Introduction: The Persistent Debate in Quantum Mechanics

Quantum mechanics (Q.M.) is a field brimming with profound and often perplexing phenomena, one of which is entanglement. The concept of entanglement has long sparked debates about the nature of physical determinism and the limitations of classical physics. This article delves into a specific scenario where a separable mixed state consisting of 99 entangled systems in the state 01, 10, and a single unentangled system in state 01, 00 poses no violation to Bell inequalities. We will explore how this mixed state aligns with the deterministic principles of quantum physics and why it does not violate Bell's inequalities.

Understanding Entanglement and Determinism

The term entanglement in quantum mechanics is often misconstrued. In reality, entanglement is more a manifestation of deterministic principles rather than a violation of them. As per classical mechanics, the state of an isolated system at a given moment indeed determines its state at every moment throughout its timespan, past, present, and future—a principle definitively championed by JosephRichardson Boscovich in 1758 and developed further by Pierre-Simon Laplace in 1814. In classical mechanics, determining a system's state is constrained by the number of independent variables known at a given moment.

Emergence of Quantum Mechanics

With the advent of quantum mechanics, the landscape shifted dramatically. For systems carrying (10^{23}) degrees of freedom, statistical mechanical methods are the only practical tools for modeling macroscopic thermodynamic variables. A small system isolated long enough to determine its state cannot retain that state indefinitely; it transitions into a quantum mechanical regime characterized by imprecise determinations of its variables. Despite this, quantum states have been sufficiently determined to extract information about remote parts of the system that could not be communicated even at the speed of light. This phenomenon is what we term entanglement.

The Role of Bell Inequalities and Quantum Mechanics

Bell inequalities are crucial in quantum mechanics for testing the predictions of quantum theory against local hidden variable theories. These inequalities name involves the measurement outcomes of entangled particles, asserting that no local hidden variable theory can reproduce all the results observed in quantum mechanics.

Why a Mixed State Does Not Violate Bell Inequalities

In the scenario where a mixed state consists of 99 entangled systems in states 01 and 10, and a single unentangled system in state 01, 00, the system does not violate Bell inequalities because it aligns with the determinism principles of quantum mechanics. Each of the 99 entangled systems evolves deterministically in a manner described by quantum mechanics, and so does the single unentangled system. This coherence across the entire system ensures that the final state respects the constraints imposed by Bell's inequalities.

Analyzing the Mixed State

Let's delve deeper into the structure of this mixed state. The 99 entangled systems in states 01 and 10 can be described as a completely entangled bipartite state, meaning their states are correlated in a specific pattern that cannot be explained by classical physics. The single unentangled system, in contrast, remains in a clear and straightforward state 01, 00. This setup ensures that the overall system does not violate Bell inequalities.

Deterministic Interpretation

From a deterministic perspective, each system in the mixed state follows a precise and predictable path determined by quantum mechanical principles. The entangled and unentangled states coexist without contradiction because quantum mechanical determinism encompasses both types of states. The unentangled system, while clearly defined, does not disrupt the overall coherence of the system's state, which remains consistent with quantum predictions.

Conclusion and Future Perspectives

Understanding why a mixed state of 99 entangled systems and one unentangled system does not violate Bell inequalities requires a deep dive into the principles of quantum mechanics. It is crucial to recognize that entanglement itself is a manifestation of determinism, albeit a non-local one, and that the predictability of quantum states aligns with the deterministic principles of classical mechanics. Future research in this area could further illuminate the nuanced relationship between entanglement, determinism, and the limits of classical physics.