Code Subspaces and Approximate Bulk Operators

A semiclassical code subspace is a restricted linear space of bulk-effective states on which one boundary operator can represent a bulk observable. Reconstruction is an intertwining relation on that domain, exact or bounded in a named norm. It is not equality of operators on the full microscopic Hilbert space. The restriction permits redundant boundary representations without duplicating an independent quantum system.

Required background. Quantum Information supplies isometric encodings, operator norms, erasure correction, and logical algebras; Gauge Theory and String-Theory Preparation supplies gauge constraints and the holographic boundary setting.

Helpful background. Semiclassical Gravity supplies the $1/N$ or $G$ expansion; Quantum Fields in Curved Spacetime supplies effective bulk observables; Black-Hole Information supplies entanglement wedges and islands.

Encodings and logical operators

Assume a finite-dimensional regulator. Let $\mathcal H_{\mathrm{code}}$ contain the selected semiclassical states and let

$$V:\mathcal H_{\mathrm{code}} \longrightarrow \mathcal H_{\mathrm{phys}}$$

be an isometry, $V^\dagger V=I_{\mathrm{code}}$. The code projector is $P=VV^\dagger$. For a logical operator $O$ on the code, a physical representative $\widetilde O$ obeys

$$\widetilde O V=VO,$$

which implies the compressed relation

$$P\widetilde O P=VOV^\dagger.$$

The converse requires code preservation, $(I-P)\widetilde O P=0$; without it, the compressed equation permits an unphysical leakage component. Subject to the full intertwining relation, only the action of $\widetilde O$ on encoded inputs is fixed. Its action on $(I-P)\mathcal H_{\mathrm{phys}}$ is arbitrary, so multiple physical operators can represent the same logical operator on the code.

In holography, $\mathcal H_{\mathrm{code}}$ is chosen around a common semiclassical background with bounded excitations and backreaction. A bulk effective-field-theory operator is logical; different boundary regions may support different representatives. This interpretation and its erasure-code examples are developed in Almheiri, Dong, and Harlow 2015, §§2–4.

Approximate equality and the error domain

An approximate representative needs a norm and state family. One useful operator-norm statement is

$$\left\lVert\widetilde O V-VO\right\rVert_{\mathrm{op}} \le\varepsilon.$$

It implies

$$\left\lVert (\widetilde O V-VO)\lvert\psi\rangle \right\rVert \le\varepsilon$$

for every normalized $\lvert\psi\rangle\in\mathcal H_{\mathrm{code}}$. A weaker state-dependent bound may hold only for a set $\mathcal S$:

$$\sup_{\lvert\psi\rangle\in\mathcal S} \left\lVert (\widetilde O V-VO)\lvert\psi\rangle \right\rVert \le\varepsilon_{\mathcal S}.$$

The second does not control arbitrary superpositions outside $\mathcal S$. Nor does a small state-vector error automatically bound long products of high-norm operators; errors can accumulate with circuit depth. The gravitational order, such as $O(1/N)$ or $O(e^{-S})$, must be separated from numerical or operational error.

For a bounded observable, the intertwining error controls matrix elements. If $\lvert\phi\rangle$ and $\lvert\psi\rangle$ are normalized code states, Cauchy–Schwarz gives

$$\left\lvert \langle\phi\rvert V^\dagger\widetilde O V\lvert\psi\rangle -\langle\phi\rvert O\lvert\psi\rangle \right\rvert \le\varepsilon.$$

This includes off-diagonal coherences, unlike a check on expectation values for a preferred basis alone. For two approximate representatives, the product error also depends on their norms; schematically it is bounded by $\varepsilon_1\lVert O_2\rVert+\varepsilon_2\lVert\widetilde O_1\rVert$. Consequently, a useful code claim states an operator class and norm cap, not only a per-generator tolerance.

Operator-algebra codes replace the full logical matrix algebra by a von Neumann algebra $\mathcal M$. Central observables can be represented on both complementary regions. This is the natural setting for fluctuating area or charge sectors; it avoids forcing a gauge-constrained Hilbert space into a naïve tensor factor. The progression from conventional and subsystem erasure correction to operator-algebra correction, together with its holographic role, is developed in Harlow 2017, §§3–5.

