The Same Title?

The surprising thing about this is that the subject is so simple, it really should have been figured out a long time ago. It's just qubits, not "entangled qubits" or systems of qubits, or anything more complicated.

I suspect that the result would have been easier to compute using the density operator techniques I prefer. Then the "effect" discussed in the second article is just a mixed density matrix state.

And the subject reminds me of the "mutually unbiased bases" that are being discussed by a few of my collaborators recently.

Dave: I was amused by the parallelism as well, but quickly decided that I like the idea! I'm generally in favor of any device that adjusts the incentive system to favor collaboration and openness -- and the competitive, winner-take-all (where "winner" means "first to publish") style of academic publishing could use some tweaking.

The conventional solution to this would be (a) join forces and write a 5-author paper, or (b) simulpublish, with different titles. As a potential reader of the work (and this is the most important perspective!), I like this solution better. It beats (a) because I get two clear perspectives on the same work, instead of a muddled juxtaposition of them. It beats (b) by making very clear to me the fact that these two papers address exactly the same question (and arrive at the same solution) in different ways. Two snaps up!

Carl: First, sorry if my comment is redundant to you; I have no idea what your background is. That said: I'm not entirely sure what you mean in the second paragraph, but "effect" is a technical term for an element of a POVM (generalized quantum measurement). So, if E is an effect and I is the identity, then E satisfies 0 &le E &le I -- but by no means does it satisfy Tr(E) = 1, so it's not a density matrix.

It's very tempting to treat effects as unnormalized density matrices. Sometimes it's even quite useful. In this problem, however, it gets you in trouble because the convex space of effects is quite a bit bigger than that of states. It's a cone in R4, rather than a sphere in R3.

Anyway, if that's all old hat to you, then please feel free to ignore it! By way of conclusion, as somebody who's worked on this problem occasionally for a couple of years, I'll gently disagree with its description as "simple". Many information-theoretic problems with classical analogues are simple for qubits and difficult for larger systems -- but this problem has no classical analogue, and (IMHO) is quite tricky even for a single qubit.

Dave, Hmmmm.

My understanding, from quickly scanning the second paper, is that an "effect" is an operator. Every operator has an interpretation as a state in density operator theory, it just, like you said, (a) might not be normalized, and in addition, (b) might not be Hermitian.

The pure density matrix states form a 2-sphere (Bloch sphere) and use up two real dimensions out of the 8. These are Hermitian. The product of any two such matrices is a real multiple of a unique not necessarily Hermitian density matrix (i.e. what the mathematicians call a primtive idempotent), which therefore require 4 degrees of freedom to specify.

An alternative way of defining non Hermitian pure density matrices is to choose an axis u (2 degrees of freedom) choose a perpendicular v (1 degree of freedom) and choose a real number k (the 4th degree of freedom). Then (1+cosh(k)sigma\cdot u + i sinh(k) sigma\cdot v)/2 is a normalized non Hermitian pure density matrix (sigma is the vector of Pauli spin matrices).

To unnormalize these, you can mulitply by an arbitrary complex number, this gives 2 more degrees of freedom for a total of 6. But these are still complex multiples of (not necessarily Hermitian) density matrices.

To get the remaining two degrees of freedom, add in a complex multiple of unity. What you get is a mixed, not necessarily Hermitian, density matrix. I think these give you all 8 complex degrees of freedom in 2x2 linear operators, but I've never had cause to actually need this.

Now I've assumed from this argument that any operator should have a density operator interpretation. I guess I'm not sufficiently interested in effects to figure out whether it would be advantageous there. I've got my own swamp to drain.