=\pm
1}\left(\frac{\mathbbm{1}+JS}{\sqrt{d(d+J)}}\right)^{\otimes\,2},$ and

$\displaystyle G_{L({\cal H})_{s}}(U)=\frac{1}{2}\left(1-\left|\frac{1-\langle
S,\,{\cal U}(S)\rangle}{d^{2}-1}\right|^{2}\right)$ (9)

Here $J=\pm 1$ is labeling the symmetric/anti-symmetric representation of the
permutation group generated by $S.$

4) ${\cal H}=\mathbf{C}^{d}\otimes\mathbf{C}^{d},$ ${\cal
A}=\mathbf{C}\mathbf{Z}_{2}$ and ${\cal A}^{\prime}=L({\cal H})_{s}.$ Here,
$\Omega_{\cal
A}=\frac{1}{2}\left(\mathbbm{1}^{\otimes\,2}+S^{\otimes\,2}\right),$ $d({\cal
A}^{\prime})=\mathrm{Tr}\,\Omega_{\cal A}=d^{2}(d^{2}+1)/2,$ whence

$\displaystyle G_{\mathbf{C}\mathbf{Z}_{2}}(U)=\frac{1}{2}\frac{d^{4}-|\langle
S,\,{\cal U}(S)\rangle|^{2}}{d^{2}(d^{2}+1)}$ (10)

Consistently with Prop. (1) both functions vanish iff $\langle S,\,{\cal
U}(S)\rangle=d^{2}\Leftrightarrow{\cal U}(S)=S.$ That is to say that non-
scrambling $U$ are such that $[U,\,S]=0$ i.e., $U\in L({\cal H})_{s}.$

5) Let $|\psi\rangle\in{\cal H}$ and $\Pi=|\psi\rangle\langle\psi|.$ We define
${\cal A}_{LE}=\mathbf{C}\\{\mathbbm{1},\,\Pi\\}$ i.e., the unital *-closed
algebra generated by the projection $\Pi.$ The commutant ${\cal
A}_{LE}^{\prime}$ is the algebra of operators leaving the subspace
$\mathbf{C}|\psi\rangle$ and its orthogonal complement invariant. One has,
$\Omega_{{\cal A}_{LE}}=\Pi^{\otimes\,2}+(\mathbbm{1}-\Pi)^{\otimes\,2},$
$d({\cal A}_{LE}^{\prime})=(d-1)^{2}+1.$

$\displaystyle G_{{\cal A}_{LE}}(U)=\frac{2(1-{\cal L}^{2})[d-2(1-{\cal
L}^{2})]}{(d-1)^{2}+1},$ (11)

where ${\cal L}:=|\langle\psi|U|\psi\rangle|$ is the Loschmidt echo. Notice,
$G_{{\cal A}_{LE}}(U)=\frac{2}{d}(1-{\cal L}^{2})+O(1/d^{2})$ and that
$2(1-{\cal L}^{2})=\|\Pi-{\cal U}(\Pi)\|_{2}^{2},$ i.e., the distance between
the algebras ${\cal A}_{LE}^{\prime}$ and its image ${\cal U}({\cal
A}_{LE}^{\prime}),$ as captured by the GAAC [see Eq. (3)], in high dimension
is directly related to the Hilbert-Schmidt distance between the states $\Pi$
and ${\cal U}(\Pi).$ From Eq. (11) one can see that the GAAC is a monotonic
decreasing function of $\cal L$ for $d>4$ and that ${\cal L}=1\Rightarrow
G_{{\cal A}_{LE}}(U)=0.$ For $d=2$ one is back to 2). The case ${\cal L}=0$
corresponds to $U\Pi U^{\dagger}=\mathbbm{1}-\Pi.$

Upper bounds and Expectations.— What are the bounds to algebra scrambling as
measured by the GAAC? Now we would like to answer this question and to see
whether and how those bounds might be saturated.

######  Proposition 3.

i)

$\displaystyle G_{\cal A}(U)\leq\mathrm{min}\\{1-\frac{1}{d({\cal
A})},\,1-\frac{1}{d({\cal A}^{\prime})}\\}=:G_{UB}({\cal A})$ (12)

ii) if $d({\cal A}^{\prime})\leq d({\cal A})$ then the bound above is achieved
iff $\mathbb{P}_{{\cal A}^{\prime}}{\cal U}\mathbb{P}_{{\cal
A}^{\prime}}={\cal T}$ where ${\cal T}\colon
X\mapsto\mathrm{Tr}(X)\frac{\mathbbm{1}}{d}.$ iii) If ${\cal A}^{\prime}$ is
Abelian the bound $1-\frac{1}{d({\cal A}^{\prime})}$ is always achieved. iv)
In the collinear case ii) and iii) above hold true with ${\cal
A}\leftrightarrow{\cal A}^{\prime}.$

The saturation condition $\mathbb{P}_{{\cal A}^{\prime}}{\cal
U}\mathbb{P}_{{\cal A}^{\prime}}={\cal T}$ is quite transparent and intuitive:
maximal scrambling is achieved when, from the point of view of the commutant,
the dynamics generated by $\cal U$ is just full depolarization. Physical
degrees of freedom supported in ${\cal A}^{\prime}$ are, quite properly, fully
_scrambled_.

Let us now briefly discuss Prop. (3) for the physical cases 1–5). In the
bipartite example 1), if $d_{A}=d_{B},$ then (12) is achieved for $U=S$ (swap)
[Styliaris  _et al._(2021)Styliaris, Anand, and Zanardi]. In the maximal
abelian case 2) the bound $1-d^{-1}$ is saturated by those $U$’s such that
$|\langle i|U|j\rangle|=d^{-1/2},\,(\forall i,j)$ [Zanardi  _et
al._(2017a)Zanardi, Styliaris, and Campos Venuti]. In case 3) the bound
$\frac{1}{2}$ is achieved for $\langle S,\,{\cal U}(S)\rangle=1,$ which
amounts to the condition ii). On the other hand, in case 4) from Eq. (10) we
see that $\langle S,\,{\cal
U}(S)\rangle=0\Rightarrow\mathrm{max}_{U}\,G_{\mathbf{C}\mathbf{Z}_{2}}(U)=\frac{1}{2}(1+1/d^{2})^{-1}<\frac{1}{2}$
i.e., bound (12) is _not_ always achieved. 5) The bound $1/2$ is achieved for
$d=2$ only (${\cal A}^{\prime}$ is abelian). For $d>4$ the maximun is for
${\cal L}=0$ and it is $O(1/d).$

The next general question that we would like to address is: what is the
typical value of the GAAC for generic unitaries? To answer this question we
perform an average of (4) over random, Haar distributed, unitaries.

