Quantum scrambling of observable algebras

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.

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 [8,9] (see also recent developments in [10,11]). As Paolo Zanardi: e-mail: zanardi@usc.edu 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 [12].
We will show that specific instances of our construction allow one to recover apparently different concepts including operator entanglement [13,14], averaged bipartite OTOCs [15,16], coherence generating power [17][18][19] and Loschmidt echo [20,21]. 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 1 .
Preliminaries.-In this section we introduce the main formal ingredients utilized in this paper and set the notation. Let H = span{|m } d m=1 be a ddimensional Hilbert space and L(H) its the full operator algebra (see 2 for further notation). In the following by the notation C{X} we will denotes the vector space spanned by the X s.
The key formal ingredients of this investigation are hermitian-closed unital subalgebras A ⊂ L(H) and their commutants A := {X ∈ L(H) / [X, Y ] = 0, ∀Y ∈ A}. The intersection A ∩ A =: Z(A) is the center of the algebra A. The fundamental structure theorem of these objects states that the Hilbert space breaks into a direct sum of d Z := dim Z(A) orthogonal blocks and each of them has a tensor product bi-partite structure: Moreover, 1 n J ⊗ 1 d J }, namely the center of A is spanned by the projections over the H J blocks.
Associated to any algebra A we have an orthogonal (super) projection CP-map: P † A = P A , P 2 A = P A and Im P A = A. Such maps can be written in the where the e α are a suitable orthogonal basis of A . Notice that In terms of the decomposition (1) one has P A (X) = J Factors: Z(A) = C1, in this case H ∼ = C n1 ⊗ C d1 namely the algebra A endows H with a bipartition into virtual subsystems [8,9]. The case in which H = clearly falls in this category.
Super-Selection: A ⊂ A this is when the commutant is an Abelian algebra. This implies n J = 1, (∀J) and therefore the Hilbert space breaks into d J -dimensional super-selection sectors i.e., . If A is a maximal abelian subalgebra one has A = A and n J = d J = 1, (∀J). This is the case that is relevant to the study of quantum coherence [26] and its dynamical generation [17,27].
When the d Z -dimensional (integer-valued) vectors d := (d J ) J , and n := (n J ) J are proportional to each other i.e., d = λn one has that d 2 = d(A)d(A ). If this is the case we shall say that the pair (A, A ) 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 The geometrical meaning of GAAC should be evident from Eq. (2): the larger G A (U ) the smaller is the intersection between A and its unitarily evolved image Remark.-In the RHS of Eq. (2) we use the A (and not A) as the dynamics U is in the Heisenberg picture. Symmetries mapped out of A by U is equivalent to 3 This can be seen from the fact that given two projectors P , and Q of rank d one has: dim(V P ∩ V Q ) ≤ tr(P Q) ≤ d. Where V P/Q are the images of P/Q. The lower (upper) bound is achieved when P and Q commute (coincide).
states mapped out of A by U † . This choice is somewhat arbitrary (See Prop. 1).
Algebraically, (2) measures how much the symmetries of the generalized quantum subsystem associated to A are dynamically broken by the channel U. Let us now start by further unveiling the geometrical nature of GAACs. First notice that, using the algebra superprojections, one can define a distance between two algebras A and B: D(A, B) := P A − P 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 A and its image U(A ). 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
where In the above proposition, all the (Hilbert-Schmidt) scalar products and norms are ordinary operators ones. Moreover, the Ω's operator can be expressed in the same way if the bases e α 's and f α 's are replaced by unitarily equivalent ones. The connection between Eqs (2) and (4) is given by Interestingly, the no-scrambling condition G A (U ) = 0 using Prop. 2 can be expressed by the operator fixedpoint equations U ⊗ 2 (Ω A ) = Ω 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) (a similar expression hold for the f α 's). This expression suggests how one could measure the GAAC by resorting to process tomography for U. Notice also that operational protocols to measure the GAAC were already discussed, for the cases 1) and 2) here below, in [16] and [17] 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 [13] and coherence generating power (CGP) [17,28], from a single vantage point. The first (second) concept is obtained when 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, Finally, the fifth illustrates in which sense even the concept of Loschmidt echo, a valuable tool in the study of quantum chaos [29][30][31][32], is comprised by the GAAC. This last connection is perhaps unsurprising as the Loschmidt echo is indeed a measure of autocorrelation of a dynamicaly evolving state which is precisely what 1 − G 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
In this case one finds that where Eq. (7) coincides exactly with the averaged OTOC discussed in [16] i.e., (here E denotes the Haar average over the unitary groups of A and A .). Remarkably, this quantity was shown to be equal to the operator entanglement [13,33] of the unitary U.
The latter concept has found important applications to a variety of quantum information-theoretic problems [14,[34][35][36][37]. More recently, it has been shown that operator entanglement requires exponentially scaled computational resources to simulate [38].
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 A B the algebra of operators which are diagonal with respect to an orthonormal basis B : This expression coincides with the coherence generating power (CGP) of U introduced in [17,39]. 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 Π i.e., [17,28]. The fact that the CGP is related to the distance between maximal abelian subalgebras was already established in [27]. CGP has been applied to the detection of the localization transitions in many-body systems [39], detection of quantum chaos in closed and open systems [40].
Here, L(H) s denotes the algebra of symmetric operators i.e., commuting with the swap S. One can readily , and Here J = ±1 is labeling the symmetric/antisymmetric representation of the permutation group generated by S.

