Unital Channel Eigenvalue Majorization
I’ve decided that, starting today, I will once a month post a mathematical result that I find interesting and/or useful, but I feel sadly gets less attention than it deserves (Update: well, that lasted about four months). I will try to present all relevant preliminaries along with the result to provide context, so hopefully the results and proofs will be accessible to someone at the upper undergraduate level.
Since I’m a quantum kinda guy, it seems natural that the first such lemma deals with quantum information science. In particular, it helps quantify the behaviour of unital quantum channels acting on density operators. Before delving into the result, let’s begin with…
Quantum Preliminaries
Given the complex matrix space Mn, a quantum channel E is defined to be a completely positive, trace-preserving map. That is, it is a map of the form
where {Aj} ∈ Mn is a family of matrices. Trace-preservation of E is equivalent to the requirement that
In many physical situations we are interested in unital quantum channels; that is, channels that satisfy E(I) = I. Such channels in general exhibit much nicer behaviour than arbitrary quantum channels, and this month’s lemma will show one particular instance of this fact.
The Hardy-Littlewood-Polya Theorem
The proof of the lemma relies on a classical result known as the Hardy-Littlewood-Polya Theorem. The result explains how doubly stochastic matrices act on vectors. Since it seems to be surprisingly difficult to find on Wikipedia and other popular (read: non-research) websites, I will state it here.
Theorem [Hardy-Littlewood-Polya]. Let x,y ∈ Rn be real vectors. Then x majorizes y if and only if y = Dx for some doubly stochastic matrix D ∈ Mn.
It might be worth mentioning that the “if” direction of the proof is borderline trivial; the real meat and potatoes of the theorem is the “only if” direction.
The Lemma Itself
The lemma makes precise something that feels quite natural when thought of physically: a unital channel (that is, a completely positive, trace-preserving map E for which E(I) = I) can only increase the impurity (or “mixedness”) of quantum states. It has several simple consequences that are of great use when dealing with unital channels, and furthermore its proof makes excellent use of classical machinery. It was originally due to Uhlmann [1,2], but has recently appeared in [3]. The proof provided in the PDF attached at the end of this post is from the latter source.
Lemma [Unital Channel Eigenvalue Majorization]. Suppose ρ = E(σ) for a unital channel E. Then the ordered spectrum r of ρ is majorised by the ordered spectrum s of σ.
One particularly useful corollary of this lemma is presented here, and its proof is omitted (and dare I say left as an exercise for the reader?)
Corollary. If E is a unital quantum channel and ρ is a positive operator, then rank(E(ρ)) ≥ rank(ρ).
Related Links
References
- A. Uhlmann, Commun. Math. Phys. 54, 21 (1977).
- I. Bengtsson, and K. Zyczkowski, Geometry of quantum states, Cambridge University Press (2006).
- D. W. Kribs, R. W. Spekkens, Phys. Rev. A 74, 042329 (2006). arXiv:quant-ph/0608045v2
Should you put x,y real vectors in Theorem [Hardy-Littlewood-Polya]?
@Minghua – Absolutely – it is fixed now. Thanks for the correction!