Advanced Quantum Mechanics

I. KRAUS OPERATORS

A. Show that the two sets of Kraus operators given by {|+X)(+x|,|-x)(-xx|} and {1/√2, σ_x/√2} implement the same CP map. Hint: Consider the action of the Kraus operators on a density operator (expressed in terms of the Bloch vector) and show that both sets transform the state the same way.

B. Let Λ1 and Λ2 be two completely positive trace preserving maps (CPTPMs), and 0 ≤ q ≤ 1. Show that the convex combination Λ₃ = qΛ₁ + (1 - q)Λ₂ is also a completely positive trace preserving map. Hint: Show that Λ₃ satisfies the properties of a CPTPM. There is no need to invoke a Kraus representation.

C. Generalize the above result to any finite set of CPTPMs {Λ_j}_j=1^N and {qj ≥ 0}_j=1^N where ∑ jqj = 1. i.e. show that ∑q_jΛ_j is a CPTPM.

D. Let {K_j}_j=1^N be a set of Kraus operators for a CPTPM. The set {Kj}_j=1^N is related to {K_j}_j=1^N by

K'_j = ∑_ku_jkK_k

where u_jk are (complex) elements of a unitary matrix u.

Show that {K'_j}_j=1^N is a Kraus operator set, i.e. ∑_j(K'_j)^†K'_j = II, and that {K'_j}_j=1^N and {K'_j}_j=1^N implement the same completely positive trace-preserving map, i.e. ∑_jK_jρK^†_j = ∑_jK'_jρ(K'_j)^† for all density operators.

Hint: Use u^†u = II = uu^†, equivalent to ∑_ju*_jmu_jn = δmn. Take care to recognize what symbols represent (complex) numbers, or operators in your calculations.

II. UNIVERSAL NOT

We define UNOT by its action on pure states, Λ^UNOT (|ψ)(ψ|) = |ψ^⊥)(ψ^⊥|

where (ψ|ψ^⊥) = 0

A. Show that Λ^UNOT is a trace preserving positive map and work out its action on the Bloch vector. Hint: De-compose ρ into a pure state mixture and use linearity.

B. We can define Λ^UNOT by its action on an operator basis instead, if O^{^} = ∑_jc_jb^{^}_j is an arbitrary operator expressed in an operator basis b^{^}_j, then Λ^UNOT(O^{^}) = ∑_jc_jΛ^UNOT(b^{^}_j) through linearity. We need to find out how the map acts on the basis elements, and we use the action of the map on states to do so.

If we choose the Pauli basis {σ₀ = II, σ₁ = σ_x, σ₂ = σ_y, σ₃ = σ_z} show that Λ^UNOT(σ_j) = (-1)(1-δ^0,j) σ_j, for j = 0, . . . , 3.

Hint: Consider the action of Λ^UNOT on a pure qubit |ψ)(ψ| = ½(II+r. σ^-), |r| = 1 and what it implies for the transformation of the Pauli operators.  

C. We can define the trivial extension of Λ^UNOT⊗II by its action on an operator basis for two-qubits, {σ_j ⊗ σ_k}_j,k=0³ with 16 elements. What is Λ^UNOT⊗ II{σ_j ⊗ σ_k}?

D. The Bell singlet state is |ψ^-)(ψ^-| = ¼(σ₀ ⊗ σ₀ -∑_j=1³  σ_j⊗σ_j), where |ψ^-) = 1/√2(|01)-|10)).

Express Λ^UNOT⊗II(|ψ^-)(ψ^-|) in the computational basis and show that it is not a positive operator, hence Λ^UNOT is not completely positive.