[ITensors] [BUG] Add check for flux when adding ITen... | ITensor/ITensorMPS.jl

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Description of bug

Currently you are allowed to add two ITensors or MPS which have different flux, which is not well defined. It should throw an error.

See https://itensor.discourse.group/t/how-to-deal-with-entangled-state-of-different-quantum-number/94.

Minimal code demonstrating the bug or unexpected behavior

Minimal runnable code

julia> using ITensors

julia> s = siteinds("S=1/2", 3; conserve_qns=true);

julia> A1 = randomITensor(QN(), s[1]', dag(s[1]));

julia> A2 = randomITensor(QN("Sz", 2), s[1]', dag(s[1]));

julia> @show A1;
A1 = ITensor ord=2
Dim 1: (dim=2|id=12|"S=1/2,Site,n=1")' <Out>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1
Dim 2: (dim=2|id=12|"S=1/2,Site,n=1") <In>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1
NDTensors.BlockSparse{Float64, Vector{Float64}, 2}
 2×2
Block(1, 1)
 [1:1, 1:1]
 -0.7467320137298004

Block(2, 2)
 [2:2, 2:2]
 -0.2159039892715488

julia> @show A2;
A2 = ITensor ord=2
Dim 1: (dim=2|id=12|"S=1/2,Site,n=1")' <Out>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1
Dim 2: (dim=2|id=12|"S=1/2,Site,n=1") <In>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1
NDTensors.BlockSparse{Float64, Vector{Float64}, 2}
 2×2
Block(1, 2)
 [1:1, 2:2]
 -0.5224630234648842

julia> @show A1 + A2;
A1 + A2 = ITensor ord=2
Dim 1: (dim=2|id=12|"S=1/2,Site,n=1")' <Out>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1
Dim 2: (dim=2|id=12|"S=1/2,Site,n=1") <In>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1
NDTensors.BlockSparse{Float64, Vector{Float64}, 2}
 2×2
Block(1, 1)
 [1:1, 1:1]
 -0.7467320137298004

Block(2, 2)
 [2:2, 2:2]
 -0.2159039892715488

Block(1, 2)
 [1:1, 2:2]
 -0.5224630234648842

julia> psi1 = MPS(s, ["Dn", "Dn", "Dn"]);

julia> psi2 = MPS(s, ["Up", "Dn", "Dn"]);

julia> psi1 + psi2
MPS
[1] ((dim=2|id=12|"S=1/2,Site,n=1") <Out>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1, (dim=1|id=84|"Link,l=1") <Out>
 1: QN("Sz",-2) => 1)
[2] ((dim=2|id=494|"S=1/2,Site,n=2") <Out>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1, (dim=1|id=155|"Link,l=2") <Out>
 1: QN("Sz",-1) => 1, (dim=1|id=84|"Link,l=1") <In>
 1: QN("Sz",-2) => 1)
[3] ((dim=2|id=221|"S=1/2,Site,n=3") <Out>
 1: QN("Sz",1) => 1
 2: QN("Sz",-1) => 1, (dim=1|id=155|"Link,l=2") <In>
 1: QN("Sz",-1) => 1)

julia> flux(psi1)
QN("Sz",-3)

julia> flux(psi2)
QN("Sz",-1)

Version information

Output from versioninfo():

julia> versioninfo()
Julia Version 1.7.2
Commit bf53498635 (2022-02-06 15:21 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Xeon(R) E-2176M  CPU @ 2.70GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, skylake)
Environment:
  JULIA_EDITOR = vim

Output from using Pkg; Pkg.status("ITensors"):

julia> using Pkg; Pkg.status("ITensors")
      Status `~/.julia/environments/v1.7/Project.toml`
  [9136182c] ITensors v0.3.10

Start from the ITensor and MPS addition paths shown by the minimal Julia example, then inspect how flux is obtained for each operand. Reproduce additions with mismatched and matching flux values; done when mismatched ITensors and MPS throw an error while valid additions remain supported.

Written by the indexing model from the issue text.

[ITensors] [BUG] Add check for flux when adding ITensors and MPS

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