Lean, informalized

Nielsen & Chuang · Chapter 10, Quantum error-correction · p. 495

Problem 10.1 - Channel equivalence is an equivalence relation, and a recovery for one channel gives one for every equivalent channel

Codes and recoveries carried across equivalent channels.

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The Exercise

Two channels ℰ1\mathcal{E}_1 and ℰ2\mathcal{E}_2 on a quantum system SS are equivalent, written ℰ1∼ℰ2\mathcal{E}_1 \sim \mathcal{E}_2, if there are unitaries UU and VV of SS such that ℰ2(ρ)=Uℰ1(VρV†)U†\mathcal{E}_2(\rho) = U\,\mathcal{E}_1\bigl(V\rho V^\dagger\bigr)\,U^\dagger(1)✦ for every state ρ\rho of SS.

The problem is to show that this is an equivalence relation, and to carry error correction across it: a code on which some recovery undoes the errors of ℰ1\mathcal{E}_1 should give a code on which a recovery undoes those of ℰ2\mathcal{E}_2. When the recovery for ℰ1\mathcal{E}_1 is a projective measurement followed by a unitary chosen according to the outcome, the recovery for ℰ2\mathcal{E}_2 is to be found in the same form.

A Quantum System

A quantum system SS has a finite dimensional complex Hilbert space as its state space.✦ A state ρ\rho of SS is a positive operator of unit trace on the state space of SS.✦ A unitary UU of SS is an operator on the state space of SS with U†U=UU†=1U^\dagger U = UU^\dagger = 1✦; it evolves a state ρ\rho of SS into the state UρU†U\rho U^\dagger.✦ Its adjoint U†U^\dagger is again a unitary of SS✦, and so is the product UVUV with a second unitary VV of SS, which applies VV first and then UU.✦

Channels and Codes

A channel ℰ\mathcal{E} on SS is a map taking each state of SS to a state of SS; a recovery ℛ\mathcal{R} on SS is a map of the same kind. A code is a set CC of states of SS, and a recovery ℛ\mathcal{R} corrects a channel ℰ\mathcal{E} on the code CC if ℛ(ℰ(ρ))=ρfor every ρ∈C.\mathcal{R}\bigl(\mathcal{E}(\rho)\bigr) = \rho \quad \text{for every } \rho \in C.(2)✦ Evolving each state of CC by V†V^\dagger gives the code V†CVV^\dagger C V, the set of states V†ρVV^\dagger \rho V with ρ∈C\rho \in C.

Recovery by Measurement

A measurement on SS with a finite set of outcomes is a family of operators MiM_i on the state space of SS, one for each outcome ii, satisfying the completeness relation ∑iMi†Mi=1\sum_i M_i^\dagger M_i = 1.✦ When the outcome is not read, the measurement takes a state ρ\rho of SS to the unread postmeasurement state ∑iMiρMi†\sum_i M_i \rho M_i^\dagger.✦

Two constructions give new measurements from a measurement with operators MiM_i: conjugating each operator by a unitary UU of SS gives the measurement with operators UMiU†UM_iU^\dagger✦, and following outcome ii by a unitary WiW_i of SS gives the measurement with operators WiMiW_iM_i✦.

A recovery ℛ\mathcal{R} on SS by projective measurement and conditional unitaries consists of a measurement on SS whose operators PiP_i are orthogonal projectors, together with a unitary WiW_i of SS for each outcome ii, applied when ii is found. Its action on a state ρ\rho is the unread state of the measurement with operators WiPiW_iP_i: ℛ(ρ)=∑iWiPiρPiWi†.\mathcal{R}(\rho) = \sum_i W_i P_i\,\rho\,P_i W_i^\dagger .(3)✦

Let ℛ\mathcal{R} be a recovery on SS by projective measurement and conditional unitaries, with projectors PiP_i and unitaries WiW_i, and let UU and VV be unitaries of SS. The transformed recovery ℛ′\mathcal{R}' of ℛ\mathcal{R} is the recovery on SS that measures the projectors UPiU†UP_iU^\dagger and on outcome ii applies the unitary V†WiU†V^\dagger W_i U^\dagger.✦

The Claim

Theorem 1. Equivalence of channels on SS is an equivalence relation: for any channels ℰ\mathcal{E}, ℰ1\mathcal{E}_1, ℰ2\mathcal{E}_2, ℰ3\mathcal{E}_3 on SS, ℰ∼ℰ,ℰ1∼ℰ2⇒ℰ2∼ℰ1,ℰ1∼ℰ2 and ℰ2∼ℰ3⇒ℰ1∼ℰ3.\mathcal{E} \sim \mathcal{E}, \qquad \mathcal{E}_1 \sim \mathcal{E}_2 \Rightarrow \mathcal{E}_2 \sim \mathcal{E}_1, \qquad \mathcal{E}_1 \sim \mathcal{E}_2 \text{ and } \mathcal{E}_2 \sim \mathcal{E}_3 \Rightarrow \mathcal{E}_1 \sim \mathcal{E}_3 .(4)✦

Theorem 2. Let ℰ1\mathcal{E}_1 and ℰ2\mathcal{E}_2 be channels on SS related by unitaries UU and VV as in (1), and let a recovery ℛ1\mathcal{R}_1 correct ℰ1\mathcal{E}_1 on a code CC. Then the recovery σ↦V†ℛ1(U†σU)V\sigma \mapsto V^\dagger \mathcal{R}_1(U^\dagger \sigma U)\,V corrects ℰ2\mathcal{E}_2 on the code V†CVV^\dagger C V: V†ℛ1(U†ℰ2(σ)U)V=σfor every σ∈V†CV.V^\dagger\,\mathcal{R}_1\bigl(U^\dagger\,\mathcal{E}_2(\sigma)\,U\bigr)\,V = \sigma \quad \text{for every } \sigma \in V^\dagger C V.(5)✦

For a recovery ℛ\mathcal{R} of the form (3), expanding the transformed recovery with U†U=1U^\dagger U = 1 gives ℛ′(σ)=V†ℛ(U†σU)V\mathcal{R}'(\sigma) = V^\dagger \mathcal{R}(U^\dagger \sigma U)\,V: the recovery of the previous theorem, performed as a projective measurement followed by conditional unitaries.

Theorem 3. Let ℛ\mathcal{R} be a recovery by projective measurement and conditional unitaries that corrects a channel ℰ1\mathcal{E}_1 on a code CC, and let UU and VV be unitaries of SS. Then the transformed recovery ℛ′\mathcal{R}' corrects the channel ρ↦Uℰ1(VρV†)U†\rho \mapsto U\,\mathcal{E}_1(V\rho V^\dagger)\,U^\dagger on the code V†CVV^\dagger C V: ℛ′(Uℰ1(VσV†)U†)=σfor every σ∈V†CV.\mathcal{R}'\bigl(U\,\mathcal{E}_1(V\sigma V^\dagger)\,U^\dagger\bigr) = \sigma \quad \text{for every } \sigma \in V^\dagger C V.(6)✦

What Is Checked

The three claims are stated over channels and recoveries on a quantum system SS, its states and unitaries, the equivalence (1), correction on a code (2), and recoveries of the form (3) with their transformed recoveries.

For these constructions the library shows that UρU†U\rho U^\dagger and every unread postmeasurement state are states of SS, and that the operators UMiU†UM_iU^\dagger and WiMiW_iM_i again satisfy the completeness relation. The operators UPiU†UP_iU^\dagger of the transformed recovery are again orthogonal projectors, so its measurement is projective.