Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit

Abstract

We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them; replacing classical bits by zero-bits makes teleportation asymptotically reversible. This decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applications to state merging, the quantum channel capacity, entanglement distillation, quantum identification and remote state preparation. The source of these results is the theory of approximate quantum error correction. The action of a quantum channel is reversible if and only if no information is leaked to the environment, a characterization that is useful even in approximate form. However, different notions of approximation lead to qualitatively different forms of quantum error correction in the limit of large dimension. We study the effect of a constraint on the dimension of the reference system when considering information leakage. While the resulting condition fails to ensure that the entire input can be corrected, it does ensure that all subspaces of dimension matching that of the reference are correctable. The size of the reference can be characterized by a parameter α; we call the associated resource an α-bit. Changing α interpolates between standard quantum error correction and quantum identification, a form of equality testing for quantum states. We develop the theory of α-bits, including the applications above, and determine the α-bit capacity of general quantum channels, finding single-letter formulas for the entanglement-assisted and amortised variants.

Learning the Alpha-bits of black holes

“…We begin with a basic review of the concept of universal subspace error correction and α-bits. For more detail see [10]. Because the definitions involved are quite technical, we will begin by illustrating the basic phenomenon we are trying to capture with a relatively simple example.…”

“…The existence of state-dependent operators for all states (but the absence of stateindependent operators) corresponds in the Schrödinger picture to a weakened version of the usual notion of quantum error correction, called universal subspace quantum error correction. This was studied in detail in [10]. In terms of the language introduced in [10], we say that region A encodes the zero-bits of the bulk region.…”

“…The difference now is that the dimension of the subspace which can be error-corrected is allowed to grow with the dimension of the larger space of all black hole microstates. If we again make use of terminology from [10], the region A now encodes the α-bits of the bulk region. Nontrivial realisations of universal subspace quantum error correction are only possible when the error correction is approximate.…”

Beyond toy models: distilling tensor networks in full AdS/CFT

“…40 In our construction, these fiber directions simply go along for the ride, without being subdivided, even though their radius of curvature is of the same order as that of the AdS factor, which we do subdivide. 41 Note that, as in all theories with large extra dimensions, the 10 or 11 dimensional Planck length will be parametrically larger than the effective Planck scale of the Kaluza-Klein reduced theory on AdS. This raises the question of whether ignoring the fiber direc- 40 Interestingly, it might be possible to do better in excited geometries which are not spherically symmetric.…”

“…This raises the question of whether ignoring the fiber direc- 40 Interestingly, it might be possible to do better in excited geometries which are not spherically symmetric. 41 See [80,81] for a possible idea for how to understand the fiber directions in terms of entanglement between different field degrees of freedom. If this idea is correct it would be interesting to combine it with our construction to obtain a discretization of the full space.…”