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Journal of Shaanxi Normal University

Factorizations of quantum effects

Chunyuan Deng

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

Some necessary and sufficient conditions are established respectively for an effect to be a sum of a sharp effect and an almost sharp effect,and to be a sum of two sharp effects.A necessary and sufficient condition is also given for an effect to be almost sharp.As applications,an effect can be represented as a sum of two almost sharp effects if the dimension of H is infinite or even and as a sum of at most three sharp effects if the dimension of H is odd.

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Some necessary and sufficient conditions are established respectively for an effect to be a sum of a sharp effect and an almost sharp effect,and to be a sum of two sharp effects.A necessary and sufficient condition is also given for an effect to be almost sharp.As applications,an effect can be represented as a sum of two almost sharp effects if the dimension of H is infinite or even and as a sum of at most three sharp effects if the dimension of H is odd.
Quantum effects are represented by operators on a Hilbert space satisfying 0⩽A⩽I, and sharp quantum effects are represented by projection operators. We say that an effect A is almost sharp if A=PQP for projections P and Q. We give simple characterizations of almost sharp effects. We also characterize effects that can be written as longer products of projections. For generality we first work in the formalism of von Neumann algebras. We then specialize to the full operator algebra B(H) and to finite dimensional Hilbert spaces.
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.
As is well known, every mixed or pure state of a bipartite quantum system is given by a statistical operator, which determines, in terms of its two reduced statistical operators, the subsystem states. Necessary and sufficient conditions for the existence of a composite-system state, and, separately, for the possibility of its being correlated or uncorrelated in terms of the range dimensions of the three mentioned statistical operators are derived. As a corollary, it is shown that it cannot happen that two of the mentioned dimensions are finite and the third is infinite.
The value of a multiqubit geometric phase at a single operating point does not reveal whether it was generated directly or through a connected sequence of lower-body interactions. For analytic, gapped, nondegenerate Abelian holonomies, we jointly resolve logical support and independently calibrated coupling support. Every nonzero connected response must involve an active set that connects and covers the target and must contain one factor of each active coupling, yielding the sharp onset bound νS≥τE(S)ν_S \geq τ_E(S)νS​≥τE​(S). The same selection law applies locally to projected Berry curvature. Conversely, an unrestricted finite-dimensional construction realizes all responses allowed by the connected-cover condition simultaneously, making the condition necessary and sufficient within this theorem class. Operationally, simultaneous confidence bands certify detected mixed responses while controlling false positives; a factor-of-two separation condition additionally recovers exact library-relative response minima. In a complete dictionary, nontriviality of every minimal-cover response is generically necessary and sufficient for these minima to coincide with the minimal calibrated mechanisms. Three-qubit Wilson calculations distinguish direct and pair-mediated routes with the same endpoint phase, and a synthetic Ramsey audit locates the finite-resolution boundary.