FR09-C · A state spread over five sites

Boundary Quantum Error Correction

How much of an encoded quantum state can be erased and still recovered? This five-qubit code recovers an arbitrary input after any two known erasures. The surviving three sites preserve both populations and relative phase. If only two sites survive, their state carries no information about the input.

Erase sites around a five-qubit encoding and compare the recovered state with the input. Change the input phase to test whether the quantum information survives.

Recovery fidelity $1.00$

Surviving sites $3$

Recovered purity $1.00$

Erasure-condition residual $0$

Model assumptions

An exact five-qubit stabilizer code encodes one logical qubit. Erasure means tracing out a known set of sites. Recovery uses the Petz transpose channel with a maximally mixed logical prior. The diagram represents one finite perfect tensor, with no spatial metric or gravitational dynamics.

Exact finite-code calculation

Compare all three state components

Gold is the input; teal is the recovered state. The three Pauli expectation values specify a qubit, including its relative phase.

Recovered density matrix and recovery condition

\rho_{\rm recovered}=\begin{pmatrix}0.704&0.343-0.301i\\0.343+0.301i&0.296\end{pmatrix}

$A_e^\dagger A_f=c_{ef}I$

The residual measures the largest failure of this erasure-correction condition for the displayed channel. Zero allows one recovery map to preserve every logical state, including entanglement with an external reference.

Start with the essentials

Focus question: How much of an encoded quantum state can be erased and still recovered?
One-sentence intuition: Three surviving sites jointly preserve the logical qubit, although any pair alone contains no information about it.

Core mathematical model

An isometric encoding

\begin{aligned}V|\psi\rangle&=\cos\frac{\vartheta}{2}|0_L\rangle\\&\quad+e^{i\varphi}\sin\frac{\vartheta}{2}|1_L\rangle\\V^\dagger V&=I\end{aligned}

The two logical codewords are orthonormal five-qubit states. The polar angle sets the populations; the azimuth sets the relative phase.

Erase by tracing out a site

\begin{aligned}\mathcal N_E(\rho)&=\operatorname{Tr}_E(V\rho V^\dagger)\\&=\sum_e A_e\rho A_e^\dagger\end{aligned}

The erased outcomes are summed over without conditioning on a measurement. Recovery receives only this reduced density matrix and the list of lost positions.

Test the whole logical subspace

\begin{aligned}A_e^\dagger A_f&=c_{ef}I\\\rho_{\rm out}&=\mathcal R_E(\mathcal N_E(|\psi\rangle\langle\psi|))\\F&=\langle\psi|\rho_{\rm out}|\psi\rangle\end{aligned}

The first condition tests whether the erased subsystem can distinguish logical states. When it holds, a recovery channel preserves every input and its entanglement with a reference. Fidelity compares the displayed recovered state with the chosen pure input.

Common difficulties

The information is shared across sites

Typical misconceptionEach site holds a spare copy of the input.

Better mental modelAny one or two sites have an input-independent state. The logical information is in joint correlations. Two disjoint groups of three sites cannot fit into a five-site code, so the recovery regions cannot make two independent copies.

Known erasure differs from unknown damage

Typical misconceptionRecovering two erased sites means any two unknown errors can also be corrected.

Better mental modelErasure tells the decoder which positions are missing. This distance-three code corrects two known erasures or one arbitrary unknown single-qubit error.

Run the experiment

01
Start with two lost sites
Predict whether the relative phase survives. Choose Two sites lost, then drag the gold dial.
What to observe: All three recovered components follow the input. The survivors preserve a superposition as well as a basis state.
02
Lose one more
Choose Too little survives, then move both input sliders.
What to observe: The input bars move, but every recovered component stays zero. With only two sites, the recovery returns the maximally mixed logical state and fidelity is one half.
03
Try another erased pair
Reset. Click the erased sites to restore them, then erase a different pair. Save a simulation record.
What to observe: Every pair gives exact recovery. The file contains the encoded complex amplitudes, surviving density matrix and recovered result for that input.

Boundary Quantum Error Correction

Compare all three state components