Can a quantum state over time resemble a quantum state at a single time?

Area of Science:

• Quantum Physics
• Foundations of Quantum Mechanics
• Quantum Information Theory

Background:

• Standard quantum theory treats space and time asymmetrically.
• A joint state for composite systems at different times is not prescribed.
• Defining such a state could enhance the quantum-classical probability analogy.

Purpose of the Study:

• To investigate proposals for a quantum joint state over time.
• To establish criteria for a successful quantum joint state formalism.
• To determine if existing proposals meet these criteria.

Main Methods:

• Analysis of operator-based joint states over time.
• Evaluation of proposals by Leifer and Spekkens, Fitzsimons, Jones, and Vedral.
• Assessment against five defined criteria for a quantum joint state.

Main Results:

• No existing proposal satisfies all five criteria for a quantum joint state over time.
• The criteria include Hermiticity, probabilistic mixing, classical limit, single-time marginals, and associativity.
• Dropping the Hermiticity criterion leads to an essentially unique construction.

Conclusions:

• A universally applicable quantum joint state over time, analogous to composite systems at a single time, is not achievable under current frameworks.
• The requirement of Hermiticity appears to be a key obstacle.
• A modified formalism without Hermiticity offers a viable path forward for a more unified space-time treatment in quantum mechanics.