Qbism and Many Worlds, with ChatGPT

The conversation has been condensed and edited for clarity.

User: Can quantum mechanics be derived by generalizing probability theory to complex numbers?

Assistant: In a sense, yes. Quantum mechanics can be viewed as a generalization of classical probability theory into the complex domain. In classical probability, probabilities are real numbers between 0 and 1 that sum to 1. Quantum mechanics extends this by introducing complex probability amplitudes. The state of a quantum system is described by a wavefunction, which assigns a complex amplitude to each possible outcome. The probability of an outcome is given by the square of the magnitude of its amplitude, according to Born's rule.

This use of complex numbers allows for phenomena like superposition and interference, where amplitudes can add constructively or destructively—behaviors not possible with real probabilities. Thus, quantum mechanics enriches probability theory by incorporating complex numbers, providing a framework that accounts for uniquely quantum effects.

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User: Are there interpretations of quantum mechanics that use this connection?

Assistant: Yes, several interpretations emphasize this connection. One prominent example is Quantum Bayesianism, or QBism. QBism interprets quantum states as expressions of an observer's personal beliefs or probabilities about measurement outcomes, updated using quantum rules. The complex structure of quantum mechanics reflects how these beliefs are updated, rather than depicting an objective reality.

Another approach is the information-theoretic interpretation, which views quantum mechanics fundamentally as a theory about information and its processing. In this perspective, the complex amplitudes represent informational content, and quantum mechanics generalizes classical probability by allowing operations that do not commute, represented mathematically by operators in Hilbert space.

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User: How does QBism make sense of quantum computers and the speedups they provide for specific problems?

Assistant: QBism interprets quantum computation as a powerful method for an observer to update and manage their probabilistic beliefs using the rules of quantum mechanics. Quantum computers exploit superposition and interference to process information in ways that are inaccessible to classical systems.

In QBism:

1. Efficient Information Encoding: Quantum bits (qubits) allow an observer to represent and manipulate complex probability amplitudes, encoding multiple possibilities simultaneously through superposition and entanglement.

2. Interference of Amplitudes: Quantum algorithms use interference to enhance the probabilities of correct outcomes and diminish those of incorrect ones. This is seen as manipulating probability amplitudes to refine beliefs about potential solutions efficiently.

3. Exploiting Quantum Structures: The speedups arise from utilizing the mathematical structure of quantum mechanics, which permits operations on probability amplitudes that have no classical counterpart.

While quantum states are subjective beliefs in QBism, they are updated according to the objective mathematical rules of quantum mechanics. The physical realization of quantum computers leverages these rules to perform computations that, for certain problems, are exponentially faster than classical methods.

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User: But if the benefits of quantum computers come purely from this mathematical generalization, wouldn't classical computers be able to simulate quantum computations efficiently? That doesn't seem right.

Assistant: You're correct; the physical properties of quantum systems are essential. While the mathematical framework involves complex probabilities, it's the unique physical phenomena of quantum mechanics—such as superposition, entanglement, and non-locality—that enable quantum computers to outperform classical ones.

In QBism, although quantum states represent subjective beliefs, they are constrained by the objective structure and laws of quantum mechanics. The physical implementation of quantum computers exploits these quantum effects, which cannot be efficiently simulated by classical computers due to the exponential resources required. Therefore, the computational advantages stem from both the mathematical generalization and the inherent physical properties of quantum systems.

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User: It still seems like QBism doesn't fully account for the computational advantages of quantum computers. How does QBism relate to the Many-Worlds Interpretation (MWI)? Are they compatible or fundamentally different?

Assistant: QBism and the Many-Worlds Interpretation are fundamentally different and largely incompatible.

• Ontology of the Quantum State:

  • MWI views the quantum state as a real, objective entity that describes all possible realities simultaneously. The wavefunction evolves deterministically, and all possible outcomes of quantum events actually occur in a vast multiverse of branching worlds.

  • QBism treats the quantum state as a subjective expression of an observer's personal beliefs about measurement outcomes. The wavefunction is not an objective entity but a tool for updating beliefs upon obtaining new information.

• Nature of Measurement:

  • In MWI, measurement causes the universe to branch into multiple, equally real worlds, each representing a different outcome.

  • In QBism, measurement results in a single, definite outcome for the observer, leading to an update of their beliefs without invoking any branching of realities.

Because QBism focuses on a single, observer-dependent reality and rejects the notion of objective wavefunction collapse or branching, it is fundamentally at odds with the ontological commitments of MWI. The two interpretations offer conflicting views on the nature of reality and the role of the observer, making them incompatible.

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User: I still don't see how QBism makes sense of quantum computers. If quantum computers perform computations that can't be efficiently simulated classically without considering all possible states simultaneously, doesn't that suggest that all these states must "exist" in some sense? That seems to support MWI.

Assistant: This is a thoughtful concern. Quantum computers do perform operations that, if simulated classically, would require tracking an exponential number of states, which seems to imply that all these states have some form of existence.

In QBism, however, the emphasis is on the observer's use of quantum mechanics to update their beliefs about measurement outcomes. The efficiency of quantum computation is attributed to the unique way quantum mechanics allows for the manipulation of probability amplitudes through superposition and interference within a single, indivisible reality.

While this may appear less intuitive, QBism argues that the quantum computer doesn't need to "exist" in all possible states simultaneously in an ontological sense. Instead, it leverages the mathematical structure of quantum mechanics to achieve computational advantages without requiring the existence of multiple, parallel realities. The superposition and interference are seen as tools for updating beliefs about outcomes, not as evidence of multiple existing states.

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User: Suppose we define "existence" as any process that's perfectly emulated in the universe. If my computer runs a simulation, that simulation exists in the same way I do. If quantum mechanics holds universally without collapse, and arbitrarily large quantum computers are possible, doesn't that mean all possible computational paths "exist"? Doesn't this make MWI unavoidable?

Assistant: Under that definition of existence, combined with the assumption that quantum mechanics applies universally without collapse, it does lead to the conclusion that all possible computational paths have a form of existence. Each possible outcome of a quantum computation would be as real as any process simulated on your computer.

This perspective aligns closely with the Many-Worlds Interpretation. In MWI, the universal wavefunction encompasses all possible states, and every possible outcome of a quantum event exists in its own branch of the multiverse. If we accept that every computational path in a quantum computer is perfectly emulated and thus exists, then MWI becomes a compelling and perhaps unavoidable interpretation.

Therefore, given your definitions and assumptions, MWI appears to be inescapable because it straightforwardly accounts for the existence of all possible computational outcomes within the framework of quantum mechanics.