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 \(\alpha \); we call the associated resource an \(\alpha \)-bit. Changing \(\alpha \) interpolates between standard quantum error correction and quantum identification, a form of equality testing for quantum states. We develop the theory of \(\alpha \)-bits, including the applications above, and determine the \(\alpha \)-bit capacity of general quantum channels, finding single-letter formulas for the entanglement-assisted and amortised variants.
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Notes
An ebit is another term for a Bell pair of two qubits.
The authors are ashamed to note that they were unaware of the existence of co-cobits until six months after this paper had first appeared on ar**v.
Technically, our construction requires the use of shared randomness to achieve this rate but it can be eliminated by block coding.
As proved in Lemma 12, the \(\alpha \)-bit capacity is a continuous function of \(\alpha \), as is the entanglement-assisted \(\alpha \)-bit capacity. However, the amortised \(\alpha \)-bit capacity has a discontinuity at \(\alpha = 1\). As a result, it remains an open question to find and prove amortised universal subspace error correction capacities for subspace dimensions that grow sublinearly but faster than \(d^\alpha \) for any \(\alpha < 1\).
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Acknowledgements
We thank David Ding, Aram Harrow, Michael Walter, and Andreas Winter for valuable discussions. This work was supported by AFOSR (FA9550-16-1- 0082), CIFAR and the Simons Foundation.
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Communicated by M. M. Wolf
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Hayden, P., Penington, G. Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit. Commun. Math. Phys. 374, 369–432 (2020). https://doi.org/10.1007/s00220-020-03689-1
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DOI: https://doi.org/10.1007/s00220-020-03689-1