######  Proposition 4.

$\displaystyle i)\qquad\overline{G_{\cal A}(U)}^{U}=\frac{(d^{2}-d({\cal
A}^{\prime}))(d({\cal A}^{\prime})-1)}{d({\cal A}^{\prime})(d^{2}-1)}$ (13)

ii) $\mathbf{Prob}_{U}\left[|G_{\cal A}(U)-\overline{G_{\cal
A}(U)}^{U}|\geq\epsilon\right]\leq\exp[-\frac{d\epsilon^{2}}{4K^{2}}].$

iii) In the collinear case $G_{UB}({\cal A})-\overline{G_{\cal
A}(U)}^{U}=O(1/d)$ and $\mathbf{Prob}_{U}\left[G_{UB}({\cal A})-G_{\cal
A}(U)\geq d^{-1/3}\right]\leq\exp[-\frac{d^{1/3}}{16K^{2}}].$

In ii) and iii) one can choose $K\geq 40.$

As a sanity check, note that Eq. (13) implies that the GAAC of any $U$
vanishes when $d({\cal A}^{\prime})=d^{2}$ i.e., ${\cal
A}=\mathbf{C}\mathbbm{1},$ or $d({\cal A}^{\prime})=1$ i.e., ${\cal A}=L({\cal
H}).$ In fact in these cases $\cal A$ is obviously invariant under the action
of _any_ $U.$ Point ii) is a direct application of the Levy Lemma on measure
concentration: in high dimension (13) is the _typical_ value of the GAAC.
Finally, point iii) shows that, in the collinear case, the average value of
the GAAC converges to the bound (12) when $d$ grows and that the GAAC is, with
overwhelming probability, close to $G_{UB}({\cal A}).$

Time dynamics: infinite averages and fluctuations.—  In this final section we
will consider a one-parameter group of unitary channels
$\\{U_{t}:=e^{-iHt}\\}_{t}$ generated by an Hamiltonian $H$. The idea is that
the behavior of the infinite-time average $\overline{G_{\cal
A}(U_{t})}^{t}:=\lim_{T\to\infty}T^{-1}\int_{0}^{T}dt\,G_{\cal A}(U_{t})$
contains information about the “chaoticity" of the dynamics as seen from the
physical degrees of freedom in the algebra. These calculations greatly extends
the corresponding results, for the bipartite averaged OTOC, reported in
[Styliaris  _et al._(2021)Styliaris, Anand, and Zanardi].

######  Proposition 5.

$\overline{G_{\cal A}(U_{t})}^{t}\leq\overline{G_{\cal
A}(U_{t})}^{NRC}\leq\overline{G_{\cal A}(U)}^{U}$ where

$\displaystyle\overline{G_{\cal A}(U_{t})}^{NRC}=1-\frac{1}{d({\cal
A}^{\prime})}\sum_{\alpha=0,1}\left[\|R^{(\alpha)}\|_{2}^{2}-\frac{1}{2}\|R_{D}^{(\alpha)}\|_{2}^{2}\right],$
(14)

$R^{(0)}_{lk}:=\|\mathbb{P}_{{\cal
A}^{\prime}}(|\Psi_{l}\rangle\langle\Psi_{k}|)\|_{2}^{2},$
$R^{(1)}_{lk}:=\langle\mathbb{P}_{{\cal
A}^{\prime}}(\Pi_{l}),\,\mathbb{P}_{{\cal A}^{\prime}}(\Pi_{k})\rangle,$ and
$(R_{D}^{(\alpha)})_{lk}:=\delta_{lk}R^{(\alpha)}_{lk},$ $(l,k=1,\ldots,d).$
Moreover, the first inequality above becomes an equality if $H$ fulfills the
so-called Non Resonance Condition (NRC).

_Remark.–_ The NRC condition amounts to to say $E_{l}+E_{k}=E_{n}+E_{m}$ iff
$l=n,\,k=m$ or $l=m,\,k=n$ In words: the Hamiltonian spectrum and its gaps are
non-degenerate. This fact holds true for _generic_ (non-interacting)
Hamiltonians.

The result above, which holds for any observable algebra $\cal A,$ has the
very same structure of the corresponding one proved for the averaged bipartite
OTOC (see Prop. 4 in [Styliaris  _et al._(2021)Styliaris, Anand, and
Zanardi]). The matrices $R^{(\alpha)},\,(\alpha=0,1)$ encode the connection
between the algebra and the _full_ system of eigenstates of $H.$

A further simplification occurs, as usual, for the collinear situation
$\mathbf{d}=\lambda\mathbf{n}$: $\lambda R^{(0)}_{lk}=\langle\mathbb{P}_{{\cal
A}}(\Pi_{l}),\,\mathbb{P}_{{\cal A}}(\Pi_{k})\rangle=:R^{(1)}_{lk}({\cal
A}^{\prime}).$ In this case Eq. (14) can be written in way in which $\cal A$
and ${\cal A}^{\prime}$ appear symmetrically and the following upper bound
holds:

$\displaystyle\overline{G_{\cal A}(U_{t})}^{NRC}\leq 1-\frac{1}{d({\cal
A}^{\prime})}-\frac{1}{d({\cal A})}+\frac{1}{d\,d({\cal A}^{\prime})}.$ (15)

This bound is saturated iff $\mathbb{P}_{{\cal
A}^{\prime}}(\Pi_{l})=\mathbb{P}_{{\cal
A}}(\Pi_{l})=\frac{\mathbbm{1}}{d},\,(\forall l).$ Namely, Hamiltonians whose
eigenstates are fully scrambled by the two algebra projections correspond to
maximal infinite-time averaged GAAC. For these Hamiltonians infinite-time
averages of arbitrary observables are, from the point of view of $\cal A$ and
${\cal A}^{\prime},$ _completely_ randomized 444Indeed, for any observable
$\mathbb{P}_{\cal A}(\overline{A(t)}^{t})=\mathbb{P}_{\cal
A}(\sum_{l}A_{l}\Pi_{l})=\sum_{l}A_{l}\mathbb{P}_{\cal
A}(\Pi_{l})=\frac{\mathbbm{1}}{d}\sum_{l}A_{l}=\frac{\mathrm{Tr}(A)\mathbbm{1}}{d}={\cal
T}(A)$ where $A_{l}:=\mathrm{Tr}(A\Pi_{l})$. Same holds for ${\cal
A}^{\prime}.$. Conceptually, this seems a natural way of characterizing
chaoticity relative to the distinguished algebra of observables.

