Материал: искусственный интеллект

156 S. Gogioso and C. M. Scandolo

Proof. Consider two objects (DD(H), dec) and (DD(K), dec), where and are special commutative †-Frobenius algebras associated with orthonormal bases (|ψx )x X and (|φy )y Y of H and K respectively. The morphisms from (DD(H), dec) to (DD(K), dec) in Split(DD(fHilb)) are exactly the maps of density hypercubes DD(H) → DD(K) in the following form:

¯

F

(40)

F

We can expand the definition of decoherence maps to see that these morphisms correspond to generic matrices Mxy of non-negative real numbers, with matrix composition as sequential composition, Kronecker product as tensor product, and the R+-linear structure of matrix addition.

=

(41) The discarding maps obtained by decoherence of the environment structure for DD(fHilb) yield the usual environment structure for classical systems:

Hence CK is equivalent to the probabilistic theory R+-Mat of classical systems.

Proposition 6. Let Split(DD(fHilb)) be the Karoubi envelope of DD(fHilb), and write Split(DD(fHilb))Q for the full subcategory of Split(DD(fHilb)) spanned

by objects in the form (DD(H), hypdec). There is an R+-linear monoidal equivalence of categories between Split(DD(fHilb))Q and the probabilistic theory CPM(fHilb) of quantum systems and CP maps between them. Furthermore, trace-preserving CP maps correspond to the maps in Split(DD(fHilb))Q nor-

malised with respect to the discarding maps (DD(H),hypdec ) := DD(H) ◦ hypdec , which we can write explicitly as follows:

Proof. We can define an essentially surjective, faithful monoidal functor from Split(DD(fHilb)) to the category CPM(fHilb) of quantum systems and CP maps

by setting (DD(H), hypdec) → H on objects and doing the following on morphisms:

In order to show monoidal equivalence we need to show that the functor is also full, i.e. that every CP map can be obtained from a map of Split(DD(fHilb)) in this way. Because of compact closure, it is actually enough to show that all states can be obtained this way. Consider a finite-dimensional Hilbert space H and a

(44) Hence the monoidal functor defined above is full, faithful and essentially surjective, i.e. an equivalence of categories. Furthermore, it is R+-linear and it respects discarding maps.

BPossibility of Extension for the Theory of Density Hypercubes

The theory of density hypercubes presented in this work is fully-fledged1 but incomplete: as shown by Eq. 18, the hyper-decoherence maps are not normalised (i.e. they are not “deterministic”, in the parlance of OPTs/GPTs)

1In the sense that it contains all the features necessary to consistently talk about operational scenarios, such as preparations, measurements, controlled transformations, reversible transformation, test, non-locality scenarios, etc.

158 S. Gogioso and C. M. Scandolo

When it comes to this work, however, this is not much of a problem: all we need to show is that an extension of our theory can exists in which the “tree-on-a- bridge” e ect above can be completed to the discarding map, and our results— both hyper-decoherence to quantum theory and higher-order interference—will automatically apply to any such extension.

Let (|ψx )x X be the orthonormal basis associated with the special commutative †-Frobenius algebra . The e ect needed to complete (DD(H),hypdec ) to the discarding map DD(H) is itself an e ect in CPM(fHilb), which can be written explicitly as follows:

Because it is an e ect in CPM(fHilb), which has R+ as its semiring of scalars, it is in particular non-negative on all states in DD(fHilb), showing that: (i) hyper-decoherence maps are sub-normalised; (ii) our theory does not satisfy the no-restriction condition; (iii) an extension to a theory with normalised hyperdecoherence is possible. This shows that our results on hyper-decoherence have physical significance. Furthermore, let |1 , ..., |d be an orthonormal basis of Cd, and let correspond to the Fourier basis for the finite abelian group Zd:

Choosing k := d, in particular, shows that the orthonormal basis above contains

the state 1 |ψ used in Sect. 4. Then the e ect defined in Eq. 45 also shows that

√d +

the computation of P[+|U ] in Sect. 4 can be done as part of a bonafide measurement in any such extended theory, and hence that our higher-order interference result has physical significance.

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