Lean, informalized

Nielsen & Chuang · Chapter 8, Quantum noise and quantum operations · p. 381

Exercise 8.23 - Amplitude damping on both modes of a dual-rail qubit, and operation elements for it on the code

A damped dual-rail qubit stays intact or leaves the code.

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The Exercise

A dual-rail qubit is carried by two modes that share one excitation. Its code is spanned by |01⟩|01\rangle and |10⟩|10\rangle, with the excitation in the second mode or in the first, and a logical state is a|01⟩+c|10⟩a|01\rangle + c|10\rangle with |a|2+|c|2=1|a|^2 + |c|^2 = 1. Each mode undergoes amplitude damping with the same parameter γ\gamma, independently of the other. The exercise asks what becomes of the encoded state, and which operation elements describe the process on the code.

Two Damped Modes

Each mode is a qubit with state space HH and computational basis |0⟩|0\rangle, |1⟩|1\rangle. The projector PiP_i onto the basis vector |i⟩|i\rangle is the operator |i⟩⟨i||i\rangle\langle i| on HH.✦ The lowering operator σ−\sigma_- is the operator |0⟩⟨1||0\rangle\langle 1| on HH, which takes the excited state |1⟩|1\rangle to the ground state |0⟩|0\rangle and annihilates |0⟩|0\rangle.✦

The operation elements E0E_0 and E1E_1 of amplitude damping with parameter γ\gamma on one mode are the operators E0=P0+1−γP1,E1=γσ−E_0 = P_0 + \sqrt{1-\gamma}\,P_1, \qquad E_1 = \sqrt{\gamma}\,\sigma_-(1) on HH. The element E0E_0 describes the case in which no quantum of energy is lost to the environment, and E1E_1 the case in which one is.

The two modes together have state space H⊗HH\otimes H, with basis vectors |jk⟩=|j⟩⊗|k⟩|jk\rangle = |j\rangle\otimes|k\rangle. The tensor product E⊗FE\otimes F of operators EE and FF on HH is the operator on H⊗HH\otimes H that applies EE to the first mode and FF to the second: (E⊗F)(|u⟩⊗|v⟩)=E|u⟩⊗F|v⟩(E\otimes F)\bigl(|u\rangle\otimes|v\rangle\bigr) = E|u\rangle\otimes F|v\rangle(2)✦ for all |u⟩|u\rangle and |v⟩|v\rangle in HH.

The amplitude damping channel ℰAD\mathcal{E}_{AD} with parameter γ\gamma takes an operator ρ\rho on HH to E0ρE0†+E1ρE1†E_0\rho E_0^\dagger + E_1\rho E_1^\dagger. The dual-rail amplitude damping channel ℰAD⊗ℰAD\mathcal{E}_{AD}\otimes\mathcal{E}_{AD} with parameter γ\gamma is amplitude damping applied to each mode separately: the map that takes an operator ρ\rho on H⊗HH\otimes H to (ℰAD⊗ℰAD)(ρ)=∑j,k=01(Ej⊗Ek)ρ(Ej⊗Ek)†.(\mathcal{E}_{AD}\otimes\mathcal{E}_{AD})(\rho) = \sum_{j,k=0}^{1} (E_j\otimes E_k)\,\rho\,(E_j\otimes E_k)^\dagger .(3)✦

The dual-rail codeword |ψ⟩|\psi\rangle with complex amplitudes aa and cc is the vector |ψ⟩=a|01⟩+c|10⟩|\psi\rangle = a\,|01\rangle + c\,|10\rangle(4)✦ of H⊗HH\otimes H.

Operation Elements on the Code

The no-loss element E0drE_0^{dr} with parameter γ\gamma is the multiple of the identity II on H⊗HH\otimes H given by E0dr=1−γI.E_0^{dr} = \sqrt{1-\gamma}\; I .(5)✦ The loss elements F01F_{01} and F10F_{10} are the operators F01=γ|00⟩⟨01|,F_{01} = \sqrt{\gamma}\,|00\rangle\langle 01| ,(6) F10=γ|00⟩⟨10|F_{10} = \sqrt{\gamma}\,|00\rangle\langle 10|(7) on H⊗HH\otimes H. The combined error element E1drE_1^{dr} is the sum of the two loss elements: E1dr=F01+F10=γ(|00⟩⟨01|+|00⟩⟨10|).E_1^{dr} = F_{01} + F_{10} = \sqrt{\gamma}\,\bigl(|00\rangle\langle 01| + |00\rangle\langle 10|\bigr) .(8)✦

The Claim

Theorem 1. Let aa and cc be complex amplitudes. The state |00⟩|00\rangle is orthogonal to the codeword |ψ⟩|\psi\rangle: ⟨00|ψ⟩=0.\langle 00|\psi\rangle = 0 .(9)✦ If moreover |a|2+|c|2=1|a|^2 + |c|^2 = 1 and 0≤γ≤10 \le \gamma \le 1, damping both modes takes the state of the codeword to (ℰAD⊗ℰAD)(|ψ⟩⟨ψ|)=(1−γ)|ψ⟩⟨ψ|+γ|00⟩⟨00|.(\mathcal{E}_{AD}\otimes\mathcal{E}_{AD})\bigl(|\psi\rangle\langle\psi|\bigr) = (1-\gamma)\,|\psi\rangle\langle\psi| + \gamma\,|00\rangle\langle 00| .(10)✦

Theorem 2. Let 0≤γ≤10 \le \gamma \le 1 and let aa and cc be complex amplitudes. The operation elements of a process satisfy the completeness relation when the sum of E†EE^\dagger E over them is the identity; the no-loss element and the two loss elements satisfy it on the codeword: ((E0dr)†E0dr+F01†F01+F10†F10)|ψ⟩=|ψ⟩.\bigl((E_0^{dr})^\dagger E_0^{dr} + F_{01}^\dagger F_{01} + F_{10}^\dagger F_{10}\bigr) |\psi\rangle = |\psi\rangle .(11)✦ If moreover γ>0\gamma > 0, the no-loss element and the combined error element violate it on the codeword |01⟩|01\rangle: ((E0dr)†E0dr+(E1dr)†E1dr)|01⟩≠|01⟩.\bigl((E_0^{dr})^\dagger E_0^{dr} + (E_1^{dr})^\dagger E_1^{dr}\bigr)|01\rangle \neq |01\rangle .(12)✦

What Is Checked

The claims are stated over the channel (3), built through the tensor products (2) from the elements (1) of each mode and so from P0P_0, P1P_1 and σ−\sigma_-; over the codeword (4); and over the operators (5) to (8) on the two modes. About these constructions the library proves the two theorems above, for any qubit state space HH with its computational basis.