Spring 2025 Research Questions

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The purpose of these notes is to lay out the beginnings of a research program which aims to explore a particular brand of non-classical theories inspired by the QBist representation of quantum mechanics induced by a 3-design measurement. For background, read https://arxiv.org/abs/2412.13505. The crucial feature of such theories is that the validity of a set of probability assignments on the outcomes of a privileged reference device is equivalent to the assumption of a lower bound on the variance of a natural class of observables. In other words, the shape of state space is ultimately determined by an uncertainty principle: the variance of any natural observable cannot be made smaller than a certain amount, which morally speaking is a way of respecting the uncreatedness of outcomes before measurement.

The ultimate goal is to motivate a re-derivation of quantum theory along QBist principles. As one proceeds, however, there are many forks in the road, places where in order to motivate our assumptions, it would be useful to explore the consequences of alternative choices, both analytically and computationally. What follows mingles assumptions, proofs, conjectures, and open questions.

There is a measure-and-reprepare reference device which is assumed to be informationally complete. Denote by the outcomes of the measurement, and the corresponding preparations. The reference device may be characterized in its own terms by a stochastic matrix Let us put some initial restrictions on this matrix. We’ll take it to be symmetric ( ), and hence bistochastic. We also assume . Probabilities for arbitrary measurements may be calculated according to the QBist Born Rule where is short-hand for a column vector of probabilities , and is short-hand for a row vector of conditional probabilities , being some arbitrary outcome, and being some arbitrary preparation. Here is a Born matrix, satisfying . We assume that we may take where is the matrix of all 1’s, and that is a vector of quasi-probabilities, possibly negative, but summing to 1.

  1. Since , we must have so that .

  2. Notice that . Since , it follows that where , that is, is some linear combination of the columns of . Denoting , we have In particular, for probability vectors , which shows that on probability vectors in its column space, acts as a depolarizing channel with parameter , mixing with the flat probability vector.

  3. Since , using the spectral decomposition , we have Since is bistochastic, is an eigenvector with eigenvalue 1. Acting on the left and on the right with , we obtain , or . We conclude the possible eigenvalues of are . We assumed that Thus where , and so

  4. projects onto . From , we have so that is idempotent and thus a projector. is invertible, and thus projects onto . In fact since , it is an orthogonal projection. Consistency with the QBist Born rule requires that and so we may without loss of generality consider only probability vectors .

  5. For an unbiased quantum 3-design, we have , and the rank of is .

Let be an observable, which here means an assignment of numerical values to the outcomes of the reference measurement. We assume that a probability vector is valid if and only if the second moment of any observable is lower bounded. Further, we assume that like the second moment itself, the lower bound is linear in and quadratic in . In other words, for some three-index tensor .

  1. Rewriting the lower bound, we have Since on and projects into , we have . If we assume is symmetric in the first two indices, then , and the question of the validity of is transformed into the question of the positive-semidefiniteness of .

  2. What assumptions can we marshal to narrow down the reasonable choices of , what natural symmetries might we impose? On the other hand, suppose we fix some . What is the best way, analytically and computationally, of charting out the state-space, the set of valid probability vectors? In particular,

    1. Under what conditions is the state-space self-dual? Must any such state-space be self-dual? In other words, suppose we have a valid probability vector . Bayes’ Rule tells us that

      Self-duality would mean that given some valid , Conversely, if we have a valid response function , i.e. such that , then Note that in particular, . If we assume that in fact , then For an unbiased quantum 3-design, .

    2. An extremal probability vector is a valid probability vector which cannot be written as a convex mixture of valid probability vectors (in ). Can we easily characterize the extremal states of the theory directly in terms of ? In terms of the eigenstructure of ? When is the variance bound saturated?

    3. Which state-spaces can support a a maximal simplex with vertices which lie on its boundary, that is, a SIC? What is the maximal size of a simplex of perfectly distinguishable states?

  3. Let us assume that for some constant . This is the case in quantum mechanics. In what sense can we say that this the simplest possible choice? If we do make this choice, we have for some arbitrary vector , where . Then In particular, for a probabilities , we have so that, by assumption, a valid probability vector must satisfy the lower-bound on its second moment, where e.g. and .

    For an unbiased quantum 3-design, .

  4. Let us now consider . On the one hand, . Let us assume that , so that . Then

    Since , let . We arrive at last at