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POVM simulation

This notebook shows how we can check how much depolarizing noise a qubit positive operator-valued measure (POVM) can take before it becomes simulable by projective measurements. The general method is described in arXiv:1609.06139. The question of simulability by projective measurements boils down to an SDP problem. Eq. (8) from the paper defines the noisy POVM that we obtain subjecting a POVM $\mathbf{M}$ to a depolarizing channel $\Phi_t$:

\[\left[\Phi_t\left(\mathbf{M}\right)\right]_i := t M_i + (1-t)\frac{\mathrm{tr}(M_i)}{d} \mathbb{1}.\]

If this visibility $t\in[0,1]$ is one, the POVM $\mathbf{M}$ is simulable.

We will use Convex.jl to solve the SDP problem.

using Convex, SCS, LinearAlgebra

For the qubit case, a four outcome qubit POVM $\mathbf{M} \in\mathcal{P}(2,4)$ is simulable if and only if

\[M_{1}=N_{12}^{+}+N_{13}^{+}+N_{14}^{+},\]

\[M_{2}=N_{12}^{-}+N_{23}^{+}+N_{24}^{+},\]

\[M_{3}=N_{13}^{-}+N_{23}^{-}+N_{34}^{+},\]

\[M_{4}=N_{14}^{-}+N_{24}^{-}+N_{34}^{-},\]

where Hermitian operators $N_{ij}^{\pm}$ satisfy $N_{ij}^{\pm}\geq0$ and $N_{ij}^{+}+N_{ij}^{-}=p_{ij}\mathbb{1}$, where $i<j$ , $i,j=1,2,3,4$ and $p_{ij}\geq0$ as well as $\sum_{i<j}p_{ij}=1$, that is, the $p_{ij}$ values form a probability vector. This forms an SDP feasibility problem, which we can rephrase as an optimization problem by adding depolarizing noise to the left-hand side of the above equations and maximizing the visibility $t$:

\[\max_{t\in[0,1]} t\]

such that

\[t\,M_{1}+(1-t)\,\mathrm{tr}(M_{1})\frac{\mathbb{1}}{2}=N_{12}^{+}+N_{13}^{+}+N_{14}^{+},\]

\[t\,M_{2}+(1-t)\,\mathrm{tr}(M_{2})\frac{\mathbb{1}}{2}=N_{12}^{-}+N_{23}^{+}+N_{24}^{+},\]

\[t\,M_{3}+(1-t)\,\mathrm{tr}(M_{3})\frac{\mathbb{1}}{2}=N_{13}^{-}+N_{23}^{-}+N_{34}^{+},\]

\[t\,M_{4}+(1-t)\,\mathrm{tr}(M_{4})\frac{\mathbb{1}}{2}=N_{14}^{-}+N_{24}^{-}+N_{34}^{-}\]

.

We organize these constraints in a function that takes a four-output qubit POVM as its argument:

function get_visibility(K)
    noise = real([tr(K[i]) * I(2) / 2 for i in 1:size(K, 1)])
    P = [[ComplexVariable(2, 2) for i in 1:2] for j in 1:6]
    q = Variable(6, Positive())
    t = Variable(1, Positive())
    constraints = Constraint[isposdef(P[i][j]) for i in 1:6 for j in 1:2]
    push!(constraints, sum(q) == 1)
    push!(constraints, t <= 1)
    append!(constraints, [P[i][1] + P[i][2] == q[i] * I(2) for i in 1:6])
    push!(
        constraints,
        t * K[1] + (1 - t) * noise[1] == P[1][1] + P[2][1] + P[3][1],
    )
    push!(
        constraints,
        t * K[2] + (1 - t) * noise[2] == P[1][2] + P[4][1] + P[5][1],
    )
    push!(
        constraints,
        t * K[3] + (1 - t) * noise[3] == P[2][2] + P[4][2] + P[6][1],
    )
    push!(
        constraints,
        t * K[4] + (1 - t) * noise[4] == P[3][2] + P[5][2] + P[6][2],
    )
    p = maximize(t, constraints)
    return solve!(p, SCS.Optimizer; silent = true)
end
get_visibility (generic function with 1 method)

We check this function using the tetrahedron measurement (see Appendix B in arXiv:quant-ph/0702021). This measurement is non-simulable, so we expect a value below one.

function dp(v)
    return I(2) + v[1] * [0 1; 1 0] + v[2] * [0 -im; im 0] + v[3] * [1 0; 0 -1]
end
b = [
    1 1 1
    -1 -1 1
    -1 1 -1
    1 -1 -1
] / sqrt(3)
M = [dp(b[i, :]) for i in 1:size(b, 1)] / 4;
p = get_visibility(M)
Problem statistics
  problem is DCP         : true
  number of variables    : 14 (103 scalar elements)
  number of constraints  : 24 (274 scalar elements)
  number of coefficients : 78
  number of atoms        : 151

Solution summary
  termination status : OPTIMAL
  primal status      : FEASIBLE_POINT
  dual status        : FEASIBLE_POINT
  objective value    : 0.8165

Expression graph
  maximize
   └─ positive variable (id: 182…310)
  subject to
   ├─ PSD constraint (convex)
   │  └─ vcat (affine; real)
   │     ├─ hcat (affine; real)
   │     │  ├─ …
   │     │  └─ …
   │     └─ hcat (affine; real)
   │        ├─ …
   │        └─ …
   ├─ PSD constraint (convex)
   │  └─ vcat (affine; real)
   │     ├─ hcat (affine; real)
   │     │  ├─ …
   │     │  └─ …
   │     └─ hcat (affine; real)
   │        ├─ …
   │        └─ …
   ├─ PSD constraint (convex)
   │  └─ vcat (affine; real)
   │     ├─ hcat (affine; real)
   │     │  ├─ …
   │     │  └─ …
   │     └─ hcat (affine; real)
   │        ├─ …
   │        └─ …
   ⋮

p.optval
0.8165487342796737

This value matches the one we obtained using PICOS.

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