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Measurement

Measurement is modeled in two ways:

  • as Positive Operator Valued Measures (POVMs),
  • measurements with post-selection.

In both cases a measurement is treated as a special case of a quantum channel (operation).

Positive Operator Valued Measure measurement

A POVM measurement is defined as follows. Let $\mu:\Gamma\to\mathrm{P}(\mathcal{X})$ be a mapping from a finite alphabet of measurement outcomes to the set of linear positive operators. If $\sum_{\xi\in\Gamma} {\mu(\xi)=\mathbb{I}_{\mathcal{X}}}$ then $\mu$ is a POVM measurement. The set of positive semi-definite linear operators is defined as $\mathrm{P}(\mathcal{X})=\{X\in \mathrm{L}(\mathcal{X}): \langle\psi|X|\psi\rangle\geq 0 \text{ for all } |\psi\rangle\in\mathcal{X}\}$. POVM measurement models the situation where a quantum object is destroyed during the measurement process and quantum state after the measurement does not exists.

We model POVM measurement as a channel $\theta:\mathrm{L}(\mathcal{X})\to \mathrm{L}(\mathcal{Y})$, where $\mathcal{Y}=\mathrm{span}\{|\xi\rangle\}_{\xi\in\Gamma}$ such that $\theta(\rho) = \sum_{\xi\in\Gamma} \mathrm{Tr}(\rho\, \mu(\xi))|\xi\rangle\langle\xi|$. This channel transforms the measured quantum state into a classical state (diagonal matrix) containing probabilities of measuring given outcomes. Note that in QuantumInformation.jl $\Gamma=\{1,2,\ldots,|\Gamma|\}$ and POVM measurements are represented by the type POVMMeasurement{T} <: AbstractQuantumOperation{T} where T<:AbstractMatrix{<:Number}. Predicate function ispovm() verifies whether a list of matrices is a proper POVM.

julia> ρ=proj(1.0/sqrt(2)*(ket(1,3)+ket(3,3)))3×3 Matrix{ComplexF64}:
 0.5+0.0im  0.0+0.0im  0.5+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.5+0.0im  0.0+0.0im  0.5+0.0im
julia> E0 = proj(ket(1,3))3×3 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im
julia> E1 = proj(ket(2,3))+proj(ket(3,3))3×3 Matrix{ComplexF64}:
 0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  1.0+0.0im
julia> M = POVMMeasurement([E0,E1])POVMMeasurement{Matrix{ComplexF64}}
    dimensions: (3, 2)
    ComplexF64[1.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]
    ComplexF64[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 1.0 + 0.0im]
julia> ispovm(M)true
julia> M(ρ)2×2 LinearAlgebra.Diagonal{Float64, Vector{Float64}}:
 0.5   ⋅
  ⋅   0.5

Measurement with post-selection

When a quantum system after being measured is not destroyed one can be interested in its state after the measurement. This state depends on the measurement outcome. In this case the measurement process is defined in the following way.

Let $\mu:\Gamma\to \mathrm{L}(\mathcal{X}, \mathcal{Y})$ be a mapping from a finite set of measurement outcomes to set of linear operators called effects. If $\sum_{\xi\in\Gamma} {\mu(\xi)^\dagger \mu(\xi)=\mathbb{I}_{\mathcal{X}}}$ then $\mu$ is a quantum measurement. Given outcome $\xi$ was obtained, the state before the measurement, $\rho$, is transformed into sub-normalized quantum state $\rho_\xi=\mu(\xi)\rho\mu(\xi)^\dagger$. The outcome $\xi$ will be obtained with probability $\mathrm{Tr}(\rho_\xi)$.

julia> PM = PostSelectionMeasurement(E1)PostSelectionMeasurement{Matrix{ComplexF64}}
    dimensions: (3, 3)
    ComplexF64[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 1.0 + 0.0im]
julia> iseffect(PM)true
julia> PM(ρ)3×3 Matrix{ComplexF64}:
 0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.5+0.0im

In QuantumInformation this kind of measurement is modeled as CP-TNI map with single Kraus operator $\mu(\xi)$ and represented as PostSelectionMeasurement{T} <: AbstractQuantumOperation{T} where T<:AbstractMatrix{<:Number}. Measurement types can be composed and converted to Kraus operators, superoperators, Stinespring representation operators, and dynamical matrices.

julia> α = 0.30.3
julia> K0 = ComplexF64[0 0 sqrt(α); 0 1 0; 0 0 0]3×3 Matrix{ComplexF64}:
 0.0+0.0im  0.0+0.0im  0.547723+0.0im
 0.0+0.0im  1.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im       0.0+0.0im
julia> K1 = ComplexF64[1 0 0; 0 0 0; 0 0 sqrt(1 - α)]3×3 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im      0.0+0.0im
 0.0+0.0im  0.0+0.0im      0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.83666+0.0im
julia> Φ = KrausOperators([K0,K1])KrausOperators{Matrix{ComplexF64}}
    dimensions: (3, 3)
    ComplexF64[0.0 + 0.0im 0.0 + 0.0im 0.5477225575051661 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]
    ComplexF64[1.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.8366600265340756 + 0.0im]
julia> ρ=proj(1.0/sqrt(2)*(ket(1,3)+ket(3,3)))3×3 Matrix{ComplexF64}:
 0.5+0.0im  0.0+0.0im  0.5+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.5+0.0im  0.0+0.0im  0.5+0.0im
julia> (PM∘Φ)(ρ)3×3 Matrix{ComplexF64}:
 0.0+0.0im  0.0+0.0im   0.0+0.0im
 0.0+0.0im  0.0+0.0im   0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.35+0.0im