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 and in particular for a probability vector , we have For an unbiased quantum 3-design, , , , and so that is the probabilistic representation of taking the Jordan product with :

5. What further symmetries can we impose in order to fix the constants, and on what grounds?

   1. Let and so that Suppose we demand commutativity: . We have from which we conclude that .

   2. By Bayes’s rule, we have . If we assume self-duality, If we assume , then For an unbiased quantum 3-design, a pure-state probability-assignment satisfies .

   3. Let us now consider , or If we demand that and moreover that (in quantum theory, this is the demand that ) then Summing over on both sides, we have and substituting , we find that , which fixes Now

   4. We can go further using by substituting in . so that For an unbiased quantum 3-design, .

      Can we go in reverse and show that if and equal the required values (and ), then ? Can we have an independent characterization of the pure states of the theory and thus show that the pure states are completely characterized by these considerations?

   5. Supposing we can pin down the state-space as an intersection of 2-norm and 3-norm spheres and the privileged subspace , we could hope to interrogate self-duality by studying -norm cones. Hölder’s inequality tells us that for real numbers such that , we have for with equality when for . In the case of , equality holds if are have only one non-zero component in the same place. This can be used to establish the dual of cone whose base is a -norm sphere. Moreover, the dual of an intersection of cones is the Minkowski sum of their duals: , and the same goes for an intersection with a subspace. This would be the jumping off place for some analysis.

   6. What if we demand that in fact is a Jordan product matrix? Recall that the Jordan product is completely characterized by its commutativity and the condition that . For us, this means on the one hand, , and on the other hand, . We’ve already explored commutativity. Direct calculation of is not particularly illuminating. But let’s think instead about structure-coefficients.

      A Euclidean Jordan algebra is a vector space equipped with an bilinear product with a “compatible” inner product . The Jordan product satisfies Let be a basis for . We define structure-coefficients to satisfy What symmetries must satisfy? Since , we have in particular We conclude . A compatible inner product satisfies . On the one hand, On the other hand, We conclude that . Thus in fact must be totally symmetric. Finally, on the one hand, while on the other hand, We conclude that In our case, in analogy to quantum theory, since we ought to take , that is, If satisfies the required identity, then our theory has a Jordan algebraic structure. Writing this out in terms of is laborious, and not obviously illuminating. I can give expressions on request.

   7. One assumption we could make is homogeneity, which is automatically a feature of Euclidean Jordan algebras. Essentially, any full-rank state can be mapped to any other by some invertible map that preserves the structure of the “cone.” Quantum mechanically the essential insight is that if is full-rank, then we can contemplate the Kraus update map . Clearly if , we end up with the identity. Then if we want to end up at some arbitrary state , we can apply the update map .

      It turns out for an element of a Jordan algebra , the map is represented by , where takes the Jordan product with . This is called the quadratic representation of . Quantum mechanically, on the one hand, and on the other hand, from which one can derive The expression can be worked out in terms of probabilities, but again without an obvious insight awaiting.

6. What is the size of a maximal simplex of perfectly distinguishable states: in other words, how can we determine ?

   Since we have Incidentally, it is worth considering in this context the symmetry which isn’t necessarily taken for granted. Assuming this implies In any case, we can then consider, for example, In other words, we can ask: what does the null-space of look like? From , we find that must statisfy What if we further assume are pure? How many solutions do such equations have?

   Finally, in 3-design quantum theory, we have which implies the existence of an orthonormal basis measurement of . What milage can we get out of that?

7. What can we say about “time evolution,” or more specifically, the continuous symmetries of the pure-states? On the one hand, to the extent that we have defining conditions on pure-states, we can derive conditions for a quasi-stochastic matrix to preserve pure-states. On the other hand, assuming quantum mechanics already, to represent the time derivative in the von Neumann equation, we need a probabilistic representation of the commutator . This can be derived from the imaginary part of the triple products. To what extent can these imaginary parts be nailed down? We have Moreover, and since , the real-part determines the imaginary-part up to sign: The choice of signs is not arbitrary since must be antisymmetric. Will any appropriate choice of signs work?

Finally, supposing we can prove most of everything we want to prove on the basis of these considerations, there remains the possibility that we end up with perfectly good theories that act quite like quantum mechanics, but the ’s just don’t correspond to quantum 3-design probabilities. It feels like we’re missing some simple, powerful, killer assumption. Could we assume the existence of a SIC? (Can we glue SICs together to form 3-designs? Are there SIC fiducials whose orbit under the Clifford group gives a 3-design?) More generally, is there some way of characterizing 3-designs themselves in QBist terms? In this connection, perhaps it is worth looking back at the literature on shadow estimation, especially Huangjun Zhu’s recent papers. There incidentally, the focus is not so much on a lower bound on the variance of an observable (according to the reference device), but on an upper bound. David Gross has interesting things to say about the symmetries of a 3-design set:

But what does this mean for us?