we know that for all ρ, σ ∈ D(H) with T (ρ, σ) ≤ ε,
|H(ρ) − H(σ)| ≤ f(ε). If two vectors p∗, q∗
∈ P with TV(p∗, q∗)
≤ ε achieve |Hcl(p∗)− Hcl(q∗)| = f(ε), then ρ∗ := diag(p∗) and σ∗ := diag(q∗) satisfy
|H(ρ∗
)− H(σ∗)| = f(ε).
That is, a tight bound on the classical level yields a tight bound on the quantum level.
3.2
A first look at majorization over the 1-norm ball
Fix ε > 0 and a state ρ ∈ D(H), with d := dim H. The main result of this chapter is that the ε-ball around ρ, the ε-ball in trace distance, Bε(ρ), admits a minimum and maximum in the majorization order. Note that since majorization is a partial order, a priori one does not know that there are states in Bε(ρ)comparable to every other state in Bε(ρ). Theorem 3.2.1. Let ρ∈ D and ϵ > 0. Then there exist two states ρ∗ε and ρ∗,ε in Bε(ρ) (which are defined by Equation (3.16) in Section 3.3 and Equation (3.42) in Section 3.6
respectively) both of which commute with ρ and satisfy
ρ∗ε ≺ ω ≺ ρ∗,ε, ∀ω ∈ Bε(ρ). (3.6)
Moreover, ρ∗ε is the unique state in Bε(ρ) satisfying the left-hand relation of (3.6), and ρ∗,ε is the unique state in Bε(ρ) satisfying the right-hand relation of (3.6) up to unitary equivalence.
This result can be specialized to the case of probability vectors. Let p ∈ Pd and ϵ > 0. Then there exist two probability vectors p∗ε and p∗,ε in Bε(p) (which are defined by Equation (3.13) in Section 3.3 and Equation (3.40) in Section 3.6 respectively) which satisfy
p∗ε ≺ q ≺ p∗,ε, ∀q ∈ Bε(p). (3.7)
Moreover, p∗ε is the unique element of Bε(p) satisfying the left-hand relation of (3.7), and p∗,ε is the unique element of Bε(p) satisfying the right-hand relation of (3.7) up to
permutations.
0 0.05 0.1 0.15 0.2
spec ρ
spec ρ
∗εγ
−γ
+spec ρ
∗,εγ
−γ
+Fig. 3.1 An example of the majorization-minimizer and majorization-maximizer with d = 2. The spectrum of a quantum state ρ is on the left, the spectra of ρ∗ε in the center and ρ∗,ε on the right, with ε = 0.03. The states are constructed in Section 3.3 and Section 3.6respectively, and have the structure of raising or lowering some eigenvalues of ρ according to a prescribed formula.
In this section, we consider the ε-ball Bε(ρ)around a state ρ ∈ D(H), and motivate the construction of maximal and minimal states in the majorization order given in (3.6). As discussed in Section 3.1, we prove Theorem 3.2.1 by reducing it to the classical case of discrete probability distributions on d symbols, and then constructing explicit states ρ∗ε (in Section3.3), and ρ∗,ε (in Section 3.6), whose eigenvalues are respectively given by the probability distributions which are minimal and maximal in majorization order.
As discussed in Section 3.1, it suffices to consider the simplex of probability vectors Pd, equipped with the total variation distance, instead of the set of density operators D(H) equipped with the trace distance. Note that Pd is the polytope (i.e. the convex hull of finitely many points) generated by (1, 0, . . . , 0) and its permutations. Recall the total variation ball,
Bε(p) = {w = (wi)di=1 ∈ Pd: 1 2∥w − p∥1 ≤ ε}. The set {x ∈ Rd :∥x∥ 1 ≤ 1} can be written {x ∈ Rd:
∥x∥1 ≤ 1} = conv{e1,−e1, . . . , ed,−ed},
where conv(A) denotes the convex hull of a set A, and e1, . . . , ed are the vectors of the standard basis (e.g. ej = (0, . . . , 0, 1, 0, . . . , 0) with 1 in the jth entry), and is therefore a
3.2 A first look at majorization over the 1-norm ball 17 polytope, called the d-dimensional cross-polytope (see e.g. [Mat02, p. 82]). As a translation and scaling of the d-dimensional cross-polytope, the set {w ∈ Rd: 1
2∥w − p∥1 ≤ ε} is a polytope as well. As Bε(p)is the intersection of this set and Pd, it too is a polytope. See Figure 3.2a for an illustration of Bε(p) in a particular example.