For example: In the bipartite case 1) with $d_{A}=d_{B}$ the bound (15) is
achieved if the (non-degenerate) Hamiltonian has a fully-entangled eigenstates
[Styliaris  _et al._(2021)Styliaris, Anand, and Zanardi]. In the maximal
abelian algebra case 2) the bound saturation corresponds to Hamiltonians with
eigenstates that have maximum coherence with respect to the basis associated
with ${\cal A}$ [Zanardi  _et al._(2017a)Zanardi, Styliaris, and Campos
Venuti]. In both these two important physical situations, the RHS of Eq. (15)
is equal to $(1-\frac{1}{d})^{2}$ and $\overline{G_{\cal
A}(U)}^{U}-\overline{G_{\cal A}(U_{t})}^{NRC}=O(\frac{1}{d^{2}});$ whereby,
assuming that NRC holds, using iii) in Prop. (4) and the Markov inequality,
one can bound temporal fluctuations:

$\displaystyle\mathbf{Prob}_{t}\left[G_{UB}({\cal A})-G_{\cal
A}(U_{t})\geq\epsilon\right]\leq O(\frac{1}{d\,\epsilon}),$ (16)

one sees e.g., by choosing $\epsilon=d^{-1/3},$ that Hamiltonians achieving
bound (15) have, in high dimension, highly suppressed temporal fluctuations
below the value (12).

In [Styliaris  _et al._(2021)Styliaris, Anand, and Zanardi] this concentration
phenomenon has been numerically observed for the bi-partite case in chaotic
many-body systems and _not_ in integrable systems. For the same type of
physical systems, suppression of temporal variance of CGP has been noticed in
[Anand  _et al._(2021)Anand, Styliaris, Kumari, and Zanardi]. These findings
were used to suggest that both the bi-partite averaged OTOC and CGP can be
used as diagnostic tools to detect some aspects of quantum chaotic behavior.
The results above show how this picture may extend to the general algebraic
setting developed in this paper.

In fact,we would like to _define_ ${\cal A}$-chaotic the dynamics generated by
$U_{t}$’s such that the (relative) difference between its infinite-time
average and the Haar-average of the GAAC is approaching zero sufficiently fast
as the system dimension grows. More formally,

$\displaystyle 1-\overline{G_{\cal A}(U_{t})}^{t}/\overline{G_{\cal
A}(U)}^{U}=:\epsilon=O(d^{-\gamma})\quad(\gamma\geq 1).$ (17)

In particular, in the collinear case, this condition would allow one to prove
the “equilibration" result for the GAAC (16). The intuition behind this
definition is quite simple: if Eq. (17) holds the long time behavior of the
GAAC gets, as the system dimension grows, quickly indistinguishable from the
one of a typical Haar random unitary i.e., a “fully chaotic" one.

Before concluding, we would like to illustrate $\cal A$-chaos with the simple
Loschmidt case 5). Here one has $\epsilon=\overline{{\cal
L}^{2}_{t}}^{t}+O(1/d)$ where ${\cal L}_{t}=|\langle\psi|U_{t}|\psi\rangle|.$
The infinite-time average is given by the purity of the Hamiltonian dephased
state $\overline{{\cal L}^{2}_{t}}^{t}=\|\overline{{\cal
U}_{t}(|\psi\rangle\langle\psi|)}^{t}\|_{2}^{2}$ [Venuti and Zanardi(2010)]
Whence the "chaoticity" condition is achieved if this purity is $O(1/d)$ which
in turn implies that the dephased state is $O(1/d)$ away from the maximally
mixed state. This condition is known to be a sufficient one to bound time-
fluctuations of the expectation value of observables with initial state
$|\psi\rangle$ [Reimann(2008)]. Namely, ${\cal A}_{LE}$-chaos amounts to
temporal-equilibration [Venuti and Zanardi(2010)].

Conclusions.—  In this paper we have proposed a novel approach to quantum
scrambling based on algebras of observables. A quantitative measure of
scrambling is introduced in terms of anti-correlation between the whole
commutant algebra and its (unitarily) evolved image. This quantity, which we
named the Geometric Algebra Anti Correlator (GAAC), has also a clear
geometrical meaning as it describes the distance between the two algebras or,
equivalently, the degree of self-orthogonalization induced by the dynamics.

We explicitly computed the GAAC for several physically motivated cases and
characterized its behavior in terms of typical values, upper bounds and
temporal fluctuations. We have shown that the GAAC formalism provides an
unified mathematical and conceptual framework for concepts like operator
entanglement, averaged bipartite OTOC, coherence generating power and
Loschmidt echo.

Finally, we suggested an approach to quantum chaos in terms of the behavior of
infinite-time average of the GAAC for large system dimension. To assess the
effectiveness of such an approach is one of the challenges of future
investigations.

##  1 Acknowledgments

I acknowledge discussions with Namit Anand and partial support from the NSF
award PHY-1819189. This research was (partially) sponsored by the Army
Research Office and was accomplished under Grant Number W911NF-20-1-0075. The
views and conclusions contained in this document are those of the authors and
should not be interpreted as representing the official policies, either
expressed or implied, of the Army Research Office or the U.S. Government. The
U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation herein.

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##  Appendix A Supplemental Material

###  A.1 Proof of Prop 1

_i)_ It is a direct computation: $D^{2}({\cal A}^{\prime},\,{\cal U}({\cal
A}^{\prime}))=\|\mathbb{P}_{{\cal A}^{\prime}}-\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}\|_{HS}^{2}=\|\mathbb{P}_{{\cal
A}^{\prime}}\|_{HS}^{2}+\|\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}\|_{HS}^{2}-2\,\langle\mathbb{P}_{{\cal
A}^{\prime}},\,\mathbb{P}_{{\cal U}({\cal A}^{\prime})}\rangle.$ Now,
$\|\mathbb{P}_{{\cal A}^{\prime}}\|_{HS}^{2}=\|{\cal U}\mathbb{P}_{{\cal
A}^{\prime}}{\cal U}^{\dagger}\|_{HS}^{2}=\|\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}\|_{HS}^{2}=d({\cal A}^{\prime}).$ Whence, by dividing $D^{2}$ by
$2\,d({\cal A}^{\prime})=2\,\|\mathbb{P}_{{\cal A}^{\prime}}\|_{HS}^{2},$ Eq.
(3) follows.

