Density Hypercubes, Higher Order Interference and Hyper-decoherence: A Categorical Approach

Abstract

In this work, we use the recently introduced double-dilation construction by Zwart and Coecke to construct a new categorical probabilistic theory of density hypercubes. By considering multi-slit experiments, we show that the theory displays higher-order interference of order up to fourth. We also show that the theory possesses hyperdecoherence maps, which can be used to recover quantum theory in the Karoubi envelope.

Appendices

A Proofs

Proposition 4

The process theory ${\mathrm{DD}({\mathrm {fHilb}})}$ is causal, in the following sense: for every object ${\mathrm{DD}(H)}$, the only effect ${\mathrm{DD}(H)} \rightarrow \mathbb {R}^+$ in DD($\mathrm {fHilb}$) which yields the scalar 1 on all normalised states of ${\mathrm{DD}(H)}$ is the “forest” discarding map of density hypercubes .

Proof

Seen as an effect in $\mathrm {CPM({\mathrm {fHilb}})}$, any such effect must take the form of a sum $\sum _{x \in X} p_x \vert a_x \rangle \langle a_x \vert$, where $p_x \in \mathbb {R}^+$ and $(\vert a_x \rangle )_{x \in X}$ is an orthonormal basis for ${H} \otimes {H}$ which satisfies an additional condition due to the symmetry requirement for effects in DD($\mathrm {fHilb}$). If we write $\sigma _{{H},{H}}$ for the symmetry isomorphism ${H} \otimes {H} \rightarrow {H} \otimes {H}$ which swaps two copies of H in ${\mathrm {fHilb}}$, the additional condition on the orthonormal basis implies that for each $x \in X$ there is a unique $y \in X$ such that $\sigma _{{H},{H}} \vert a_x \rangle = e^{i\theta _{x}}\vert a_y \rangle$ and $p_x = p_y$; we define an involutive bijection $s: X \rightarrow X$ by setting s(x) to be that unique y. For each $x \in X$, consider the normalised state $\rho _x := \frac{1}{2}(\vert a_x \rangle \langle a_x \vert + \vert a_{s(x)} \rangle \langle a_{s(x)} \vert )$ in $\mathrm {CPM({\mathrm {fHilb}})}$, which we can realise in the sub-category DD($\mathrm {fHilb}$) by considering the classical structure on $\mathbb {C}^2$ corresponding to orthonormal basis $\vert 0 \rangle , \vert 1 \rangle$ and the vector $\vert r_x \rangle := \frac{1}{\root 4 \of {2}}(\vert a_x \rangle \otimes \vert 0 \rangle + \vert a_{s(x)} \rangle \otimes \vert 1 \rangle )$:

(38)

Now observe that the requirement that our effect yield 1 on all normalised states implies, in particular, that the following equation must hold:

(39)

As a consequence, our effect is written $\sum _{x \in X} \vert a_x \rangle \langle a_x \vert$, which is exactly the “forest” discarding map of density hypercubes on ${\mathrm{DD}(H)}$.

Proposition 5

Let $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ be the Karoubi envelope of DD($\mathrm {fHilb}$), and write $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}_K$ for the full subcategory of $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ spanned by objects in the form . There is an $\mathbb {R}^+$-linear monoidal equivalence of categories between $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}_K$ and the probabilistic theory $\mathbb {R}^+\text {-}\mathrm {Mat}$ of classical systems. Furthermore, classical stochastic maps correspond to the maps in $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}_K$ normalised with respect to the discarding maps defined as and which we can write explicitly as follows:

(16)

Proof

Consider two objects and , where and are special commutative $\dagger$-Frobenius algebras associated with orthonormal bases $(\vert \psi _x \rangle )_{x \in X}$ and $(\vert \phi _y \rangle )_{y \in Y}$ of H and K respectively. The morphisms from to in $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ are exactly the maps of density hypercubes ${\mathrm{DD}(H)} \rightarrow {\mathrm{DD}(K)}$ in the following form:

(40)

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

(41)

The discarding maps obtained by decoherence of the environment structure for DD($\mathrm {fHilb}$) yield the usual environment structure for classical systems:

(42)

Hence $\mathcal {C}_K$ is equivalent to the probabilistic theory $\mathbb {R}^+\text {-}\mathrm {Mat}$ of classical systems.

Proposition 6

Let $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ be the Karoubi envelope of DD($\mathrm {fHilb}$), and write $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}_Q$ for the full subcategory of $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ spanned by objects in the form . There is an $\mathbb {R}^+$-linear monoidal equivalence of categories between $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}_Q$ and the probabilistic theory ${\mathrm {CPM({\mathrm {fHilb}})}}$ of quantum systems and CP maps between them. Furthermore, trace-preserving CP maps correspond to the maps in $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}_Q$ normalised with respect to the discarding maps , which we can write explicitly as follows:

(18)

Proof

We can define an essentially surjective, faithful monoidal functor from $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ to the category $\mathrm {CPM({\mathrm {fHilb}})}$ of quantum systems and CP maps by setting on objects and doing the following on morphisms:

(43)

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 $\mathrm {Split({\mathrm{DD}({\mathrm {fHilb}})})}$ 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 classical structure on it, and write $(\vert \psi _x \rangle )_{x \in X}$ for the orthonormal basis of H associated to . The most generic mixed quantum state on $\mathcal {H}$ takes the form $\rho = \sum _{y \in Y} p_y \vert \gamma _y \rangle \langle \gamma _y \vert$, where $(\vert \gamma _y \rangle )_{y \in Y}$ is some orthonormal basis of H and $p_y \in \mathbb {R}^+$. Let be the classical structure associated with the orthonormal basis $(\vert \gamma _y \rangle )_{y \in Y}$, and define the states , where $\sqrt{\langle \psi _x \vert \gamma _y \rangle } \in \mathbb {C}$ is such that $\sqrt{\langle \psi _x \vert \gamma _y \rangle }^2 = \langle \psi _x \vert \gamma _y \rangle \in \mathbb {C}$. If we write , then the desired state $\rho$ can be obtained as follows:

(44)

Hence the monoidal functor defined above is full, faithful and essentially surjective, i.e. an equivalence of categories. Furthermore, it is $\mathbb {R}^+$-linear and it respects discarding maps.

B Possibility 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)

(18)

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” effect 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 $(\vert \psi _x \rangle )_{x \in X}$ be the orthonormal basis associated with the special commutative $\dagger$-Frobenius algebra . The effect needed to complete to the discarding map is itself an effect in ${\mathrm {CPM({\mathrm {fHilb}})}}$, which can be written explicitly as follows:

(45)

Because it is an effect in ${\mathrm {CPM({\mathrm {fHilb}})}}$, which has $\mathbb {R}^+$ as its semiring of scalars, it is in particular non-negative on all states in ${\mathrm{DD}({\mathrm {fHilb}})}$, 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 hyper-decoherence is possible. This shows that our results on hyper-decoherence have physical significance. Furthermore, let $\vert 1 \rangle ,...,\vert d \rangle$ be an orthonormal basis of $\mathbb {C}^d$, and let correspond to the Fourier basis for the finite abelian group $\mathbb {Z}_{d}$:

(46)

Choosing $k:=d$, in particular, shows that the orthonormal basis above contains the state $\frac{1}{\sqrt{d}}\vert \psi _+ \rangle$ used in Sect. 4. Then the effect defined in Eq. 45 also shows that the computation of $\mathbb {P}[+|U]$ 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.