Quantum Key Recycling with 8-state encoding (The Quantum One-Time Pad is more interesting than we thought)

Abstract

Perfect encryption of quantum states using the Quantum One-Time Pad (QOTP) requires two classical key bits per qubit. Almost-perfect encryption, with information-theoretic security, requires only slightly more than 1. We slightly improve lower bounds on the key length. We show that key length [Formula: see text] suffices to encrypt [Formula: see text] qubits in such a way that the cipherstate’s [Formula: see text]-distance from uniformity is upperbounded by [Formula: see text]. For a stricter security definition involving the [Formula: see text]-norm, we prove sufficient key length [Formula: see text], where [Formula: see text] is a small probability of failure. Our proof uses Pauli operators, whereas previous results on the [Formula: see text]-norm needed Haar measure sampling. We show how to QOTP-encrypt classical plaintext in a nontrivial way: we encode a plaintext bit as the vector [Formula: see text] on the Bloch sphere. Applying the Pauli encryption operators results in eight possible cipherstates which are equally spread out on the Bloch sphere. This encoding, especially when combined with the half-keylength option of QOTP, has advantages over 4-state and 6-state encoding in applications such as Quantum Key Recycling (QKR) and Unclonable Encryption (UE). We propose a key recycling scheme that is more efficient and can tolerate more noise than a recent scheme by Fehr and Salvail. For 8-state QOTP encryption with pseudorandom keys, we do a statistical analysis of the cipherstate eigenvalues. We present numerics up to nine qubits.