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Appendix: Proof of Eq. (3)
In this appendix, we show Eq. (3). Due to the triangle inequality and the invariance of the trace norm under a
unitary operation,
3m 3m
1 σ Ψ′ Ψ′ = 1 σ XxjZzj P ψ 0 ⊗m + ⊗m P† XxjZzj
2 (cid:13) (cid:13) −| ih | (cid:13) (cid:13)1 2(cid:13) (cid:13) (cid:13) − (cid:0)Oj=1 j j (cid:1) (cid:0) ⊗ ⊗ (cid:1) (cid:0)Oj=1 j j (cid:1)(cid:13) (cid:13) (cid:13)1
3m 3m
= 1 P† XxjZzj σ XxjZzj P ψ 0 ⊗m + ⊗m
2(cid:13) (cid:13) (cid:0)Oj=1 j j (cid:1) (cid:0)Oj=1 j j (cid:1) − ⊗ ⊗ (cid:13) (cid:13)1
(cid:13) (cid:13)
3m 3m
1
P† XxjZzj σ XxjZzj P ρ
≤ 2(cid:13) (cid:13) (cid:0)Oj=1 j j (cid:1) (cid:0)Oj=1 j j (cid:1) − before (cid:13) (cid:13)1
(cid:13)1 (cid:13)
⊗m ⊗m
+ ρ ψ 0 + ,
before
2(cid:13) − ⊗ ⊗ (cid:13)1
(cid:13) (cid:13)
(cid:13) (cid:13)
where ρ is the state before measuring Λ ,Λ .
before 0 1
{ }
From the monotonicity of the trace distance under a CPTP map, the first term is upperbounded as
3m 3m
1 1
P† XxjZzj σ XxjZzj P ρ G ρ √2ǫ.
2(cid:13) (cid:13) (cid:0)Oj=1 j j (cid:1) (cid:0)Oj=1 j j (cid:1) − before (cid:13) (cid:13)1 ≤ 2k σ − k1 ≤
(cid:13) (cid:13)
As is shown below, the second term is upperbounded as
1 ⊗m ⊗m 2
ψ 0 + ρ +ǫ. (A.1)
2(cid:13) ⊗ ⊗ − before (cid:13)1 ≤ r3
(cid:13) (cid:13)
(cid:13) (cid:13)
6
Therefore, we have shown Eq. (3).
Let us show Eq. (A.1). Note that
ρ = 1 1 P†σ E σ P ψ 0 ⊗m + ⊗m P†σ E†σ P,
before (3m)!43m α k α ⊗ ⊗ α k α
PX,α,k (cid:0) (cid:1)
where σ is a 3m-qubit Pauli operator,and E is a Kraus operator. Let us decompose each Kraus operator in terms
α k
of Pauli operators as E = Ckσ . Since
k β β β
P
I = E†E
k k
Xk
= Ck∗Ckσ σ
β γ β γ
kX,β,γ
= Ck 2I + Ck∗Ckσ σ ,
| β| β γ β γ
Xk,β kX,β6=γ
we obtain
Ck 2 =1.
| β|
Xk,β
Then,
ρ = 1 1 CkCk∗P†σ σ σ P ψ 0 ⊗m + ⊗m P†σ σ σ P
before (3m)!43m β γ α β α ⊗ ⊗ α γ α
P,αX,k,β,γ (cid:0) (cid:1)
= 1 Ck 2P†σ P ψ 0 ⊗m + ⊗m P†σ P
(3m)! | β| β ⊗ ⊗ β
PX,k,β (cid:0) (cid:1)
= 1 D P†σ P ψ 0 ⊗m + ⊗m P†σ P
β β β
(3m)! ⊗ ⊗
XP,β (cid:0) (cid:1)
= D ψ 0 ⊗m + ⊗m + 1 D P†σ P ψ 0 ⊗m + ⊗m P†σ P
0 β β β
⊗ ⊗ (3m)! ⊗ ⊗
(cid:0) (cid:1) PX,β6=0 (cid:0) (cid:1)
ρ +ρ .