Observable Algebras and Channels

M. D. Maia

Salimah H Meghani

This paper proves a finite-dimensional collapse-and-obstruction theorem for the relation between Heisenberg observable evolution and Schrödinger state evolution. The starting datum is a completion-locked boundary-channel datum: a boundary Hilbert space, a locked boundary observable algebra, a Heisenberg comparison on boundary observables, and an admissible state-side channel. The basic duality law is the trace pairing Tr[X Φ(ρ)] = Tr[Φ*(X) ρ], restricted, when appropriate, to a locked boundary algebra A∂. The theorem proves that this pairing undergoes Heisenberg–Schrödinger collapse exactly in the full-algebra reversible case: A∂ = B(H∂) and the Heisenberg dual is a ∗-automorphism, equivalently the state-side channel is unitary conjugation. Outside this reversible full-algebra case, the same expectation pairing is a boundary-channel law rather than a unitary full-algebra equivalence. Three rigorous obstructions certify noncollapse relative to the locked boundary datum: a proper boundary algebra is state- and channel-compressive; a nonmultiplicative Heisenberg dual cannot be implemented by unitary conjugation on the full algebra; and a many-to-one bulk/history realization can induce identical boundary laws from inequivalent channels or histories. Thus finite-dimensional closed-system unitary quantum mechanics is identified as the special case in which the boundary algebra is the whole kinematic algebra and the observable-state interface has no quotient, no proper boundary restriction, and no channel-valued loss of reversibility. The theorem has immediate consequences for measurement theory. Nonselective projective measurement is a boundary-channel operation: it fixes the commutative measured algebra while removing off-boundary coherences. Selective measurement is conditioning on a boundary record, not state-uniform retuning of the unconditioned boundary law. Bipartite no-signaling is likewise a fixed-boundary theorem: a remote completely positive trace-preserving channel fixes the local boundary algebra in the Heisenberg picture. Random-unitary symmetry-selection channels and finite rigidity-gap quotient selections supply two realization mechanisms. Their conserved observables are determined by the active commutant, while their low-temperature or equilibrium limits are controlled by explicit channel-norm bounds. License note: Distributed under CC BY-NC-ND 4.0.

Alexei Kitaev

Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$-idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We prove that any finite-dimensional $\varepsilon$-$C^*$ algebra $A$ is $O(\varepsilon)$-isomorphic to some genuine $C^*$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $η$-idempotent UCP map $Φ$, the $O(η)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$, and an encoding channel.

Stanley Gudder

If $H_1$ and $H_2$ are finite-dimensional Hilbert spaces, a channel from $H_1$ to $H_2$ is a completely positive, linear map $\mathcal{I}$ that takes the set of states $\mathcal{S}(H_1)$ for $H_1$ to the set of states $\mathcal{S}(H_2)$ for $H_2$. Corresponding to $\mathcal{I}$ there is a unique dual map $\mathcal{I}^*$ from the set of effects $\mathcal{E}(H_2)$ for $H_2$ to the set of effects $\mathcal{E}(H_1)$ for $H_1$. We call $\mathcal{I}^*(b)$ the effect $b$ conditioned by $\mathcal{I}$ and the set $\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))$ the conditioned set of $\mathcal{I}$. We point out that $\mathcal{I}^c$ is a convex subeffect algebra of the effect algebra $\mathcal{E}(H_1)$. We extend this definition to the conditioning $\mathcal{I}^*(B)$ for an observable $B$ on $H_2$ and say that an observable $A$ is in $\mathcal{I}^c$ if $A=\mathcal{I}^*(B)$ for some observable $B$. We show that $\mathcal{I}^c$ is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.