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Sample Essay: Quantum Decoherence and the Measurement Problem

Written by Richard H.

 

1. Introduction

Realistic quantum systems are not isolated from the environment but rather continuously interact with it (Schlosshauer, 2005). Treating this interaction within the traditional framework of quantum mechanics results in the concept of decoherence (Zurek, 2003). Decoherence results in the suppression of interference between the alternative states which has been experimentally observed. Furthermore, it provides insight on why the superposition resulting from measurement appears as a classical mixture (Bassi et al., 2013). Meanwhile, the quantum measurement problem remains a major conceptual question in quantum mechanics as the precise mechanics of the non-deterministic evolution following a measurement is unclear (Schlosshauer, 2005). However, some misconceptions may exist regarding decoherence and whether it fully resolves the measurement problem (Joos and Zeh, 1985; Dieks, 1989; Adler, 2003). The aim of the present paper is to review the concept of quantum decoherence and how it relates to the quantum measurement problem within the context of different interpretations of quantum mechanics.

2. Decoherence and the measurement problem

The section first reviews the problems of definite outcomes and preferred basis from the point of view of major interpretations of quantum mechanics. Next, the concept of decoherence is discussed as well as relevant implications of environmental interaction. Finally, it is discussed how decoherence fits within the framework of interpretations of quantum mechanics and whether it resolves the measurement problem.

2.1 Quantum measurement problem

A major property of the Schrödinger equation is its linearity (Schlosshauer, 2005). Given two solutions  and  of the equation, their linear combination  also constitutes a solution to the initial equation. While the possibility of a superposition of states is central to quantum mechanics, it seems to contradict our observations regarding the macroscopic world (Zurek, 2003). In particular, macroscopic objects are not observed in a superposition of pointer position states (Bassi et al., 2013). While the Schrödinger equation governs deterministic evolution of a system, performing a measurement is assumed to induce a probabilistic evolution with the outcomes being determined by the Born probability rule (Bassi et al., 2013).

The measurement problem can be illustrated as follows (Schlosshauer, 2005). Consider a microscopic system  represented by basis vectors  in a Hilbert space  interacting with a measurement apparatus  with its own basis vectors  in the corresponding Hilbert space . The basis vectors of the apparatus are assumed to represent pointer positions that are macroscopically distinguishable. Assume that the initial state of the microscopic system is a general superposition of its basis states  while the initial state of the apparatus is a specific ready state . The total system is represented by a state in the product space  which, due to the linearity of the Schrödinger equation, evolves into a superposition of system-apparatus states .

The example above illustrates the problems arising from examining the pre-measurement state (Zurek, 2003). Firstly, there is a problem of definite outcomes: we appear to perceive the measurement apparatus to be in a specific state  rather than a superposition of position states. Secondly, there is a problem of preferred basis which is the issue of non-uniqueness of the expansion of the final state of the combined system. Consequently, the measured observable is ill-defined. Typically, only the problem of definite outcomes is referred to as the quantum measurement problem (Schlosshauer, 2005). However, the problem of preferred basis is also important as it precludes from discussing outcomes due to set of possible outcomes not being uniquely defined.

The traditional Copenhagen interpretation posits that there is a divide between macroscopic and microscopic objects (Bohr, 1928). While the macro-world obeys the laws of classical mechanics, the dynamics of the micro-world is governed by the quantum theory (Zurek, 2006). An interaction between two worlds is described as a measurement which induces a ‘collapse’ of the micro-system from a superposition of states to one of the eigenstates of the measured observable (Zurek, 2003). However, the traditional theory does not specify the exact mechanism of the collapse and why it contradicts the linearity of the Schrödinger equation. Furthermore, the divide between microscopic and macroscopic objects is ill-defined as there appears to be no precise definition on the separation. As such, the Copenhagen interpretation does not address the measurement problem and provides no explanation on why no macroscopic superpositions are observed (Schlosshauer, 2005).

Alternative interpretations exist that aim to explain the Born rule. One of the most prominent interpretations is the many-worlds interpretation (Everett, 1957). According to this view, measurement-induced evolution is also described by the deterministic Schrödinger equation. Put differently, there is no non-deterministic evolution within this framework (Schlosshauer, 2005). The interpretation explains the realisation of only a single outcome by positing that the state continues to be in a superposition of alternatives. The alternatives correspond to the observer detecting the measurement apparatus and the quantum system in a specific state (Zurek, 2003). These alternatives are posited to exist in different branches of the Universe (Bassi et al., 2013). While this interpretation is logically consistent, some issues exist with incorporating the Born probability rule. Another interpretation of quantum mechanics, Bohmian mechanics, relies more strongly on classical mechanics and classical concepts of particles and trajectories (Bassi et al., 2013). The motion of particles in the physical space is directed by the wave function which is governed by the Schrödinger equation. The collapse following a measurement is effective and therefore experimentally verifying collapse models would falsify Bohmian mechanics (Schlosshauer, 2005).

