The Decoherent Arrow of Time and the Entanglement Past Hypothesis (2405.03418v1)

Abstract: If an asymmetry in time does not arise from the fundamental dynamical laws of physics, it may be found in special boundary conditions. The argument normally goes that since thermodynamic entropy in the past is lower than in the future according to the Second Law of Thermodynamics, then tracing this back to the time around the Big Bang means the universe must have started off in a state of very low thermodynamic entropy: the Thermodynamic Past Hypothesis. In this paper, we consider another boundary condition that plays a similar role, but for the decoherent arrow of time, i.e. the quantum state of the universe is more mixed in the future than in the past. According to what we call the Entanglement Past Hypothesis, the initial quantum state of the universe had very low entanglement entropy. We clarify the content of the Entanglement Past Hypothesis, compare it with the Thermodynamic Past Hypothesis, and identify some challenges and open questions for future research.

• The paper demonstrates that the Entanglement Past Hypothesis, by positing an initially low-entanglement state, underlies the emergence of an irreversible time arrow through decoherence.
• The paper contrasts different entropy measures, emphasizing entanglement entropy over von Neumann entropy to capture the growth of correlations in quantum systems.
• The paper examines how various Hilbert space factorizations impact entanglement dynamics, raising fundamental questions in quantum cosmology and the nature of space-time.

Overview of "The Decoherent Arrow of Time and the Entanglement Past Hypothesis"

The paper "The Decoherent Arrow of Time and the Entanglement Past Hypothesis" by Jim Al-Khalili and Eddy Keming Chen explores the symmetry and asymmetry in time with respect to decoherence and quantum entanglement. It introduces the notion of the Entanglement Past Hypothesis (EPH) as an initial boundary condition necessary to explain the decoherent arrow of time—a concept parallel to the Thermodynamic Past Hypothesis (TPH) in thermodynamics.

The authors begin by addressing the process of decoherence in quantum mechanics, which accounts for the emergence of classical order from quantum states, and they emphasize the asymmetry in time that arises with decoherence. They assert that while the combined system-environment dynamics are unitary and time-reversal invariant, the observable increase in entanglement between a system and its environment defines the irreversible arrow of time that decoherence manifests.

Key Concepts and Framework

The authors contrast several notions of entropy, emphasizing entanglement entropy as the relevant measure for understanding the decoherent arrow of time. This focus is motivated by the entanglement entropy's potential to reflect an increase in entropy even in closed quantum systems. Unlike the von Neumann entropy, which remains constant under unitary evolution, entanglement entropy can capture the growth of correlations between subsystems.

To formalize EPH, the authors outline multiple approaches:

1. Standard Formulation: EPH asserts that the universe began in a state of low entanglement at the initial temporal boundary. This formulation raises questions about the specific factorization of the Hilbert space of the universe into subsystems, a critical aspect in defining entanglement entropy.
2. Partition-Independent Formulations: To mitigate arbitrariness, the paper proposes versions of EPH that remain valid across various factorizations within certain constraints. These include ensuring that entanglement entropy is either zero or below a small threshold for a class of partitions.

Theoretical Implications and Challenges

The authors note the profound implications of adopting an EPH. Doing so offers a novel perspective on the initial conditions of the universe and the emergent nature of space-time, which may arise from quantum entanglement on cosmological scales. They argue for EPH as a potentially fundamental law, expanding on its role in ensuring consistency in the observations of decoherence and its persistence over time.

The dependency of EPH on the factorization of Hilbert space points to foundational questions in quantum theory, specifically regarding the nature of subsystems and system-environment separation. These issues resonate with broader questions about how to reconcile such dependencies with the fundamentally relational aspects of entanglement, especially when space-time itself may be emergent from those quantum descriptions.

Speculative Directions and Future Research

In addressing the connection between EPH and TPH, the authors speculate about whether these hypotheses might either derive from one another or share a common cause. They explore how cosmological entropy considerations and universal expansion might relate to entanglement dynamics, suggesting avenues for further exploration in quantum cosmology.

Moreover, the paper touches on the possibility that the emergence of space-time itself might be intimately linked with the patterns of entanglement growth from an initially unentangled state, a hypothesis aligning with some contemporary theories suggesting space-time as a macro-level manifestation of underlying quantum entanglement.

Conclusion

In conclusion, Al-Khalili and Chen's paper offers a rigorous examination of the decoherent arrow of time through the lens of entanglement entropy and postulates the Entanglement Past Hypothesis as a critical initial condition. This work opens up significant theoretical and practical implications, bridging ideas in quantum mechanics, thermodynamics, and the fabric of space-time. Future research along these lines can deepen our understanding of the universe's beginnings and the foundational nature of time's arrow.