_ii)_ Since $D$ is a metric from Eq. (3) one has $G_{\cal
A}(U)=0\Leftrightarrow{\cal U}({\cal A}^{\prime})={\cal
A}^{\prime}\Leftrightarrow{\cal U}({\cal A})={\cal A}.$ Last equivalence is
obtained by taking the commutant of both sides, using ${\cal
A}^{\prime\prime}={\cal A}$ (double commutant theorem) and that ${\cal
U}({\cal A})^{\prime}={\cal U}({\cal A}^{\prime})$ (true for unitary auto-
morphisms).

###  A.2 Proof of Prop 2

_i)_ Let us write the Algebra projections in the Kraus form $\mathbb{P}_{{\cal
A}^{\prime}}(X)=\sum_{\alpha}e_{\alpha}Xe^{\dagger}_{\alpha}$ and
$\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}(X)=\sum_{\alpha}(Ue_{\alpha}U^{\dagger})X(Ue_{\alpha}U^{\dagger})^{\dagger}.$
Here, because of the structure theorem (1), one can choose the _orthogonal_
basis of ${\cal A}$ given by

$\displaystyle
e_{\alpha}=\frac{1}{\sqrt{d_{J}}}\mathbbm{1}_{n_{J}}\otimes|l\rangle\langle
m|,\in{\cal A}\qquad\alpha:=(J,l,m)\,(l,m=1,\ldots,d_{J}).$ (18)

(note that $|\\{\alpha\\}|=\sum_{J}d_{J}^{2}=d({\cal A})$ and that the set
$\\{e_{\alpha}\\}_{\alpha}$ is closed under hermitian conjugation). Hence
$\langle\mathbb{P}_{{\cal A}^{\prime}},\,\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}\rangle=\mathrm{Tr}_{HS}\left(\mathbb{P}_{{\cal
A}^{\prime}}\mathbb{P}_{{\cal U}({\cal
A}^{\prime})}\right)=\sum_{\alpha,\beta}|\mathrm{Tr}\left(e_{\alpha}Ue_{\beta}U^{\dagger}\right)|^{2}=\sum_{\alpha,\beta}|\langle
e_{\alpha},\,{\cal U}(e_{\beta})\rangle|^{2}.$ Here we’ve used that ${\cal
T}=\sum_{i}T_{i}XT_{i}^{\dagger}\Rightarrow\mathrm{Tr}_{HS}\,{\cal
T}=\sum_{i}|\mathrm{tr}\,T_{i}|^{2}.$ Because of the definition (2) this
proves Eq. (6).

On the other hand, if $\Omega_{\cal A}=\sum_{\alpha}e_{\alpha}\otimes
e_{\alpha}^{\dagger}$ one has $\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega_{\cal A})\rangle=\sum_{\alpha,\beta}\langle
e_{\alpha}\otimes e_{\alpha}^{\dagger},\,Ue_{\beta}U^{\dagger}\otimes
Ue^{\dagger}_{\beta}U^{\dagger}\rangle=\sum_{\alpha,\beta}\langle
e_{\alpha},\,Ue_{\beta}U^{\dagger}\rangle\langle
e^{\dagger}_{\alpha},\,Ue^{\dagger}_{\beta}U^{\dagger}\rangle=\sum_{\alpha,\beta}|\langle
e_{\alpha},\,{\cal U}(e_{\beta})\rangle|^{2}.$ Moreover, from (18) one has

$\displaystyle\Omega_{\cal
A}=\sum_{J}\frac{\mathbbm{1}_{n_{J}}^{\otimes\,2}}{d_{J}}\otimes\sum_{l,m=1}^{d_{J}}|lm\rangle\langle
ml|=:\sum_{J}\frac{\mathbbm{1}_{n_{J}}^{\otimes\,2}\otimes S_{d_{J}}}{d_{J}}.$
(19)

Therefore, $\|\Omega_{\cal A}\|_{2}^{2}=\mathrm{Tr}\left(\Omega^{2}_{\cal
A}\right)=\mathrm{Tr}\left(\sum_{J}\frac{(\mathbbm{1}_{n_{J}}\otimes\mathbbm{1}_{d_{J}})^{\otimes\,2}}{d_{J}^{2}}\right)=\sum_{J}(n_{J}d_{J})^{2}/d_{J}^{2}=\sum_{J}n_{J}^{2}=d({\cal
A}^{\prime}).$ This completes the proof of the first equality in Eq. (4). Now,
if $S$ is the swap operator $S\Omega_{\cal
A}S=\sum_{\alpha}e_{\alpha}^{\dagger}\otimes
e_{\alpha}=\sum_{\alpha}e_{\alpha}\otimes e_{\alpha}^{\dagger}=\Omega_{\cal
A},$ i.e.,$[S,\,\Omega_{\cal A}]=0.$ Since $[U^{\otimes\,2},\,S]=0$ and
$\|\Omega_{\cal A}\|_{2}^{2}=\|S\Omega_{\cal
A}\|_{2}^{2}=\|{\tilde{\Omega}_{\cal A}}\|_{2}^{2}$ the second equality in Eq.
(4) follows. Also, since $S(\mathbbm{1}_{n_{J}}^{\otimes\,2}\otimes
S_{d_{J}})=S_{n_{J}}\otimes\mathbbm{1}_{d_{J}}^{\otimes\,2},$ ($S_{n_{J}}$ is
a swap operator defined over the $\mathbf{C}^{n_{J}}$ factors) one finds
$\tilde{\Omega}_{\cal
A}=:\sum_{J}\frac{S_{n_{J}}\otimes\mathbbm{1}_{d_{J}}^{\otimes\,2}}{d_{J}}=\sum_{\gamma}f_{\gamma}\otimes
f^{\dagger}_{\gamma},$ where $\gamma:=(J,p,q),\,p,q=1,\ldots,n_{J}$ and
$f_{\gamma}=\frac{1}{\sqrt{d_{J}}}|p\rangle\langle
q|\otimes\mathbbm{1}_{d_{J}}\in{\cal A}^{\prime}.$ This is an _orthonormal_
basis of ${\cal A}^{\prime}.$

By direct computation $\mathrm{Tr}_{1}\left(S\Omega_{\cal
A}(X\otimes\mathbbm{1}\right)=\sum_{J}\mathrm{tr}_{d_{J}}(X)\otimes\frac{\mathbbm{1}_{d_{J}}}{d_{J}}=\mathbb{P}_{{\cal
A}^{\prime}}(X)$ which proves Eq. (5).

