We formulate a self-contained operator-theoretic framework for a dyadic system in which admissible states are represented on a bipartite Hilbert space H+⊗H−H+⊗H−. The theory is built from three axioms: (A1) a dyadic state space, (A2) a sector observable, and (A3) a zero-sum balance principle expressed as an exchange-odd constraint. From these assumptions we define a balance generator D=Q⊗I−I⊗QD=Q⊗I−I⊗Q, derive the entropy-maximizing state under the balance constraint, and show that the resulting Gibbs-type maximizer factorizes across the dyad. We then prove a sharp admissibility result: if both eλQeλQ and e−λQe−λQ are trace class for some λ>0λ>0, then the underlying Hilbert space must be finite-dimensional. Consequently, the simultaneous dual-trace condition is a finite-dimensional boundary condition rather than an infinite-dimensional asymptotic regime. The paper concludes with an application of the spectral theorem to admissible operators up to unitary equivalence and a discussion of how the framework can be weakened to admit genuinely infinite-dimensional models.
We formulate a self-contained operator-theoretic framework for a dyadic system in which admissible states are represented on a bipartite Hilbert space H+⊗H−H+⊗H−. The theory is built from three axioms: (A1) a dyadic state space, (A2) a sector observable, and (A3) a zero-sum balance principle expressed as an exchange-odd constraint. From these assumptions we define a balance generator D=Q⊗I−I⊗QD=Q⊗I−I⊗Q, derive the entropy-maximizing state under the balance constraint, and show that the resulting Gibbs-type maximizer factorizes across the dyad. We then prove a sharp admissibility result: if both eλQeλQ and e−λQe−λQ are trace class for some λ>0λ>0, then the underlying Hilbert space must be finite-dimensional. Consequently, the simultaneous dual-trace condition is a finite-dimensional boundary condition rather than an infinite-dimensional asymptotic regime. The paper concludes with an application of the spectral theorem to admissible operators up to unitary equivalence and a discussion of how the framework can be weakened to admit genuinely infinite-dimensional models.
We establish a sharp quantitative stability theorem for the von Neumann entropy under a natural class of convex constraints induced by orthogonal block decompositions of finite-dimensional Hilbert spaces. Given a block–convex set of block-diagonal quantum states, we prove that any state whose entropy is within ε of the minimal value must lie within O(ε)O(\sqrt{\varepsilon})O(ε) in trace norm of the manifold of entropy minimizers. The stability exponent is shown to be optimal, and purely quadratic stability is characterized precisely in the confining (fixed-support) regime. The proof relies on a canonical decomposition of entropy into classical (Shannon) and conditional components, combined with sharp relative entropy estimates and geometric properties of the marginal polytope. As an application, we obtain quantitative non-concentration bounds for spectral measures associated with finite spectral decompositions of self-adjoint operators. The framework is entirely finite-dimensional, independent of arithmetic or dynamical assumptions, and provides a general geometric stability principle for entropy minimization under structured convex constraints.
We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing uniform exponential convergence guarantees for the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models - including Schrödinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians - and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation.
Quantum information decoupling is a pillar of quantum information theory, underlying quantum communication, error correction, and information recovery. However, its existing formulations rely on random unitary operations tied to finite-dimensional assumptions on quantum systems. Here we establish a decoupling framework for arbitrary separable, possibly infinite-dimensional systems. For finite-dimensional inputs and arbitrary separable reference/output systems, we derive error bounds on one-shot decoupling for completely positive maps in terms of sandwiched Rényi conditional entropies. For partial-trace decoupling with finite-dimensional references, the error exponent is optimal up to the known critical rate. To handle infinite-dimensional inputs, we assume finite entropy of the manipulated system. We construct finite-rank projections on independent and identically distributed (IID) states to restrict randomization to a projected finite-dimensional subspace, while ensuring high success probability by including auxiliary components in the discarded subsystem at asymptotically negligible cost. This leads to an infinite-dimensional IID partial-trace decoupling protocol achieving optimal asymptotic dimension rates. As an application, we construct an infinite-dimensional quantum-state-merging protocol, a mother protocol of quantum information theory. Under finite entropy of Alice's marginal, it achieves the same quantum-communication and total-cost rates as in finite dimensions, governed by mutual information and conditional entropy. These results show that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions. More broadly, our framework provides foundational tools for exploring quantum information regardless of whether we model the physical world using finite- or infinite-dimensional spaces.
We present a self-contained axiomatic framework in which a fundamental entity is defined not by intrinsic materiality but as a finite, quantized information node governed by an internal duality. The central postulate—the Axiom of Operational Parity—asserts that an individual entity is mathematically identical to an interacting pair of itself and its conjugate: S≡S×SˉS≡S×Sˉ. Working in a finite-dimensional Hilbert space H=CdH=Cd with d≥3d≥3, we introduce a strictly positive quantization operator QQ encoding geometric bounds, an antiunitary conjugation CC, and a command vector ∣v⟩∣v⟩ satisfying two orthogonality conditions. From these we construct a pair of positive rank-one entity operators (S,Sˉ)(S,Sˉ) with equal trace ττ and orthogonal support. The entity's global state is postulated to be the maximally entangled Bell state of the active and conjugate sectors. Temporal persistence is governed by a dissipative binary recurrence—a deterministic discrete-time linear evolution coupling S(n)S(n) and Sˉ(n)Sˉ(n). We prove that positivity is preserved if and only if the coupling parameters satisfy ∣A−B∣≤1−γ∣A−B∣≤1−γ, and that the trace decays as Tr(S(n))=τ(1−γ)nTr(S(n))=τ(1−γ)n. An intrinsic collapse criterion based on trace threshold is introduced. The main result, the Finite Lifetime Theorem, establishes that every entity returns to the zero state in a finite number of steps, with the explicit formula Nmax=⌈logα/log(1−γ)⌉Nmax=⌈logα/log(1−γ)⌉, depending only on the dissipation constant γγ and the collapse threshold αα. We provide a categorical framing via dagger compact closed categories, distinguish rigorously between proved results and postulates, and identify open problems including the dimensional obstruction to realizing the type axiom in finite-dimensional operator models.
We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite dimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighbourhood of every product state lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still trace-norm dense in state space, even under the constraint of a finite mean energy.
Traditional formulations of quantum dynamics emphasize bulk evolution governed by unbounded differential operators. In this work, we introduce a unified operator-theoretic framework demonstrating that physical interaction, measurement, and macroscopic reality are fundamentally boundary-resolved phenomena. We define "boundary mechanics" as the spectral constraint structure governing admissible interactions across an interface, and "boundary dynamics" as the time evolution of interface-localized couplings. By embedding these concepts into bipartite open quantum systems (System-Bath architectures), we prove that classical macroscopic events emerge strictly as boundary-induced decoherence projections (einselection). Furthermore, we establish that dynamical coherence is preserved subject to a strict spectral admissibility condition, providing a necessary and physically robust sufficient criterion characterized by the non-vanishing of the regularized Birman-Schwinger determinant: D_boundary(z) = det2(I + K(z)) != 0. This work goes beyond standard decoherence theory by demonstrating that decoherence alone is insufficient as a criterion for classicality; it must be supplemented by a spectral stability condition at the interface. This yields a closed formalism in which macroscopic observable dynamics are governed entirely by spectrally admissible interface interactions, rigorously bridging Zurek's decoherence theory with the Kato-Rellich theorem and Birman-Schwinger perturbation theory.