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Tensor-product channel error bound

Statement

For a positive integer $L$ and a finite family of matching channel pairs $Φ_{j}, Ψ_{j} : L (H_{j}) \to L (K_{j})$ , $j = 1, \dots, L$ , on nonzero finite-dimensional complex Hilbert spaces, $d_{⋄} (⨂_{j = 1}^{L} Φ_{j}, ⨂_{j = 1}^{L} Ψ_{j}) \leq \sum_{j = 1}^{L} d_{⋄} (Φ_{j}, Ψ_{j}) .$ For every nonzero finite-dimensional reference space $R$ , every density operator $ρ$ on $(⨂_{j = 1}^{L} H_{j}) \otimes R$ , and every finite POVM on $(⨂_{j = 1}^{L} K_{j}) \otimes R$ , the total variation distance between the outcome laws after applying $(⨂_{j} Φ_{j}) \otimes {id}_{R}$ and $(⨂_{j} Ψ_{j}) \otimes {id}_{R}$ to $ρ$ is at most the same sum. No product-input assumption is made; $ρ$ may be entangled across the component inputs and reference.

Facts & Assumptions

Given: The finite channel pairs, product spaces, nonzero finite reference space $R$ , input density operator $ρ$ , and final finite POVM specified in the Statement.

[F1]

End-to-end channel and output error contract: $d_{⋄}$ and $D$ each include a factor $1 / 2$ ; the raw diamond norm does not.

[F2]

Channel Tensor Error Bound: for two matching channel pairs, the raw diamond distance between tensor products is at most the sum of the raw local diamond errors; the diamond norm is multiplicative under tensor products.

[F3]

Finite-dimensional channels compose and tensor: tensor products and compositions of finite-dimensional channels are channels.

[F4]

Hybrid bound for a composed channel circuit: for matching finite-dimensional channels, every finite reference input and final POVM has outcome total variation at most the normalized diamond distance between the channels.

Proof

1.1basegivenF1algebra

For $L = 1$ the channel inequality is equality by [F1].

1.2ihF1F2F3algebra

Assume the bound for $L$ factors. By [F3], the first $L$ channel factors in each tensor product form channels. Apply [F2] to this prefix channel pair and the final channel pair. Dividing the raw diamond inequality by $2$ gives the normalized distance of the full tensor products at most the prefix normalized distance plus $d_{⋄} (Φ_{L + 1}, Ψ_{L + 1})$ . The induction hypothesis bounds the prefix term by the first $L$ summands.

2.1step 1.1step 1.2discharge-induction: step 1.2

Steps 1.1 and 1.2 prove the channel inequality for every finite $L$ by induction.

3.1givenF3F4step 2.1discharge-induction: step 2.1∎

By [F3], the product maps $Φ : = ⨂_{j} Φ_{j}$ and $Ψ : = ⨂_{j} Ψ_{j}$ are channels. Apply [F4] to the one-stage pair $Φ, Ψ$ , with the declared reference $R$ , input $ρ$ , and final POVM. Its outcome total variation is at most $d_{⋄} (Φ, Ψ)$ , which step 2.1 bounds by $\sum_{j} d_{⋄} (Φ_{j}, Ψ_{j})$ .

Depends on

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Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)