## SchattenNorms.jl

Julia package providing Schatten norms, including completely bounded norms, such as Kitaev's diamond norm (ubiquitous in, e.g., quantum information processing)
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June 2015

# SchattenNorms.jl

Simple implementation of Schatten norms. This package includes the complete bounded versions of the induced norms for linear transformations of matrices (i.e., superoperators), implemented as described in

Semidefinite programs for completely bounded norms, John Watrous, Theory of Computing, Volume 5 (2009), pp. 217–238. (preprint)

Simpler semidefinite programs for completely bounded norms, John Watrous, Chicago Journal of Theoretical Computer Science Volume 8 (2013), p. 1-19. (preprint)

This package only supports the completely bounded norms for p=1 and p=∞ (which are dual). It is not clear if there is an efficient way to compute the completely bounded norms for other p.

## Norms implemented

Taking σᵢ(M) to be the ith singular value of a matrix M, we have

Function name Mathematical meaning
`snorm(M, r)` ‖X‖ᵣ = ∑ᵢ (σᵢ(M))ʳ
`cbnorm(M, r)` supᵢ {‖M⊗1ᵢ(X)‖ᵣ : ‖X‖ᵣ=1}

Some useful aliases and relatd calls are

Alias function Equivalent call Common name
`trnorm(M)` `snorm(M,1)` trace norm
`nucnorm(M)` `snorm(M,1)` nuclear norm
`fnorm(M)` `snorm(M,2), snorm(M)` Frobenius norm (default for `snorm`)
`specnorm(M)` `snorm(M,Inf)` spectral norm.
`cbnorm(M)` `cbnorm(M,Inf)` completely bounded norm usually refers to p=∞, so this is the default
`dnorm(M)` `cbnorm(M,1)` diamond norm

For the special case where `M` is the difference between CPTP maps, or the difference between superoperators corresponding to unitary maps, use `ddist` described below.

## Utility functions

Function name Common name Mathematical meaning
`worstfidelity(u, v)` Worst case output state (Jozsa) fidelity min {❘⟨ψ ❘ v⁺ u ❘ψ⟩❘² : ❘⟨ψ❘ψ⟩❘² = 1}
`ddistu(U,V)` Diamond norm distance between unitary maps `dnorm(liou(U)-liou(V))`
`ddist(E,F)` Diamond norm distance between CPTP maps `dnorm(E-F)`

Despite the mathematical equivalences between `ddist`/`ddistu` and `dnorm`, `ddist`/`ddistu` are much faster and more accurate.

## Usage

```julia> snorm([1 0; 0 -1])
1.4142135623730951

julia> snorm([1 0; 0 -1],1.0)
2.0

julia> snorm([1 0; 0 -1],Inf)
1.0

julia> U,_,_ = svd(randn(2,2)); V,_,_ = svd(randn(2,2)); # unitary invariance

julia> isapprox(snorm([1 0; 0 -1]),snorm(U*[1 0; 0 -1]*V))
true

julia> isapprox(snorm([1 0; 0 -1],1.0),snorm(U*[1 0; 0 -1]*V,1.0))
true

julia> isapprox(snorm([1 0; 0 -1],Inf),snorm(U*[1 0; 0 -1]*V,Inf))
true

julia> R = randn(3,3); snorm(R,1) >= snorm(R,2) >= snorm(R,Inf)
true

julia> p = sort(abs(randn(3))+1.0); snorm(R,p[1]) >= snorm(R,p[2]) >= snorm(R,p[3])
true

julia> snorm(R,1.0) == snorm(R,1) == trnorm(R) == nucnorm(R)
true```

## Dependencies

SCS.jl and Convex.jl, for the completely bounded norms.

## TODO

• The implementation of the completely bounded 1 and ∞ norms is somewhat tailored to transformations between isomorphic spaces. It should be easy to make it more general.

• The distance between two quantum channels (i.e., trace preserving, completely positive linear maps of operators) is "easier" to compute than the completely bounded 1 norm (the diamond norm). Adding a function just for that would be nice.

• The diamond norm distance between two unitary maps is also much easier to compute -- see, e.g., Lecture 20 for John Watrous's 2011 Quantum Information course -- so a customized function would be nice.

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