QBuki Notes on Reconstruction

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