When Can Errors Be Undone?

Quantum Error Correction

Juan Pablo Yamamoto Zazueta

The Setup

Key Concept

Entanglement Fidelity

Fₑ = ⟨Ψ(RQ)| ρ″(RQ) |Ψ(RQ)⟩

          Fₑ = 1  :  |Ψ(RQ)⟩ perfectly recovered

          Fₑ < 1  :  Entanglement partially or fully lost

Classical Setup

The Markov Chain & Data Processing

Perfect error correction is possible when
H(X:Y) = H(X) ⟺ H(X|Y) = 0

Information lost cannot be recovered afterwards

Classical Setup

Fano's Inequality

H(X|X̂) ≤ 1 + Pᴇ log N

H(X|Y) > 0

Pᴇ bounded away from 0

Imperfect recovery

⟺

H(X|Y) = 0

Bob can design decoder
with Pᴇ = 0
Perfect recovery

          Pᴇ = 0  ⟺  H(X|Y) = 0  ⟺  H(X:Y) = H(X)
        

Measuring Information

Coherent Information

The right measure for quantum information

I(Q;R) = S(R:Q) − S(R) = S(Q) − S(RQ)

Mutual information

S(R:Q) = 2S(Q)

double-counts

Coherent information

I(Q;R) = S(Q) ✓

for |Ψ(RQ)⟩ pure

            Range:  −S(Q)  ≤  I(Q;R)  ≤  +S(Q)
          

Coherent Information

Why Mutual Information Fails

Both circles count the same entanglement, once from each side

S(R:Q) = S(R) + S(Q) − S(RQ)

for a pure |Ψ(RQ)⟩

                S(RQ) = 0

                S(R) = S(Q)

∴ S(R:Q) = 2S(Q) ✕

the problem

We are counting the information carried by R twice, while it is supposed to be just a bystander.

Coherent Information

Subtract the Constant

I(Q;R) = S(R:Q) − S(R) = S(Q) − S(RQ)

              value for pure |Ψ(RQ)⟩. Target is S(Q)
            

                S(R:Q)
              

2S(Q) ✕

                S(Q)
              

target

                I(Q;R)
              

= S(Q) ✓

Coherent Information

A Quantum Signature

classical analog

                −H(X|Y) ≤ 0   always

                There is no notion of "order", information is either present or absent.

quantum setting: sign

                I > 0: S(Q) > S(RQ)

                Entanglement survives: the joint state is more ordered than Q alone.

                I < 0: S(RQ) > S(Q)

                Entanglement is lost: the joint state is more disordered.

Quantum Error Correction Condition

              I(Q′;R) = S(Q)
            

No coherent information may be lost in the channel

Example

A Concrete Code

Code for three-qubit phase-flip

ℰ(ρ) = (1−p)ρ + p ZρZ (per qubit)
Z|+⟩ = |−⟩ Z|−⟩ = |+⟩

codewords

                |0𞁞⟩ = |+++⟩
|1𞁞⟩ = |−−−⟩
              

logical state

                |ψʟ⟩ = α|0𞁞⟩ + β|1𞁞⟩

                α and β are the relevant information

Example

The Noise Channel & Codewords

the channel (per qubit)

                ℰ(ρ) = (1−p)ρ + p ZρZ
              

Z in the two bases

                    standard basis
                  

                    Z|0⟩ =  |0⟩
Z|1⟩ = −|1⟩
                  

|0⟩, |1⟩ unaffected

                    conjugate basis
                  

                    Z|+⟩ = |−⟩
Z|−⟩ = |+⟩
                  

Acts as a bit flip

We reduce the problem of decoherence in the standard basis to detecting bit flip errors in another basis.

code subspace

                |0𞁞⟩ = |+ + +⟩
|1𞁞⟩ = |− − −⟩
              

logical state

                |ψʟ⟩ = α|0𞁞⟩ + β|1𞁞⟩
              

                α and β are the amplitudes carrying the relevant quantum information
              

code subspace projector

                Πc = |0𞁞⟩⟨0𞁞| + |1𞁞⟩⟨1𞁞|
              

2-dimensional subspace of ℋ⊗3 ≅ ℂ⁸

Example

Four Orthogonal Subspaces

Π𞁞 ⊥ Z⁽¹⁾Π𞁞Z⁽¹⁾ ⊥ Z⁽²⁾Π𞁞Z⁽²⁾ ⊥ Z⁽³⁾Π𞁞Z⁽³⁾

dimension count

                4 subspaces × dim 2 = 8
= dim ℋ⊗3 ✓
              

direct sum decomposition

                ℋ⊗ℋ⊗ℋ = Π𞁞

                           ⊕ Z⁽¹⁾Π𞁞Z⁽¹⁾

                           ⊕ Z⁽²⁾Π𞁞Z⁽²⁾

                           ⊕ Z⁽³⁾Π𞁞Z⁽³⁾

The state after noise is in exactly one of the four subspaces, which are distinguishable by orthogonality.

Example

Error Detection & Correction

                error
              

                operator
              

                correction
              

                None
              

                𝟙
              

                D₀ = Π𞁞
              

                Qubit 1
              

                Z⁽¹⁾
              

                D₁ = Π𞁞Z⁽¹⁾
              

                Qubit 2
              

                Z⁽²⁾
              

                D₂ = Π𞁞Z⁽²⁾
              

                Qubit 3
              

                Z⁽³⁾
              

                D₃ = Π𞁞Z⁽³⁾
              

error identification projectors

                {Π𞁞, Z⁽¹⁾Π𞁞Z⁽¹⁾, Z⁽²⁾Π𞁞Z⁽²⁾, Z⁽³⁾Π𞁞Z⁽³⁾}
              

key point

The error identification projectors tell us which subspace the state is in, not the amplitudes α and β that it carries

                It can be implemented as a single measurement without the need for cloning (ancilla qubits)
              

Fₑ = 1 after 𝒟(ℰ(|ψʟ⟩⟨ψʟ|))

The Connection

Information & Isolation

What does the environment learn?

ρ⁽ᴿᴱ⁾′ = ρ⁽ᴿ⁾ ⊗ σ⁽ᴱ⁾′

R and E have no correlation

The state of the environment is independent of Q's input

equivalent to

                I(Q′;R) = S(Q)
              

The environment learns nothing if and only if the channel is invertible.

Information & Isolation

The Environment Learns Nothing

the conceptual trick

R can be measured first because it does not participate in the interaction. Each conditional Q state |ψα⟩ must have the same effect on E, making them independent.

Perfect quantum error correction is possible if and only if E is the same for every input.

why this works

If E cannot distinguish between Q states, it has learned nothing and the interaction can be undone.

Information & Isolation

A Wonderful Theorem

ρ⁽ᴿᴱ⁾′ = ρ⁽ᴿ⁾ ⊗ σ⁽ᴱ⁾′

                Perfect Error Correction = Perfect Privacy
              

Summary

When can errors be undone?

              Fₑ = 1
            

Perfect recovery

⟺

              I(Q′;R) = S(Q)
            

No coherent info lost

⟺

              ρ⁽ᴿᴱ⁾′ = ρ⁽ᴿ⁾ ⊗ σ⁽ᴱ⁾′
            

R and E are a product state

⟺

              σα⁽ᴱ⁾′ independent of α
            

E learns nothing

"Quantum mechanics is what happens
when nobody is looking."