[Paper Review] Discrete Quantum Processes

Stan Gudder|arXiv (Cornell University)|May 31, 2011

Quantum Mechanics and Applications9 references3 citations

TL;DR

This paper introduces discrete quantum processes as sequences of local quantum states on an L² Hilbert space, showing how they generate a quantum measure on cylinder sets and can be systematically extended to a broader class of physically relevant sets via a 'quadratic algebra'. The key contribution is a rigorous extension of quantum measures to suitable sets beyond cylinder sets, enabling the definition of a quantum integral through operator quantization of random variables.

ABSTRACT

A discrete quantum process is defined as a sequence of local states $ρ_t$, $t=0,1,2,...$, satisfying certain conditions on an $L_2$ Hilbert space $H$. If $ρ=\limρ_t$ exists, then $ρ$ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure $μ$ on the algebra of cylinder sets $\cscript$. We consider the problem of extending $μ$ to other physically relevant sets in a systematic way. To this end we show that $μ$ can be properly extended to a quantum measure $\mutilde$ on a "quadratic algebra" containing $\cscript$. We also show that a random variable $f$ can be "quantized" to form a self-adjoint operator $\fhat$ on $H$. We then employ $\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are given

Motivation & Objective

• To formalize discrete quantum processes as sequences of local states on an L² Hilbert space, particularly when a global state does not exist.
• To address the challenge of extending a quantum measure μ defined on cylinder sets to a broader class of physically meaningful sets.
• To develop a systematic extension of the quantum measure μ to a 'quadratic algebra' S that properly contains the algebra of cylinder sets C.
• To define a quantum integral ∫fdμ̃ via the quantization of random variables into self-adjoint operators, enabling computation of quantum expectations.

Proposed method

• Define a discrete quantum process as a sequence of local states ρₜ on H = L²(Ω, A, ν), where Ω represents paths or histories.
• Construct a quantum measure μ on the algebra of cylinder sets C via μ(A) = lim ⟨ρₜχ_A, χ_A⟩, using the decoherence functional Dρ(A,B) = ⟨ρχ_B, χ_A⟩.
• Introduce the class of 'suitable' sets S ⊇ C for which lim ⟨ρₜχ_A, χ_A⟩ exists and is finite, defining a new measure μ̃(A) = lim ⟨ρₜχ_A, χ_A⟩ on S.
• Show that S forms a 'quadratic algebra' and μ̃ is a valid quantum measure on S, extending μ.
• Quantize a real-valued random variable f by defining a self-adjoint operator f̂ on H, using the spectral structure of characteristic functions.
• Define the quantum integral as ∫fdμ̃ = tr(ρf̂), and compute it via eigenvalue decomposition of f̂ for simple functions, extending to general functions by limit processes.

Experimental results

Research questions

• RQ1 Can a quantum measure μ defined on cylinder sets be systematically extended to a larger class of physically relevant sets when a global state does not exist?
• RQ2 What algebraic structure underlies the collection of sets for which the limit lim ⟨ρₜχ_A, χ_A⟩ exists and defines a consistent quantum measure?
• RQ3 How can the quantum integral ∫fdμ̃ be defined and computed when f is a random variable and μ̃ is an extended quantum measure?
• RQ4 What is the role of the decoherence functional and its associated operator in constructing the local states ρₜ and the resulting quantum measure?
• RQ5 How can the quantization of a random variable f into a self-adjoint operator f̂ be used to compute the quantum integral via spectral decomposition?

Key findings

• The collection S of suitable sets forms a 'quadratic algebra' that properly contains the algebra of cylinder sets C, enabling a systematic extension of the quantum measure.
• The extended measure μ̃(A) = lim ⟨ρₜχ_A, χ_A⟩ defines a valid quantum measure on S, which generalizes the original measure μ on C.
• For a two-site quantum random walk, the event 'the particle visits the origin' and its complement are in S ∖ C, showing that S captures physically meaningful events beyond cylinder sets.
• The quantum integral ∫fdμ̃ is defined as tr(ρf̂), and for simple functions, it can be computed via eigenvalue decomposition of the quantized operator f̂.
• For a two-valued random variable f = αχ_A + βχ_B, the eigenvalues of f̂ are found explicitly using a quadratic equation, enabling direct computation of the quantum integral.
• The method generalizes to n-valued functions via inclusion-exclusion over pairwise quantizations, allowing the quantum integral to be computed as a finite sum involving eigenvalues and state overlaps.