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.

License

Apache Lincense 2.0 (summary)

Copyright

Raytheon BBN Technologies 2015

Authors

Marcus P da Silva (@marcusps)