Strict positivity and D-majorization

ABSTRACT

Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector d to majorization on square matrices with respect to a positive definite matrix D. For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of D has multiplicity one.

Keywords:

• Strict positivity
• majorization relative to d
• majorization on matrices
• preorder
• quantum channel

AMS CLASSIFICATIONS:

• 15A45
• 15B48
• 47B65
• 81P47

Acknowledgements

I would like to thank Gunther Dirr, Thomas Schulte-Herbrüggen, Michael M. Wolf, Michael Keyl, and the editor for valuable and constructive comments. Also this manuscript greatly benefited from my time in Toruń, in particular from illuminating discussions with Sagnik Chakraborty, Ujan Chakraborty and Dariusz Chruściński. Finally, I am grateful to the anonymous referee for making me aware of Ref. [25] which led to a shorter proof of Proposition 2.5. This work was supported by the Bavarian excellence network (enb) via the International PhD Programme of Excellence Exploring Quantum Matter (exqm).

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A matrix $A\in {\mathbb{R}}^{n\times n}$ is said to be column stochastic if all its entries are non-negative and $\sum _{i=1}^n{A}_{ij}=1$ for all j = 1, …, n, i.e. the entries of each column sums up to one.

2 The positive trace-preserving maps are precisely those linear maps T which satisfy $T\left(\mathbb{D}\right({\mathbb{C}}^n\left)\right)\subseteq \mathbb{D}\left({\mathbb{C}}^n\right)$ – hence this also holds for every quantum channel. As the states form a convex and compact set (Lemma 2.1) by the Brouwer fixed-point theorem [47], every such T has a fixed point in $\mathbb{D}\left({\mathbb{C}}^n\right)$.

3 For example, choose λ = y/x where y is the largest eigenvalue of Y and x>0 is the smallest eigenvalue of X.

4 Of course there are a number of classes of quantum-dynamical processes which are not Markovian but as these do not interact as nicely with strict positivity, we will sweep them under the rug here. For these more general notions of divisibility we refer the reader to Ref. [48].

5 The key here is the following well-known result: let $X\in {\mathbb{C}}^{n\times n}$ be hermitian with smallest eigenvalue x_m and largest eigenvalue x_M. Then, $-X+t\mathbb{1}\ge 0$ if and only if t ≥ x_M and $X-t\mathbb{1}\ge 0$ if and only if t ≤ x_m . This is evident due to $x_m\parallel y{\parallel }^2\le 〈y,Xy〉\le x_M\parallel y{\parallel }^2$ for all $y\in {\mathbb{C}}^n$ (cf. [49, Theorem 4.2.2]).

6 A matrix convex function (cf. Refs. [50–53]) is a map $\psi :\mathbb{R}\to \mathbb{R}$ which acts on hermitian matrices via the spectral theorem and then satisfies ψ(λA + (1 − λ)B) ≤ λψ(A) + (1 − λ)ψ(B) for all $\lambda \in [0,1],A,B\in {\mathbb{C}}^{n\times n}$ hermitian and all $n\in \mathbb{N}$ where ≤ is the partial ordering on the hermitian matrices induced by positive semi-definiteness.

7 While these equivalences are straightforward to check the main ingredients are the estimate $〈x,Dx〉=\sum _{i=1}^n{d}_i|〈e_i,x〉{|}^2\ge d_k\sum _{i=1}^n|〈e_i,x〉{|}^2=d_k\parallel x{\parallel }^2$for all $x\in {\mathbb{C}}^n$ – with equality if and only if x = λe_k for some $\lambda \in \mathbb{C}$ because d_k is the unique minimal entry of d – as well as the renowned fact that equality in the Cauchy–Schwarz inequality holds if and only if one vector is a multiple of the other.

8 An operator J on the power set $\mathcal{P}\left(S\right)$ of a set S is called closure operator or hull operator if it is extensive (X⊆ J(X)), increasing (X⊆ Y ↠ J(X)⊆ J(Y)) and idempotent (J(J(X)) = J(X)) for all $X,Y\in \mathcal{P}\left(S\right)$, cf., e.g.[54, p. 42].

9 Let $A,B\in {\mathcal{P}}_c\left(X\right)$ and d be the metric on X. The Hausdorff metric is defined via $\mathrm{\Delta }(A,B):=max\{\underset{z\in A}{max}d(z,B),\underset{z\in B}{max}d(z,A)\},$where as usual d(z, B) = min _w∈Bd(z, w) (and d(z, A) analogously), refer to Ref. [55]. Then for A, B ∈ P_c(X) and γ ≥ 0, one has Δ(A, B) ≤ γ if and only if for all a ∈ A, there exists b ∈ B with d(a, b) ≤ γ and vice versa, cf. Ref. [56, Lemma 2.3].

10 In the second-to-last row of the following computation, we will make use of this basic result (cf., [56, Lemma 2.5.(a)]): Let $(A_n{)}_{n\in \mathbb{N}}$ and $(B_n{)}_{n\in \mathbb{N}}$ be bounded sequences of non-empty compact subsets of any metric space which Hausdorff-converge to A and B, respectively. Now if A_n⊆ B_n for all $n\in \mathbb{N}$, then A⊆ B.

11 For bounded sequences $(A_n{)}_{n\in \mathbb{N}}$ of non-empty compact subsets (of some metric space) which Hausdorff-converges to A, one has x ∈ A if and only if there exists a sequence $(a_n{)}_{n\in \mathbb{N}}$ with a_n ∈ A_n which converges to x, cf. Ref. [56, Lemma 2.4].

12 Similarly, while the set of d-stochastic matrices forms a convex polytope, the set Q_D(n) has infinitely many extreme points. The argument, just as below, relies on the concatenation with suitable unitary channels from left and right – after all the bijective quantum channels the inverse of which is a channel again are precisely the unitary ones [56, Proposition 1].