_ii)_ In the collinear case $\tilde{\Omega}_{\cal
A}=\frac{1}{\lambda}{\Omega}_{{\cal
A}^{\prime}}=\frac{1}{\lambda}\sum_{J}\frac{S_{n_{J}}\otimes\mathbbm{1}_{d_{J}}^{\otimes\,2}}{n_{J}}$
where $d_{J}=\lambda n_{J},\,(\forall J).$ Inserting this in Eq. (4) and using
$\lambda^{2}\|{\Omega}_{\cal
A}\|_{2}^{2}=\lambda^{2}\sum_{J}n_{J}^{2}=\sum_{J}d_{J}^{2}=d({\cal
A})=\|{\Omega}_{{\cal A}^{\prime}}\|_{2}^{2}$ one sees that in this collinear
case $G_{\cal A}(U)=G_{{\cal A}^{\prime}}(U).$

###  A.3 Proof of Prop 3

_i)_ First, notice that for any two orthogonal projections $P$ and $Q$ one has
that
$\mathrm{Tr}\left(PQ\right)\geq\mathrm{dim}\left(\mathrm{Im}P\cap\mathrm{Im}Q\right).$
Since both ${\cal A}^{\prime}$ and ${\cal U}({\cal A}^{\prime})$ contain the
identity $\mathbbm{1}$ one has $\langle\mathbb{P}_{{\cal
A}^{\prime}},\,\mathbb{P}_{{\cal U}({\cal A}^{\prime})}\rangle\geq 1,$ from
which the bound $G_{\cal A}(U)\leq 1-1/d({\cal A}^{\prime}),$ immediately
follows.

To prove the bound $G_{\cal A}(U)\leq 1-1/d({\cal A}),$ we begin by observing
that ${\Omega}_{\cal
A}=\sum_{\alpha}\tilde{e}_{\alpha}\otimes\tilde{e}^{\dagger}_{\alpha}$ for any
basis $\tilde{e}_{\alpha}=\sum_{\beta}U_{\beta,\alpha}e_{\beta},$ where the
$e_{\beta}$’s are given by (18) and the matrix $U_{\beta,\alpha}$ is unitary.
Now $\frac{\mathbbm{1}}{\sqrt{d({\cal A})}}=\sum_{J}\sqrt{\frac{d_{J}}{d({\cal
A})}}\sum_{l=1}^{d_{J}}e_{(J,l,l)}=:\tilde{e}_{1}.$ Since
$\sum_{J,l}|\sqrt{\frac{d_{J}}{d({\cal A})}}|^{2}=\frac{1}{d({\cal
A})}\sum_{J}d_{J}^{2}=1,$ we see that on can always unitarily move to a new
basis such that $\tilde{e}_{1}=\frac{\mathbbm{1}}{\sqrt{d({\cal A})}}.$ Whence
${\Omega}_{\cal A}=\frac{\mathbbm{1}}{{d({\cal
A})}}+\sum_{\alpha>1}\tilde{e}_{\alpha}\otimes\tilde{e}^{\dagger}_{\alpha}=:\frac{\mathbbm{1}}{{d({\cal
A})}}+\Omega_{\cal A}^{\prime}$ and $\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega_{\cal A})\rangle=\frac{\mathrm{Tr}\,\Omega_{\cal
A}}{d({\cal A})}+\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega^{\prime}_{\cal A})\rangle\geq\frac{d({\cal
A}^{\prime})}{d({\cal A})}$ ( note $\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega^{\prime}_{\cal
A})\rangle=\sum_{\alpha}\sum_{\beta>1}|\langle e_{\alpha},\,{\cal
U}(\tilde{e}_{\beta})\rangle|^{2}\geq 0.$). Plugging this inequality in Eq.
(4) one finds $G_{\cal A}(U)\leq 1-1/d({\cal A}).$

In summary, $G_{\cal A}(U)\leq\mathrm{min}\\{1-1/d({\cal A}),\,1-1/d({\cal
A}^{\prime})\\}.$

_ii)_ One has that $\langle\mathbb{P}_{{\cal A}^{\prime}},\,\mathbb{P}_{{\cal
U}({\cal A}^{\prime})}\rangle=\|\mathbb{P}_{{\cal A}^{\prime}}{\cal
U}\mathbb{P}_{{\cal A}^{\prime}}\|_{HS}^{2}.$ This last norm is always larger
than the (square of the) operator norm of the CP-map ${\cal
F}:=\mathbb{P}_{{\cal A}^{\prime}}{\cal U}\mathbb{P}_{{\cal A}^{\prime}}$
which is one. The lower bound is achieved _iff_ ${\cal F}$ has rank one, but
the only rank one unital trace-preserving CP-map is the depolarizing channel
${\cal T}.$

