Elementary Proofs Of Two Theorems Involving Arguments Of Eigenvalues Of A Product Of Two Unitary Matrices

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices --- one by Childs \emph{et al.} [J. Mod. Phys., \textbf{47}, 155 (2000)] and the other one by Chau [arXiv:1006.3614]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators.

Actually, a more general version of Theorem 1 was first proven by Nudel'man and Švarcman [4] by looking into the geometric properties of certain hyperplanes related to the argument of the eigenvalues of a unitary matrix. Built on this geometric approach, Thompson [5] extended Nudel'man and Švarcman's result by giving an even more general version of Theorem 1. (Note that Nidel'man and Švarcman as well as Thompson used a different convention in which all arguments of the eigenvalues are taken from the interval [0, 2π). Nonetheless, the convention does not affect the conclusions of Theorem 1.) Later on, Agnihotri and Woodward [6] as well as Biswas [7] showed among other things the validity of Theorem 1 by means of quantum Schubert calculus. Belkale [8] obtained Theorem 1 by studying the local monodromy of certain geometrical objects.
Along a similar line of investigation, Chau [9] recently showed among other things the following theorem using Rayleigh-Schrödinger series.
Theorem 2. Let U, V be two n-dimensional unitary matrices. Then, Moreover, the equality holds if and only if Note that all existing proofs of Theorems 1 and 2 involve rather high level geometrical or analytical methods. Here, we report elementary proofs of these two theorems. One of the advantages of these elementary proofs is that one can easily deduce the necessary and sufficient conditions for equalities. Besides, it is straightforward to extend the theorem to cover the case of infinite-dimensional unitary operators.
Our elementary proofs of these two theorems rely on Lemma 2, which in turn follows from Lemma 1.
Proof. By definition, UV|φ . By taking the arguments in both sides, we obtain Note that for any normalized state ket |ψ〉, 〈ψ |U| ψ〉 and 〈ψ |V| ψ〉 are located in the convex hull formed by the vertices {e iθ ↓ k (U) } n k=1 and {e iθ ↓ k (V) } n k=1 on the complex plane ℂ, respectively. Combined with the conditions that θ Furthermore, the extremum in the R.H.S. of the above equation is attained by choos- for all |ψ〉. Proof. Any Hilbert subspace of codimension j -1 must have non-trivial intersection To prove the validity of Equation 1a for case (i), we apply Lemma 1 to obtain Separately applying Equation 6 in Lemma 2 to the two terms in the R.H.S. of Equation 7, we have Hence, Equation 1a is valid for case (i). Furthermore, the equality holds if and only if . This proves the validity of this theorem for case (i). The validity of cases (ii) and (iii) follow that of case (i). (For simplicity, we only consider the reduction from case (ii) to case (i) as the reduction from case (iii) to case (i) is similar.) Let U, V be a pair of unitary matrices satisfying the conditions of case (ii).

we can pick a number a from the non-empty open interval
It is easy to check that a (0, 1) and that 0 < a θ ↓ As a result, the pair of matrices U a and V satisfies the conditions of this theorem for case (i) where the notation U a denotes the unitary matrix j e iaθ ↓ j (U) |φ Further notice that the pair of matrices U 1-a and U a V also obeys the conditions of this theorem for case (i). Hence, θ ↓ This proves the validity of this theorem for case (ii). □ Elementary proof of Theorem 2. We may assume that |θ | ↓ 1 (U) + |θ | ↓ 1 (V) < π for the theorem is trivially true otherwise. Then, from Equations 1a and 1b in Theorem 1, we have of UV such that lim j ∞ arg〈ψ j |UV|ψ j 〉 = sup arg(UV), lim j ∞ arg〈ψ j | U|ψ j 〉 = sup arg(U) and lim j ∞ arg〈ψ j |V|ψ j 〉 = sup arg(V). In a similar fashion, the equality of Equation 11b holds if and only if there exists a sequence of eigenkets {|ψ j } ∞ j=1 of UV such that lim j ∞ arg 〈ψ j |UV|ψ j 〉 = inf arg(UV ), lim j ∞ arg 〈ψ j |U|ψ j 〉 = inf arg(U) and lim j ∞ arg 〈ψ j |V|ψ j 〉 = inf arg(V).
Outline proofs of Theorems 3 and 4. We can use the convex hull argument in Lemmas 1 and 2 to show that (1) sup arg(UV ) = sup arg 〈j |U|j 〉+sup arg 〈j |V| j 〉 where the supremum is taken over all eigenkets |j 〉 of UV, and (2) inf arg(U) ≤ arg 〈ψ |U| ψ 〉 ≤ sup arg(U) for all |ψ 〉 whenever sup arg(U) -inf arg(U) <π. Hence, Equation 11a in Theorem 3 holds in the case of sup arg(U) -inf arg(U), sup arg(V) -inf arg(V) <π. Furthermore, by examining the condition for arg 〈ψ |U| ψ 〉 = sup arg(U) in the case of sup arg(U) -inf arg(U) <π, it is straightforward to verify the validity of the necessary and sufficient conditions for equality of Equation 11a in the case of sup arg (U) -inf arg(U), sup arg(V) -inf arg(V) <π. Now, we can follow the arguments in the proofs of the remaining cases in Theorem 1 as well as in the proof of Theorem 2 to show the validity of Theorems 3 and 4. □