Question

In Quantum Computing in Hidden Variables, Scott Aaronson characterizes a hidden variable theory as a framework that transforms a unitary matrix mapping [α1…αn]^T↦[β1…βn]^T to a stochastic matrix mapping [|α1|2…|αn|2]T↦[|β1|2…|βn|2]T. This perspective, as presented by Aaronson, appears to be a conventional way of conceptualizing hidden variable theories. However, it may seem somewhat restrictive.

The essence of a hidden variable theory lies in the existence of additional degrees of freedom beyond the observable probability distribution. One might expect such a theory to map n×n unitaries to (n+k)×(n+k) stochastic matrices, introducing some additional parameters denoted by k. Aaronson's specification that k=0 constrains the stochastic matrix to operate on a representation that, for example, lacks the distinction between |+⟩ and |−⟩. This could potentially render certain issues he discusses as artifacts of this constraint rather than inherent characteristics of hidden variables in a broader sense.

The question arises: Is Aaronson's definition aligned with the perspective of physicists on hidden variable theories? If so, what rationale exists for dismissing the extension of the state vector with supplementary entries?

Answer 1

The question examines Scott Aaronson's characterization of hidden variable theories in quantum computing, which potentially limits the theory by not introducing additional parameters to distinguish different quantum states. Despite the concern, physicists may prefer such simplifications to maintain strong empirical support, paralleling the situation in superstring theory.

Step-by-step explanation:

The question pertains to hidden variable theories within quantum computing, examining the depiction of such theories as frameworks transforming unitary matrices into stochastic matrices without extending the state vectors with additional parameters. In quantum mechanics, developed by prominent scientists like Heisenberg, Bohr, De Broglie, and Schrödinger, the uncertainty principle and the wavefunction play a crucial role in describing the probabilities of quantum events.

Hidden variable theories suggest there are underlying factors that determine quantum phenomena with more certainty than quantum mechanics allows. Yet the specifics of Scott Aaronson's characterization, which does not expand the state vector, may be seen as restrictive because it does not account for additional possible degrees of freedom that could distinguish quantum states with similar probability distributions but different underlying realities. Physicists may align with Aaronson's definition in the context of wanting to remain as close to experimental data as possible, since a theory introducing extra unobservable parameters may be seen as lacking empirical support, reflecting concerns similar to those with superstring theory, where many theoretical possibilities exist that are currently unconstrained by experiment.