MajorizationExtrema

Install

This package isn't registered yet, but you can install it via

] add https://github.com/ericphanson/MajorizationExtrema.jl

This package implements methods from

E. P. Hanson and N. Datta, “Maximum and minimum entropy states yielding local continuity bounds,” Journal of Mathematical Physics, vol. 59, no. 4, p. 042204, Apr. 2018.

for which a preprint is available here: https://arxiv.org/abs/1706.02212v2.

Usage

Provides the following functions:

TV, tracedist, ≺, ≻ # distances, majorization
majmax, majmin, localbound # majorization-extrema
randprobvec, randunitary, randdm # generate random data

Compute the majorization minimizer and maximizer over total variation balls of discrete probability distributions (in finite dimensions).

using MajorizationExtrema

q = [0.25, 0.25, 0.5]
majmin(q, 0.05) # [0.275, 0.275, 0.45]
majmax(q, 0.05) # [0.2, 0.25, 0.55]

These vectors satisfy majmin(q, 0.05) ≺ p ≺ majmax(q, 0.05) for any probability distribution p within 0.05 of q in total variation distance (TV(p, q) <= 0.05), where ≺ is the majorization preorder.

These functions also work with rational numbers:

q = [1//4, 1//4, 1//2]
majmin(q, 1//20) # [11//40, 11//40, 9//20]
majmax(q, 1//20) # [1//5, 1//4, 11//20]

If you're doing extended computations with rational numbers, you may want to use Rational{BigInt} types which won't overflow:

q = big.([1//4, 1//4, 1//2])
...

The existence of majorization minimizer and maximizers allows the computation of local bounds of Schur concave or Schur convex functions on probability distributions around a specific distribution; localbound is provided as a convenience function for this. For example,

using MajorizationExtrema

function entropy(q)
    eta(x) = x <= 0 ? zero(x) : -x*log2(x)
    return sum(eta, q)
end

p = [.2, .3, .5] # a fixed probability distribution
q = randprobvec(3) # a random probability distribution
ϵ = TV(p, q) # total variation distance
abs(entropy(p) - entropy(q)) <= localbound(entropy, p, ϵ) # true

Note localbound(entropy, p, ϵ) only depends on the choice of q via ϵ; the bound holds uniformly over all probability distributions q within total variation distance ϵ of p.

These functions also work on quantum density matrices. For example,

julia> using MajorizationExtrema

julia> ρ = randdm(3)
3×3 LinearAlgebra.Hermitian{Complex{Float64},Array{Complex{Float64},2}}:
   0.235005+0.0im         0.00492117+0.00548039im  -0.0168253-0.020445im 
 0.00492117-0.00548039im    0.458451+0.0im         -0.0724026+0.0714849im
 -0.0168253+0.020445im    -0.0724026-0.0714849im     0.306544+0.0im      

julia> σ = majmin(ρ, .01)
3×3 LinearAlgebra.Hermitian{Complex{Float64},Array{Complex{Float64},2}}:
   0.242182+0.0im         0.00433537+0.00652348im  -0.0142623-0.0168348im
 0.00433537-0.00652348im     0.45091+0.0im         -0.0689234+0.0679345im
 -0.0142623+0.0168348im   -0.0689234-0.0679345im     0.306909+0.0im      

julia> σ ≺ ρ
true

(One will likely want to write using LinearAlgebra when working with quantum states in order to compute eigenvalues, etc.)