2.2 Decoherence

Central to the phenomenon of decoherence is the idea that a quantum system cannot be isolated from the environment when a measurement occurs (Zurek, 2003). The interaction with a measuring apparatus during a measurement can be modelled using the same deterministic unitary evolution postulate that is used to describe the system’s dynamics without measurements (Zurek, 1996). The mathematical formulation of decoherence relies on the reduced density matrix description of a quantum system (Dieks, 1989). As the environment interacts with the measurement device in different ways, it is likely that there are many environmental degrees of freedom that remain unobserved (Zurek, 2003). At the same time, only the pointer state of the apparatus is eventually measured which could become correlated with these unobserved degrees of freedom (Zurek, 1996). The reduced density matrix approach allows for obtaining the state of the observed component of the system by taking the reduced trace of the density operator with respect to the unobserved environmental degrees of freedom (Dieks, 1989). A major consequence of this is that the reduced density matrix will be close to a classical statistical mixture of pointer states as long as the environmental states that are correlated with pointer position states are close to being orthogonal to each other.

The concept of decoherence can be illustrated as follows (Bassi et al., 2013). Consider a two-state system which is initially in a superposition of the two states . If the initial state of the measurement apparatus  and the initial state of the environment  are known, then the initial state of the whole system is the direct product of the initial states of its subsystems: . Interaction leads to the evolution of the initial state to a superposition of macroscopically distinguishable entangled states of the environment and the measurement apparatus: .  It can be shown (Bassi et al., 2013) that the process of decoherence leads to the direct product of the two entangled states to quickly vanish: . As such, the final state of the system is a classical statistical mixture of two states with relative weights corresponding to the initial relative weights of the two-state system .

At the same time, observations show that a measurement results in the system being in only one state and not in a superposition. The model above provides an explanation for the suppression of interference between the two alternative states, but it does not explain why we do not observe a superposition of states of a macroscopic object. This is a direct consequence of the linearity of the Schrödinger equation and the decoherence approach operating within this framework (Bassi et al., 2013). As such, the measurement process remains unexplained as quantum measurements are observed to induce a loss of superposition. Nevertheless, the decoherence perspective shows that the interaction of the measurement apparatus with the environment results in the quantum probability distributions that are similar to classical mixtures of alternatives.

Following the seminal paper of Zurek (1991), the topic of decoherence gained in popularity (Schlosshauer, 2007). Notably, the phenomenon has been observed experimentally. In general, empirical studies were in line with theoretical predictions and showed that decoherence is an extremely quick process which becomes stronger with the size of the system but does not require the presence of a large environment. Early experiments involved trapped ions and microwave cavities (Brune et al., 1996; Monroe et al., 1996), double-slit experiments with massive molecules (Arndt et al., 1999), and superpositions in Josephson junctions (Mooij et al., 1999). Later studies employed more complex systems such as superconducting quantum interference devices and Bose-Einstein condensates (Schlosshauer, 2007).

2.3 Implications of decoherence

Some misconceptions may arise regarding the role of decoherence in understanding the measurement problem. In particular, it has been proposed that decoherence fully resolves the problem (Adler, 2003). At the same time, it has been argued that is not possible to explain why only one dynamically independent component of the system is observed by invoking arguments based on unitary time evolution (Joos, 1999). The discussion above shows that decoherence leads to the reduced density matrix being close to a classical mixture of pointer states from the perspective of prediction. However, this does not imply that the system actually is in one of the reduced mixture states (Dieks, 1989). This corresponds to a conceptual problem of observed unique measurements contradicting the result of a full quantum mechanical treatment of the system, namely that the system still exists in a superposition of different states (Zurek, 1996). In other words, decoherence alone cannot solve the measurement problem in the Copenhagen interpretation. At the same time, decoherence implies the existence of environmentally-induced preferred pointer states of the apparatus (Schlosshauer, 2005). This may provide a link between the state evolved as a quantum system and the classical states that are experienced in the macro-world.

While decoherence does not resolve the measurement issue by itself, combining it with the many-worlds interpretation results in a consistent explanation of the phenomenon (Bassi et al., 2013). Indeed, decoherence describes the loss of interference between the alternatives. It can be posited that these decohered states continue to exist in different branches of the Universe (Zurek, 2003). The advantage of this approach is that it does not require modifying the existing quantum theory in order to explain measurement. However, this perspective does not improve our understanding of the nature of the Born rule as it is still required to be introduced as a postulate. From the perspective of Bohmian mechanics, decoherence cannot be used to show why specific pointer states are observationally relevant (Zurek, 2003). Nevertheless, the results of decoherence allow for describing why specific classical mixtures arise as they can be linked to environmentally-induced preferred states (Schlosshauer, 2005).

3. Conclusion

The aim of the present paper was to review the idea of quantum decoherence and how it may help resolve the quantum measurement problem within the context of major interpretations of quantum mechanics. The key motivation behind studying decoherence effects is that realistic quantum systems are unlikely to be fully isolated from the environment. The decoherence framework relies on the traditional formulation of quantum mechanics and treats the interaction between the measurement apparatus and the environment as a deterministic unitary evolution. This leads to the suppression of interference between the preferred states, resulting in a superposition that appears as a classical statistical mixture. As such, decoherence alone does not solve the measurement problem as it still requires invoking the Born rule to explain why the superposition is not observed in the Copenhagen interpretation. At the same time, decoherence can be combined with alternative interpretations such as many-worlds to explain the emergence of classical pointer states.

 

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