We introduce an elementary measure of non-commutativity between two algebras of quantum operators acting on the same Hilbert space. This quantity, which we call Mutual Averaged Non-commutativity (MAN), is a simple generalization of a type of averaged Out-of-Time-Order-Correlators used in the study of quantum scrambling and chaos. MAN is defined by a Haar averaged squared norm of a commutator and for some types of algebras is manifestly of entropic nature. In particular, when the two algebras coincide the corresponding self-MAN can be fully computed in terms of the structural data of the associated Hilbert space decomposition. Properties and bounds of MAN are established in general and several concrete examples are discussed. Remarkably, for an important class of algebras, —which includes factors and maximal Abelian ones—MAN can be expressed in the terms of the algebras projections CP-maps. Assuming that the latter can be enacted as physical processes, one can devise operational protocols to directly estimate the MAN of a pair of algebras.

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21.

Importantly, A and B are not supposed to be associated with separate spatial locations.

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Here, if X is an algebra U(X){U(X)/XX}.

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This can also be immediately seen from the fact that S(AB:AB̃)=0 iff ABAB̃=AB̃ and since, by symmetry of S, even the opposite inclusion holds the AB̃=AB identity follows.

25.

See  Appendix F.

26.

The following facts hold true: (i) d(AS)=(d2)|S|. (ii) S1S2AS1AS2. (iii) AS=ASc. (iv) AS1AS2=AS1S2. (v) AS1AS2=AS1S2. (vi) PAS1PAS2=PAS2PAS1=PAS1S2.

27.

This is true for any A with abelian commutant i.e., nJ = 1 ∀J.

28.

Note that here dimH=d2 thus (26) reads 11/d4.

29.

ω(Z(A))=(PZ(A)1)|Φ+Φ+|=(PA1)(PA1)|Φ+Φ+|=(PA1)ω(A).

30.

For example, if f(ϕ)PA(ϕ̂)22=TrSPA2(ϕ̂2) then, using standard operator norm inequalities and the fact that the algebra projections are CP-maps, |f(ϕ1)f(ϕ2)|PA2(ϕ̂12ϕ̂22)1ϕ̂12ϕ̂221K|ϕ1|ϕ2 where K = O(1).

31.

In  Appendix L it is shown how MAN, in general, can be connected to average entropies of maps associated to A and B.

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