_iii)_ If ${\cal A}^{\prime}$ is Abelian one take the orhonormal basis
$f_{J}=\frac{\mathbbm{1}_{1}\otimes\mathbbm{1}_{d_{J}}}{\sqrt{d_{J}}}=\frac{\Pi_{J}}{\sqrt{d_{J}}}.$It
follows, $\langle\tilde{\Omega}_{\cal A},\,{\cal
U}^{\otimes\,2}(\tilde{\Omega}_{\cal
A})\rangle=\sum_{J,K}\frac{1}{d_{J}d_{K}}|\langle\Pi_{J},\,{\cal
U}(\Pi_{K})\rangle|^{2}.$ Now if $\Pi_{J}=\sum_{l=1}^{d_{J}}|Jl\rangle\langle
Jl|$ and $\Pi_{K}=\sum_{m=1}^{d_{K}}|Km\rangle\langle Km|,$ one has
$\langle\Pi_{J},\,{\cal
U}(\Pi_{K})\rangle=\sum_{l=1}^{d_{J}}\sum_{m=1}^{d_{K}}|\langle
Km|U|Jl\rangle|^{2}.$ Therefore, if $U$ maps the basis $|Jl\rangle$ into
mutually unbiased one i.e., $|\langle Km|U|Jl\rangle|=1/\sqrt{d},\,(\forall
J,K,l,m)$ one finds $\langle\Pi_{J},\,{\cal
U}(\Pi_{K})\rangle=\frac{d_{J}d_{K}}{d}$ whence
$\sum_{J,K}\frac{1}{d_{J}d_{K}}|\langle\Pi_{J},\,{\cal
U}(\Pi_{K})\rangle|^{2}=\sum_{JK}\frac{d^{2}_{J}d^{2}_{K}}{d_{J}d_{K}d^{2}}=1$
(here we used that, in the Abelian case, $\sum_{J}d_{J}=d$.) By Eq. (4) this
last relation implies that, for these $U$’s, the upper bound $1-1/d({\cal
A}^{\prime})$ is saturated. Notice that in this case, $\mathbb{P}_{{\cal
A}^{\prime}}{\cal
U}(\frac{\Pi_{J}}{\sqrt{d_{J}}})=\sum_{K}\frac{\Pi_{K}}{\sqrt{d_{K}}}\langle\frac{\Pi_{K}}{\sqrt{d_{K}}},\,{\cal
U}(\frac{\Pi_{J}}{\sqrt{d_{J}}})\rangle=\frac{1}{d}\sum_{K}\frac{\Pi_{K}}{\sqrt{d_{K}}}\frac{d_{J}d_{K}}{\sqrt{d_{J}}\sqrt{d_{K}}}=\sqrt{d_{J}}\frac{1}{d}\sum_{K}\Pi_{K}=\sqrt{d_{J}}\frac{\mathbbm{1}}{d}={\cal
T}(\frac{\Pi_{J}}{\sqrt{d_{J}}}),\,\forall J$ which implies $\mathbb{P}_{{\cal
A}^{\prime}}{\cal U}\mathbb{P}_{{\cal A}^{\prime}}={\cal T}.$

_iv)_ Since in the collinear case $G_{\cal A}(U)=G_{{\cal A}^{\prime}}(U)$ ii)
and iii) above holds with $\cal A$ replacing ${\cal A}^{\prime}.$

###  A.4 Proof of Prop 4

_i)_ Averaging over the Haar measure gives you a projector: $\overline{{\cal
U}^{\otimes\,2}(X)}^{U}=\overline{U^{\otimes\,2}XU^{\dagger\otimes\,2}}^{U}=:\mathbb{P}_{Haar}(X)$
over the commutant of the algebra generated by $\\{U^{\otimes\,2}\,/\,U\in
U({\cal H})\\}.$ By Schur-Weyl duality this commutant is generated by
$\mathbbm{1}$ and the swap $S$:

$\displaystyle\mathbb{P}_{Haar}(X)=\frac{1}{2}\sum_{\alpha=\pm
1}\frac{\mathbbm{1}+\alpha S}{d(d+\alpha)}\langle\mathbbm{1}+\alpha
S,\,X\rangle.$ (20)

Therefore, $\overline{\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega_{\cal A})\rangle}^{U}=\langle\Omega_{\cal
A},\,\mathbb{P}_{Haar}(\Omega_{\cal
A})\rangle=\|\mathbb{P}_{Haar}(\Omega_{\cal
A})\|_{2}^{2}=\frac{1}{2}\sum_{\alpha=\pm 1}\frac{|\langle\mathbbm{1}+\alpha
S,\,\Omega_{\cal A}\rangle|^{2}}{d(d+\alpha)}.$ Now
$\langle\mathbbm{1},\,\Omega_{\cal A}\rangle=\mathrm{Tr}\,\Omega_{\cal
A}=d({\cal A}^{\prime}),$ and $\langle S,\,\Omega_{\cal
A}\rangle=\mathrm{Tr}\,\tilde{\Omega}_{\cal
A}=\sum_{J}n_{J}d_{J}^{2}/d_{J}=\sum_{J}n_{J}d_{J}=d.$ Proving Eq. (13) is now
straightforward algebra from these equations and (4).

_ii)_ This is an application of the Levy’s Lemma for the GAAC: $U\in U({\cal
H})\mapsto G_{\cal A}(U):=\frac{\langle\Omega_{\cal A},\,{\cal
U}^{\otimes\,2}(\Omega_{\cal A})\rangle}{\|\Omega_{\cal A}\|_{2}^{2}}.$

Levy’s Lemma:

$\displaystyle|G_{\cal A}(U)-G_{\cal A}(V)|\leq
K\|U-V\|_{2}\Rightarrow\mathbf{Prob}_{U}\left[|G_{\cal A}(U)-\overline{G_{\cal
A}(U)}^{U}|\geq\epsilon\right]\leq\exp\left(-\frac{d\epsilon^{2}}{4K^{2}}\right).$
(21)

Let us show that this is Lipschitz function. $|G_{\cal A}(U)-G_{\cal
A}(V)|=\|\Omega_{\cal A}\|_{2}^{-2}|\langle\Omega_{\cal A},\,({\cal
U}^{\otimes\,2}-{\cal V}^{\otimes\,2})(\Omega_{\cal A})\rangle|\leq\|{\cal
U}^{\otimes\,2}-{\cal V}^{\otimes\,2}\|_{2,2}$ [here $\|{\cal
T}\|_{2,2}:=\sup_{\|X\|_{2}=1}\|{\cal T}(X)\|_{2}.$] If ${\cal U}-{\cal
V}=\Delta$ one has $\|{\cal U}^{\otimes\,2}-{\cal
V}^{\otimes\,2}\|_{2,2}\leq\|\Delta\|_{2,2}(\|\Delta\|_{2,2}+2)$ [where we
used $\|X\otimes Y\|_{2,2}=\|X\|_{2,2}\|Y\|_{2,2},$ and $\|{\cal
V}\|_{2,2}=1.$] Moreover, if $U-V=\delta$ then $\Delta(X)=\delta
X\delta^{\dagger}+\delta XV^{\dagger}+VX\delta^{\dagger}.$ From this one finds
$\|\Delta\|_{2,2}\leq\sup_{\|X\|_{2}=1}(\|\delta\|_{2}\|X\delta^{\dagger}\|_{2}+\|\delta
X\|_{2}+\|X\delta^{\dagger}\|_{2})\leq 4\,\|\delta\|_{2}.$ Notice also,
$\|X\delta\|_{2}=\|X(U-V)\|_{2}\leq 2\|X\|_{2},$ whence
$\|\Delta\|_{2,2}\leq\sup_{\|X\|_{2}=1}(2\|X\|_{2}+2\|X\|_{2}+2\|X\delta^{\dagger}\|_{2})\leq
8.$ Bringing everything together:

