A (complex-projective, unbiased) -design is a set of pure quantum states which satisfy 1-designs, rescaled, form measurements. 2-designs include SICs and MUBs. 3-designs will concern us here. Let us consider a measure-and-prepare device which performs measurement and conditionally prepares a state where the states form an 3-design. Note that a -design is also a -design, so that indeed, is a measurement. Moreover, from the fact that the device forms a 2-design, it is informationally-complete: the probabilities suffice to pick out a density matrix. But crucially, not all probability distributions correspond to valid states (they map to matrices with negative eigenvalues). So how can we characterize the valid probability-assignments to the reference device? Here the 3-design property comes into play.
Consider the agreement-probabilities for devices To evaluate this, note that On the one hand, which is maximized when pure. On the other hand, which again is maximized when pure. Consider the following lemma:
Proof. Let be the eigenvalues of . means that . On the one hand, implies that . On the other hand, with equality if and only if . But since the whole sum must be 1, we must have exactly one and the rest 0. Thus is a rank-1 projector, and hence a pure state. ◻
We conclude that we can characterize pure-states by the following equations with the caveat that , where . Why this last condition? The reason is that a 3-design representation is necessarily overcomplete—indeed, — and in our derivation, we’ve assumed that all probabilities are obtained from . Let be the matrix whose rows are and be the matrix whose columns are where . On the one hand, ; on the other hand, , which is a full-rank factorization and thus the columns of form a basis for the column-space of . Therefore our proof becomes if-and-only if as long as .
It is worth noting that we can motivate the restriction that on QBist grounds. Let the Born matrix be any matrix satisfying . Then the Born rule appears as a deformation of the law of total probability. In particular, . Thus for consistency’s stake, we ought to require . implies that is a projector. On what subspace, though? For a 2-design so that For a pure state , , and so letting , we arrive at the resolution of the identity , which demonstrates informational-completeness. Comparing this to , it follows that we may take . Since is full rank, projects onto .
So the contour of quantum state-space according to a 3-design is given by the intersection of the non-negative orthant, a 1-norm sphere, a 2-norm sphere, and a 3-norm sphere of prescribed radii, and a dimensional subspace: . Alternatively, we can derive a single equation picking out pure probability-assignments from the demand that . From the resolution of the identity, , we have where the real-part comes from . Let so that and therefore Our condition for pure-statehood then simplifies to which we note automatically implies .
In fact, we can do even better, and derive a condition for the validity of any state, pure or mixed. We note that are essentially the structure-coefficients for the Jordan product , The linear operator which performs the Jordan product (and which acts on vectorized states) is . The matrix does the same on e.g. probability vectors: where the tilde recalls that might not be a normalized probability distribution. Note that does not depend on any redundancy in . We find after substitution that Now clearly, . Moreover, . To see this note that if is a frame with dual elements such that , we can write an arbitrary operator , where, considering the matrix of coefficients , iff , since . We thus have a condition for statehood, pure or mixed: , which again only depends on reference device probabilities.
Finally, let us give an interpretation of this last result. Consider that if is some arbitrary observable, it follows that which is immediately equivalent to . Again substituting in yields a condition on valid where e.g. , and . If we make the simplifying assumption that , using the 2-design property, this simplifies to where e.g. . Notice that we are considering the second-moment with respect to the reference device as opposed to a von Neumann measurement (although the inequality is saturated iff ). Thus the shape of quantum state-space can be understood in terms of a kind of uncertainty principle: a valid probability-assignment to the reference device implies a certain minimum variance to any observable in .
We begin in a formless void without yet quantum mechanics.
: There is a reference device characterized by a stochastic matrix where is symmetric and hence bistochastic.
: We assume that is a Born matrix for , satisfying , and that are quasi-probabilities, possibly negative, summing to 1. Here is the matrix of all 1’s.
On the one hand, since , we must have so that . On the other hand, Noting that , we have . Letting and , we have In particular, for probabilities , in other words, for probability-assignments in its column space, acts as a depolarizing channel. We note that projects onto .
: A probability-assignment is valid if and only if for any observable , the second-moment with respect to the reference device satisfies a lower bound. Further we assume that like the second-moment itself, the lower bound is linear in and quadratic in .
We can characterize the lower bound in terms of a three-index tensor such that a valid satisfies or Let . Since on , and projects onto that subspace, we have simplicter iff is a valid state. Indeed, if we choose to be symmetric in the first two indices, then will be postive semi-definite. We’ve thus managed to translate the validity of , expressed in terms of a lower bound on the second-moment of any observable with respect to the reference device, into the postive-semidefiniteness of a certain matrix associated to .
: We assume that .
Substituting this simple form for into the expression for yields where .
Now let , , , and . Then is precisely the matrix we derived earlier, which represents taking the Jordan product with . In other words, if in fact characterizes a quantum 3-design, then .
:
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, . (Moreover, a Euclidean Jordan algebra satisfies for a choice of inner product on the underlying vector space .) Does any for arbitrary symmetric, stochastic, depolarizing , given the appropriate choices of constants, satisfy the Jordan product conditions? In other words, have we found an alternative way of characterizing (some subset of) the Euclidean Jordan algebras? A great deal of tedious algebra lies in between resolving this yes or no. Suppose the answer is yes. Recall that all EJA’s are direct sums of the simple EJA’s: , , , , and . Then the choice of quantum theory over is likely no simpler than the condition that can be represented as for a 3-design in . On the other hand, suppose the answer is no. Then the two defining conditions on the Jordan product translate into restrictions on the probabilities . This could pick out a whole class of EJA’s. Or if we’re unreasonably lucky, it might pick out quantum theory over specifically, and thus providing a characterization of 3-designs themselves entirely in terms of reference device probabilities .
Suppose instead we want to derive the Jordan structure. Answering the aforementioned question will likely suggest the best way of doing that. But we can already ask, for example: given some alone (taking the simple form or not), a) can we characterize the extremal probability distributions? b) can we characterize its dual (the space of non-negative linear functionals of the form )? Must such a state space be self-dual? Can we then show that iff is extreme , for instance?