The existence of ρ∗
ε and ρ∗,ε in Bε(ρ) satisfying (3.7) is equivalent to p ∗
ε and p∗,ε in Bε(p) satisfying
p∗ε ≺ w ≺ p∗,ε (3.8)
for all w ∈ Bε(p). Using Birkhoff’s Theorem (e.g. [NC09, Theorem 12.12]), the set of vectors majorized by a point w ∈ Pd can be shown to be given by
Mw :={p ∈ Pd: w ≻ p} = conv{π(w) : π ∈ Sd}, (3.9) where Sd is the symmetric group on d letters (see [MOA11, p. 34]). Let us illustrate this with an example in d = 3. Let us choose p = (0.21, 0.24, 0.55) and ϵ = 0.1. The simplex Pd and ball Bε(p)are depicted in Figure 3.2a. A point w = (0.14, 0.28, 0.58) ∈ Bε(p)is shown in Figure 3.2b, and the set Mw in Figure 3.2c.
x y
z
(a)Bε(p), with p in black
x y
z
(b) Some w∈ Bε(p) (white).
x y
z
(c) The set Mw (green). Fig. 3.2 In dimension d = 3, the simplex, Pd, of probability vectors is the shaded triangle shown in (a), along with the ball Bε(p) which is the hexagon shown in blue, centered at p = (0.21, 0.24, 0.55) (depicted by a black dot) with ϵ = 0.1. In (b), a point w = (0.14, 0.28, 0.58)is depicted in white, and in (c), the set Mw is shown in green.
The geometric characterization (3.9), depicted in Figure 3.2, requires that Bε(p)⊆ Mp∗,ε, and conversely, p
∗
ε ∈ Mp for each p ∈ Bε(p). Figure 3.2c shows that for the point w, Bε(p)̸⊆ Mw, implying that w ̸= p∗,ε. Moreover, one can check that that e.g. w ̸∈ Mp, and hence w ̸= p∗ε.
Next we consider Schur concave functions on Pd, in order to gain insight into the probability distributions p∗,ε and p∗ε which arise in the majorization order (3.8). In particular, let us consider Shannon entropy S(w) := −Pdi=1wilog wi of a probability distribution w = (wi)di=1. It is known to be strictly Schur concave. Hence, if p
satisfying (3.8) exists, it must satisfy
S(p∗,ε)≤ S(w)
for any w ∈ Bε(p). Thus, p∗,ε must be a minimizer of S, which is a concave function, over Bε(p), a convex set. Similarly, p∗ε must be a maximizer of S over Bε(p). Properties of maximizers of concave functions over a convex sets are well-understood; in particular, any local maximizer is a global maximizer.
The task of minimizing a concave function over a convex set is a priori more difficult; in particular, local minima need not be global minima. There is, however, a minimum principle which asserts that the minimum occurs on the boundary of the set; this is formulated more precisely in e.g. [Roc96, Chapter 32]. Since Bε(p) is a polytope, S is minimized on one of the finitely many vertices of Bε(p). This fact yields a simple solution to the problem of minimizing S over Bε(p), as described below by example, and in generality in Section3.6.
Let us return to the example of Figure 3.2. We see Bε(p)has six vertices; these are {p+π((ϵ, −ϵ, 0)) : π ∈ Sd}. The vertex which minimizes S is v := (0.21 − ϵ, 0.24, 0.55 + ϵ),
x y
z
(a) A minimumv of S over Bε(p), in white.
x y
z
(b) Maximumm of S over Bε(p), in white. Fig. 3.3 For the example of Figure 3.2, the (unique) maximum and minimum of the Shannon entropy S over Bε(p)are shown. Both v and m occur on the boundary of Bε(p). where the smallest entry is decreased and the largest entry is increased, as shown in Figure3.3b. Moreover, one can check that w ≺ v for any w ∈ Bε(p). This leads us to the conjecture that the vertex corresponding to decreasing the smallest entry and increasing the largest entry will yield p∗,ε satisfying (3.8) in general. We see in Section 3.6that this is indeed true, although in some cases more than one entry needs to be decreased.
On the other hand, finding the probability distribution p∗
ε in Bε(p) which satisfies (3.8) is more than a matter of checking the vertices of Bε(p), as shown by Figure 3.3b: in
this example, p∗
3.3 Constructing the majorization-minimizer (3.7) 19