Paper 2024/1826

Cloning Games, Black Holes and Cryptography

Abstract

The no-cloning principle has played a foundational role in quantum information and cryptography. Following a long-standing tradition of studying quantum mechanical phenomena through the lens of interactive games, Broadbent and Lord (TQC 2020) formalized cloning games in order to quantitatively capture no-cloning in the context of unclonable encryption schemes. The conceptual contribution of this paper is the new, natural, notion of Haar cloning games together with two applications. In the area of black-hole physics, our game reveals that, in an idealized model of a black hole which features Haar random (or pseudorandom) scrambling dynamics, the information from infalling entangled qubits can only be recovered from either the interior or the exterior of the black hole---but never from both places at the same time. In the area of quantum cryptography, our game helps us construct succinct unclonable encryption schemes from the existence of pseudorandom unitaries, thereby, for the first time, bridging the gap between ``MicroCrypt'' and unclonable cryptography. The technical contribution of this work is a tight analysis of Haar cloning games which requires us to overcome many long-standing barriers in our understanding of cloning games: 1. Are there cloning games which admit no non-trivial winning strategies? Resolving this particular question turns out to be crucial for our application to black-hole physics. Existing work analyzing the $n$-qubit BB84 game and the subspace coset game only achieve the bounds of $2^{-0.228n}$ and $2^{-0.114n+o(n)}$, respectively, while the trivial adversarial strategy wins with probability $2^{-n}$. We show that the Haar cloning game is the hardest cloning game, by demonstrating a worst-case to average-case reduction for a large class of games which we refer to as oracular cloning games. We then show that the Haar cloning game admits no non-trivial winning strategies. 2. All existing works analyze $1\mapsto 2$ cloning games; can we prove bounds on $t\mapsto t+1$ games for large $t$? Such bounds are crucial in our application to unclonable cryptography. Unfortunately, the BB84 game is not even $2\mapsto 3$ secure, and the subspace coset game is not $t\mapsto t+1$ secure for a polynomially large $t$. We show that the Haar cloning game is $t\mapsto t+1$ secure provided that $t = o(\log n / \log \log n)$, and we conjecture that this holds for $t$ that is polynomially large (in $n$). Answering these questions provably requires us to go beyond existing methods (Tomamichel, Fehr, Kaniewski and Wehner, New Journal of Physics 2013). In particular, we show a new technique for analyzing cloning games with respect to binary phase states through the lens of binary subtypes, and combine it with novel bounds on the operator norms of block-wise tensor products of matrices.