Towards Holography via Quantum Source-Channel Codes

While originally motivated by quantum computation, quantum error correction (QEC) is currently providing valuable insights into many-body quantum physics, such as topological phases of matter. Furthermore, mounting evidence originating from holography research (AdS/CFT) indicates that QEC should also be pertinent for conformal field theories. With this motivation in mind, we introduce quantum source-channel codes, which combine features of lossy compression and approximate quantum error correction, both of which are predicted in holography. Through a recent construction for approximate recovery maps, we derive guarantees on its erasure decoding performance from calculations of an entropic quantity called conditional mutual information. As an example, we consider Gibbs states of the transverse field Ising model at criticality and provide evidence that they exhibit nontrivial protection from local erasure. This gives rise to the first concrete interpretation of a bona fide conformal field theory as a quantum error correcting code. We argue that quantum source-channel codes are of independent interest beyond holography.

Figure 1

For a density matrix $\rho$, Schumacher compression may be performed by a unitary that concentrates the information of ${\rho }^n$ on the last $k\approxeq S(\rho )n$ qubits. As a result of the compression unitary, we may think of the remaining $n-k$ qubits as being set to zero, with high probability, and discarding them leads to minimal loss. Indeed, the compression can also be interpreted as an algorithmic cooling unitary [20] in which the discarded qubits approximate a known pure state. Precisely how small a loss will depend on $\rho$ and the choice of $k$. Conversely, traditional channel coding for a $[[n,k,d]]$ QECC may be performed by a unitary that takes as input $k$ data qubits and $n-k$ ancillary qubits, which are initially set to $|0⟩$. We may view unitary source-channel coding as the result of matching parameters and skipping the intermediate reinitialization of ancillas.

Figure 2

We consider ${\rho }_{BC}={\rho }_n^{(\beta )}$ from Eq. (12). The number of physical qubits is given by $n=|BC|$ and take $\beta =0.2n$ to maintain the global entropy $S_{BC}$ (black diamonds) approximately constant. We plot CMI of Eq. (8), associated to a small subsystem $C$, using blue, red, and green dots, corresponding to $|C|=1$, 2, 3. These provide convincing numerical evidence that the CMI converges quadratically for large $n$. We include $1/n$ (dashed blue) and $6/n^2$ (dashed green) lines as visual guides to the eye as well as Lorenzian fits to the data (solid red); we omit small $n<2|C|$ from fitting.

Figure 3

Illustration of the wormhole geometry, which corresponds to thermofield double state $|{\mathrm{\Psi }}_{\mathrm{TFD}}⟩$. In our interpretation, only ${\mathrm{CFT}}_1\equiv BC$ is physical, whereas ${\mathrm{CFT}}_2\equiv A$ is simply included for illustrative purpose. The bulk regions in green illustrate the entanglement wedge of $B$ [42, 43]; this is the region contained between $B$ and the minimal bulk surface ${\chi }_B$, separating $B$ from its boundary complement $AC$. Left: a small CFT subregion $C$ is lost (red boundary), and the entanglement wedge of $B$ (green) reaches all the way up to the wormhole throat or black hole horizon (blue). This corresponds to accurate reconstruction on $B$ of thermally active degrees of freedom. Right: the CFT region $C$ that has been lost is large (roughly half), and the entanglement wedge of $B$ (green) no longer touches the wormhole throat. In this scenario, it is impossible to recover any of the information associated to the thermally active degrees of freedom.