A three-qutrit erasure code

Consider

$$\begin{aligned} \lvert0_L\rangle&=\frac{\lvert000\rangle+\lvert111\rangle+\lvert222\rangle}{\sqrt3},\\ \lvert1_L\rangle&=\frac{\lvert012\rangle+\lvert120\rangle+\lvert201\rangle}{\sqrt3},\\ \lvert2_L\rangle&=\frac{\lvert021\rangle+\lvert102\rangle+\lvert210\rangle}{\sqrt3}. \end{aligned}$$

For any logical basis states $i,j$, tracing out either pair of qutrits gives

$$\operatorname{Tr}_{23}\lvert i_L\rangle\langle j_L\rvert =\delta_{ij}\frac{I_3}{3},$$

and the analogous identity holds for qutrit 2 or 3. One erased qutrit therefore contains no logical information, so the other two admit a recovery channel and representatives of every logical operator. This is an explicit complementary-channel check, not merely dimension counting.

To see the trace identity, inspect the three terms in each codeword. In any chosen physical position, the symbols $0,1,2$ occur exactly once, so diagonal logical matrix elements reduce to $I_3/3$. For $i\ne j$, no pair of symbols on the traced two positions coincides between the two codewords, so every off-diagonal partial trace vanishes. Linearity then shows that the erased qutrit has state $I_3/3$ for every logical density operator, including superpositions.

The code also shows redundancy without cloning. Representatives on different pairs agree on the code but are not independent copies; they act on overlapping physical degrees of freedom and can differ outside $P$.

Recovery is visible directly in this example. Label codeword terms modulo three as $\lvert x,x+i,x+2i\rangle/\sqrt3$ summed over $x$. On qutrits 2 and 3, the reversible change of basis $\lvert y,z\rangle\mapsto\lvert z-y,2y-z\rangle$ sends the encoded state to

$$\lvert i\rangle_{\mathrm{logical}} \otimes\frac1{\sqrt3}\sum_{x=0}^{2} \lvert x\rangle_1\lvert x\rangle_{\mathrm{aux}}.$$

Linearity gives the same factorization for any superposition of logical labels. The retained pair therefore contains the recoverable qutrit and an auxiliary system entangled with the erased qutrit. This is an explicit decoder, not only an entropy argument.

Proof architecture and limitations

Under the hypotheses above, exact reconstruction on a physical region follows if its complementary channel is constant on the logical algebra: the environment cannot distinguish code states or coherences relevant to that algebra. Stinespring uniqueness then supplies a recovery isometry on the retained region. The detailed complementary-recovery theorem is the next page.

The logical algebra must be fixed before this test. If only a commuting subalgebra is protected, the complementary channel may retain information about central labels while losing all phase information; that is classical rather than full quantum recovery. If the full matrix algebra is protected, the environment state must be independent of every logical density operator. Confusing these cases can turn a correct area-sector statement into a false claim of complete bulk locality.

Approximate encodings introduce a further distinction. Perturbing $V$ by a small operator can preserve inner products on a chosen finite list of states while producing a larger operator-norm error on their span. Random overlaps may be small with high probability yet fail under an adversarial superposition. A stated error probability over random states is therefore not interchangeable with a worst-case norm bound.

A discriminating counterexample comes from leaving the code. If $X$ acts entirely in $(I-P)\mathcal H_{\mathrm{phys}}$, then $\widetilde O$ and $\widetilde O+X$ have identical action on every encoded state. On a superposition containing an outside-code vector, they disagree. Thus the intertwining relation cannot determine a state-independent full-Hilbert-space operator.

Other limitations are a code dimension large enough to alter the background, gravitational dressing that changes the algebra, a center omitted from the logical specification, or an error bound stated only in expectation value when channel recovery is claimed. Generic gauge theory and explicit AdS/CFT construction remain prerequisites rather than results of this page.

Exercises

Let $\widetilde O$ represent $O$ exactly on code projector $P$, and let $X=(I-P)X(I-P)$. Show that $\widetilde O+X$ is another representative. Then take a normalized state $\lvert\Phi\rangle=(V\lvert\psi\rangle+\lvert\chi\rangle)/\sqrt2$ with $P\lvert\chi\rangle=0$ and explain when the two representatives are distinguishable.

Solution

Because $(I-P)V=0$, one has $XV=0$ and therefore $(\widetilde O+X)V=\widetilde OV=VO$. Both operators represent $O$ on the code. Acting on $\lvert\Phi\rangle$, their outputs differ by $X\lvert\chi\rangle/\sqrt2$. They are distinguishable whenever $X\lvert\chi\rangle\ne0$. The example proves that a representative’s equality is domain restricted, even when its code action is exact.

References

• Almheiri, Dong, and Harlow 2015, Ahmed Almheiri, Xi Dong, and Daniel Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” Journal of High Energy Physics 04 (2015) 163.
• Harlow 2017, Daniel Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction,” Communications in Mathematical Physics 354 (2017) 865–912.