$\displaystyle|G_{\cal A}(U)-G_{\cal
A}(V)|\leq\|\Delta\|_{2,2}(\|\Delta\|_{2,2}+2)\leq 10\,\|\Delta\|_{2,2}\leq
40\|\delta\|_{2}=40\,\|U-V\|_{2}.$ (22)

This shows that one can choose a Lipschitz constant $K\geq 40$ for $f.$

_iii)_ In this collinear case, since $d^{2}=d({\cal A})d({\cal A}^{\prime}),$
the Haar average (13) takes the form $\overline{G_{\cal
A}(U)}^{U}=(1-1/d^{2})^{-1}(1-1/d({\cal A}^{\prime}))(1-1/d({\cal A})).$
Suppose $d({\cal A}^{\prime})\leq d,$ ($d({\cal A})\geq d$) then $G_{UB}({\cal
A})=1-1/d({\cal A}^{\prime})$ One has $G_{UB}({\cal A})-\overline{G_{\cal
A}(U)}^{U}=\frac{1-1/d({\cal A}^{\prime})}{1-1/d^{2}}(1/d({\cal
A})-1/d^{2})\leq 1/d({\cal A})-1/d^{2}\leq 1/d.$ The case $d({\cal
A}^{\prime})>d,$ works exactly in the same way (with ${\cal
A}\leftrightarrow{\cal A}^{\prime}.$) This proves that $G_{UB}({\cal
A})-\overline{G_{\cal A}(U)}^{U}=O(1/d).$

Now, $G_{UB}({\cal A})-G_{\cal A}(U)=(G_{UB}({\cal A})-\overline{G_{\cal
A}(U)}^{U})+(\overline{G_{\cal A}(U)}^{U}-G_{\cal A}(U))\geq\epsilon$ implies,
for large $d$, that $\overline{G_{\cal A}(U)}^{U}-G_{\cal
A}(U))\geq\epsilon-1/d\geq\epsilon/2.$ It follows that
$\mathbf{Prob}_{U}\left[G_{UB}({\cal A})-G_{\cal
A}(U)\geq\epsilon\right]\leq\mathbf{Prob}_{U}\left[\overline{G_{\cal
A}(U)}^{U}-G_{\cal
A}(U))\geq\epsilon/2\right]\leq\exp[-\frac{d\epsilon^{2}}{16K^{2}}].$

###  A.5 Proof of Prop 5

Suppose the unitary evolution has the spectral resolution
$U_{t}=\sum_{n}\Pi_{n}e^{-iE_{n}t}$ (here $n$ ranges over the set of distinct
eigenvalues) then one has ${\cal
U}_{t}^{\otimes\,2}(X)=\sum_{k,h,p,m}\Pi_{k}\otimes\Pi_{h}\,X\,\Pi_{p}\otimes\Pi_{m}\,\exp[-it(E_{k}+E_{h}-E_{p}-E_{m})].$
Performing the infinite time average one has
$\overline{\exp[-it(E_{k}+E_{h}-E_{p}-E_{m})]}^{t}=,\delta_{0,E_{k}+E_{h}-E_{p}-E_{m}}.$
and therefore

$\displaystyle\bar{\mathbb{P}}(X):=\overline{{\cal
U}_{t}^{\otimes\,2}(X)}^{t}=\sum_{k,h,p,m}\Pi_{k}\otimes\Pi_{h}\,X\,\Pi_{p}\otimes\Pi_{m}\,\delta_{0,E_{k}+E_{h}-E_{p}-E_{m}}.$
(23)

Clearly, the map $X\mapsto\bar{\mathbb{P}}(X)$ is an orthogonal projection
whose range contains $\mathbbm{1}$ and $S$ from which
$\bar{\mathbb{P}}\geq{\mathbb{P}}_{Haar}.$ Limiting the sum to the pairs
$k=p,\,h=m$ _or_ $k=m,\,h=p$ (with $k\neq h$) one obtains the sub-projection
$\mathbb{P}_{NRC^{+}}\leq\bar{\mathbb{P}}.$ By direct inspection $\mathbbm{1}$
and $S$ also belong to the range of $\mathbb{P}_{NRC^{+}},$ which in turn,
implies ${\mathbb{P}}_{NRC^{+}}\geq{\mathbb{P}}_{Haar}.$

A further operator inequality can be obtained considering the (NB non-unique)
resolution of the projectors $\Pi_{l}$’s. In fact, if
$\Pi_{l}=\sum_{j=1}^{d_{l}}|\psi_{l,j}\rangle\langle\psi_{l,j}|=:\sum_{j=1}^{d_{l}}\Pi_{l,j}$
one has $\mathbb{P}_{NRC^{+}}\geq\mathbb{P}_{NRC}$ where

$\displaystyle\mathbb{P}_{NRC}(X)=\sum_{k,h,i,j}\Pi_{k,i}\otimes\Pi_{h,j}\,X\,\Pi_{k,i}\otimes\Pi_{h,j}+\sum_{k\neq
h,i,j}\Pi_{k,i}\otimes\Pi_{h,j}\,X\,\Pi_{h,j}\otimes\Pi_{k,i}.$ (24)

This projector equals $\bar{\mathbb{P}}$ if the Non Resonant Condition (NRC)
holds i.e., $E_{k}+E_{h}=E_{p}+E_{m}\Leftrightarrow(k=p)\,\&\,(h=m)$ _or_
$(k=m)\,\&\,(h=p)\,\&\,(k\neq h).$ Notice that NRC implies (for $k=h$) that
the Hamiltonian is non-degenerate.