5)
Let |ψ ∈ H and Π = |ψ ψ|. We define A LE = C{1, Π} i.e., the unital *-closed algebra generated by the projection Π. The commutant A LE is the algebra of operators leaving the subspace C|ψ and its orthogonal complement invariant. One has, where L := | ψ|U |ψ | is the Loschmidt echo. No- and that 2(1−L 2 ) = Π−U(Π) 2 2 , i.e., the distance between the algebras A LE and its image U(A LE ), as captured by the GAAC [see Eq. (3)], in high dimension is directly related to the Hilbert-Schmidt distance between the states Π and U(Π). From Eq. (11) one can see that the GAAC is a monotonic decreasing function of L for d > 4 and that L = 1 ⇒ G A LE (U ) = 0. For d = 2 one is back to 2). The case L = 0 corresponds to 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.

always achieved. iv) In the collinear case ii) and iii) above hold true with
The saturation condition P A UP A = T is quite transparent and intuitive: maximal scrambling is achieved when, from the point of view of the commutant, the dynamics generated by U is just full depolarization. Physical degrees of freedom supported in A 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) [16]. In the maximal abelian case 2) the bound 1 − d −1 is saturated by those U 's such that | i|U |j | = d −1/2 , (∀i, j) [17]. In case 3) the bound 1 2 is achieved for S, U(S) = 1, which amounts to the condition ii). On the other hand, in case 4) from Eq. (10) we see that S, (12) is not always achieved.

5) The bound 1/2 is achieved for d = 2 only (A is abelian). For d > 4 the maximun is for 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.
In ii) and iii) one can choose K ≥ 40. The idea is that the behavior of the infinite-time average 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 [16].  = 1, . . . , 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 (noninteracting) Hamiltonians.
The result above, which holds for any observable algebra A, has the very same structure of the corresponding one proved for the averaged bipartite OTOC (see Prop. 4 in [16]). The matrices R (α) , (α = 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 d = λn: λR lk (A ). In this case Eq. (14) can be written in way in which A and A appear symmetrically and the following upper bound holds: . (15) This bound is saturated iff P A (Π l ) = P A (Π l ) = 1 d , (∀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 A and A , completely randomized 4 . 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 [16]. 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 A [17]. In both these two important physical situations, the RHS of Eq. (15) is equal ; whereby, assuming that NRC holds, using iii) in Prop. (4) and the Markov inequality, one can bound temporal fluctuations: one sees e.g., by choosing = d −1/3 , that Hamiltonians achieving bound (15) have, in high dimension, highly suppressed temporal fluctuations below the value (12).
In [16] this concentration phenomenon has been numerically observed for the bi-partite case in chaotic 4 Indeed, for any observable P A (A(t) where A l := Tr(AΠ l ). Same holds for A . 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 [40]. 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 A-chaotic the dynamics generated by U t 's such that the (relative) difference between its infinite-time average and the Haaraverage of the GAAC is approaching zero sufficiently fast as the system dimension grows. More formally, 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 A-chaos with the simple Loschmidt case 5). Here one has This condition is known to be a sufficient one to bound time-fluctuations of the expectation value of observables with initial state |ψ [42]. Namely, A LE -chaos amounts to temporal-equilibration [41].
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.

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.