In summary, so far we have proven that $\overline{\langle\Omega_{\cal
A},\,{\cal U}_{t}^{\otimes\,2}(\Omega_{\cal
A})\rangle}^{t}=\langle\Omega_{\cal A},\,\bar{\mathbb{P}}(\Omega_{\cal
A})\rangle=\|\bar{\mathbb{P}}(\Omega_{\cal
A})\|_{2}^{2}\geq\|{\mathbb{P}}_{NRC^{+}}(\Omega_{\cal
A})\|_{2}^{2}\geq\|{\mathbb{P}}_{NRC}(\Omega_{\cal
A})\|_{2}^{2}\geq\|{\mathbb{P}}_{Haar}(\Omega_{\cal A})\|_{2}^{2}.$ These
inequalities, by Eq. (4), translate into the following ones for the GAAC:

$\displaystyle\overline{G_{\cal A}(U_{t})}^{t}\leq\overline{G_{\cal
A}(U_{t})}^{NRC^{+}}\leq\overline{G_{\cal
A}(U_{t})}^{NRC}\leq\overline{G_{\cal A}(U)}^{U}.$ (25)

Let us now consider $\overline{G_{\cal A}(U_{t})}^{NRC}$ explicitly. From now
on the labels $k,\,h=1,\ldots,d$ correspond to eigenstates (_not_
eigenspaces). First $\|{\mathbb{P}}_{NRC}(\Omega_{\cal
A})\|^{2}_{2}=\sum_{k,h}\|\Pi_{k}\otimes\Pi_{h}\,\Omega_{\cal
A}\,\Pi_{k}\otimes\Pi_{h}\|_{2}^{2}+\sum_{k,\neq
h}\|\Pi_{k}\otimes\Pi_{h}\,\Omega_{\cal
A}\,\Pi_{h}\otimes\Pi_{k}\|_{2}^{2}=\sum_{k,h}|\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{k}\psi_{h}\rangle|^{2}+\sum_{k,\neq
h}|\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{h}\psi_{k}\rangle|^{2}=\sum_{k,h}|\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{k}\psi_{h}\rangle|^{2}+\sum_{k,h}|\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{h}\psi_{k}\rangle|^{2}-\sum_{h}|\langle\psi_{h}\psi_{h}|\Omega_{\cal
A}|\psi_{h}\psi_{h}\rangle|^{2}$

Moreover, $\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{k}\psi_{h}\rangle=\sum_{\alpha}\langle\psi_{k}|e_{\alpha}|\psi_{k}\rangle\langle\psi_{h}|e^{\dagger}_{\alpha}|\psi_{h}\rangle=\sum_{\alpha}\mathrm{Tr}\left(|\psi_{h}\rangle\langle\psi_{k}|e_{\alpha}|\psi_{k}\rangle\langle\psi_{h}|e^{\dagger}\right)=\langle|\psi_{k}\rangle\langle\psi_{h}|,\,\mathbb{P}_{{\cal
A}^{\prime}}(|\psi_{k}\rangle\langle\psi_{h}|)\rangle=\|\mathbb{P}_{{\cal
A}^{\prime}}(|\psi_{k}\rangle\langle\psi_{h}|)\|_{2}^{2}=:R^{(0)}_{lk},$ and

$\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{h}\psi_{k}\rangle=\sum_{\alpha}\langle\psi_{k}|e_{\alpha}|\psi_{h}\rangle\langle\psi_{h}|e^{\dagger}_{\alpha}|\psi_{k}\rangle=\sum_{\alpha}\mathrm{Tr}\left(|\psi_{k}\rangle\langle\psi_{k}|e_{\alpha}|\psi_{h}\rangle\langle\psi_{h}|e^{\dagger}\right)=\langle|\psi_{k}\rangle\langle\psi_{k}|,\,\mathbb{P}_{{\cal
A}^{\prime}}(|\psi_{h}\rangle\langle\psi_{h}|)\rangle=\langle\mathbb{P}_{{\cal
A}^{\prime}}(|\psi_{k}\rangle\langle\psi_{k}|),\,\mathbb{P}_{{\cal
A}^{\prime}}(|\psi_{h}\rangle\langle\psi_{h}|)\rangle=:R^{(1)}_{lk}.$

Since, $\overline{G_{\cal A}(U_{t})}^{NRC}=1-d({\cal
A}^{\prime})^{-1}\|{\mathbb{P}}_{NRC}(\Omega_{\cal A})\|^{2}_{2},$ the
equations above prove Eq. (14).

Collinear case.—  Notice, $\langle\psi_{k}\psi_{h}|\Omega_{\cal
A}|\psi_{k}\psi_{h}\rangle=\langle\psi_{k}\psi_{h}|\tilde{\Omega}_{\cal
A}|\psi_{h}\psi_{k}\rangle$ since in the _collinear_ case
$\tilde{\Omega}_{\cal A}=\lambda^{-1}\Omega_{{\cal A}^{\prime}}$ and $d({\cal
A}^{\prime})\lambda^{2}=d({\cal A}),$ from Eq. (14) it follows that

$\displaystyle\overline{G_{\cal A}(U_{t})}^{NRC}=1-\frac{1}{d({\cal
A})}\sum_{l,k}|\langle\mathbb{P}_{{\cal A}}(\Pi_{l}),\,\mathbb{P}_{{\cal
A}}(\Pi_{k})\rangle|^{2}-\frac{1}{d({\cal
A}^{\prime})}\sum_{l,k}|\langle\mathbb{P}_{{\cal
A}^{\prime}}(\Pi_{l}),\,\mathbb{P}_{{\cal
A}^{\prime}}(\Pi_{k})\rangle|^{2}+\frac{1}{d({\cal
A}^{\prime})}\sum_{l}\|\mathbb{P}_{{\cal A}^{\prime}}(\Pi_{l})\|_{2}^{4}.$
(26)

The first two terms are basically H-S norms of bistochastic matrices, the
third in a diagonal contribution that could also be written as
$\frac{1}{d({\cal A})}\sum_{l}\|\mathbb{P}_{{\cal A}}(\Pi_{l})\|_{2}^{4}.$ The
quantity (26) is maximized , providing the upper-bound (15), when all the
entries of both matrices are equal to $1/d$ (maximally mixed bistochastic
matrices with constant entries, a 1D projector). This is true iff
$\mathbb{P}_{{\cal A}^{\prime}}(\Pi_{l})=\mathbb{P}_{{\cal
A}}(\Pi_{l})=\frac{\mathbbm{1}}{d},\,(\forall l).$ In terms of the Hamiltonian
dephasing map ${\cal D}_{H}(X)=\sum_{l}\Pi_{l}X\Pi_{l},$ this maximal
scrambling condition reads $\mathbb{P}_{{\cal A}^{\prime}}{\cal
D}_{H}=\mathbb{P}_{{\cal A}}{\cal D}_{H}={\cal T}$ where $\cal T$ denotes the
depolarizing channel i.e., ${\cal T}(X)=\mathrm{Tr}(X)\frac{\mathbbm{1}}{d}.$