In a monogamy-of-entanglement (MoE) game, two players who do not communicate try to simultaneously guess a referee's measurement outcome on a shared quantum state they prepared. We study the prototypical example of a game where the referee measures in either the computational or Hadamard basis and informs the players of her choice. We show that this game satisfies a rigidity property similar to what is known for some nonlocal games. That is, in order to win optimally, the players' strategy must be of a specific form, namely a convex combination of four unentangled optimal strategies generated by the Breidbart state. We extend this to show that strategies that win near-optimally must also be near an optimal state of this form. We also show rigidity for multiple copies of the game played in parallel. We give three applications: (1) We construct for the first time a weak string erasure (WSE) scheme where the security does not rely on limitations on the parties' hardware. Instead, we add a prover, which enables security via the rigidity of this MoE game. (2) We show that the WSE scheme can be used to achieve bit commitment in a model where it is impossible classically. (3) We achieve everlasting-secure randomness expansion in the model of trusted but leaky measurement and untrusted preparation and measurements by two isolated devices, while relying only on the temporary assumption of pseudorandom functions. This achieves randomness expansion without the need for shared entanglement.
In a monogamy-of-entanglement (MoE) game, two players who do not communicate try to simultaneously guess a referee's measurement outcome on a shared quantum state they prepared. We study the prototypical example of a game where the referee measures in either the computational or Hadamard basis and informs the players of her choice. We show that this game satisfies a rigidity property similar to what is known for some nonlocal games. That is, in order to win optimally, the players' strategy must be of a specific form, namely a convex combination of four unentangled optimal strategies generated by the Breidbart state. We extend this to show that strategies that win near-optimally must also be near an optimal state of this form. We also show rigidity for multiple copies of the game played in parallel. We give three applications: (1) We construct for the first time a weak string erasure (WSE) scheme where the security does not rely on limitations on the parties' hardware. Instead, we add a prover, which enables security via the rigidity of this MoE game. (2) We show that the WSE scheme can be used to achieve bit commitment in a model where it is impossible classically. (3) We achieve everlasting-secure randomness expansion in the model of trusted but leaky measurement and untrusted preparation and measurements by two isolated devices, while relying only on the temporary assumption of pseudorandom functions. This achieves randomness expansion without the need for shared entanglement.
Multipartite entanglement remains a challenging and not fully understood aspect of quantum information. Monogamy-of-Entanglement (MoE) games have been highly effective for studying limitations on the usefulness of entanglement imposed by monogamy constraints. To better reveal the extent to which multipartite entanglement can be useful, we introduce a class of quantum guessing games, termed One-at-a-Time Guessing (OTG) games. In these games, quantum players individually guess the outcomes of random measurements performed by a referee on a pre-shared entangled state. Unlike MoE games, OTG games select players individually at random according to a specified probability distribution, thereby probing each player's correlation with the referee. We show that, despite monogamy constraints, players sharing certain entangled states can moderately outperform those relying only on classical uncertainty. This advantage arises even in simple OTG games involving only Pauli measurements on qubits, where optimal entanglement increases the winning probability by at least 4%. This contrasts with MoE games, where shared entanglement has been shown in several settings to provide only limited (if any) advantage over classical strategies. We further establish a majorization property: the value of an OTG game respects the majorization ordering of the player-selection probability distribution. We also analyze in detail a two-player OTG game in which the referee measures one of the three Pauli observables on a qubit, and show that it is optimally played using a specific parameterized family of three-qubit W-like states. These results suggest that OTG games provide a useful framework for investigating the usefulness of multipartite entanglement in multiparty quantum correlations.
By leveraging the no-cloning principle of quantum mechanics, unclonable cryptography enables us to achieve novel cryptographic protocols that are otherwise impossible classically. Two most notable examples of unclonable cryptography are quantum copy-protection and unclonable encryption. Most known constructions rely on the quantum random oracle model (as opposed to the plain model), in which all parties have access in superposition to a powerful random oracle. Despite receiving a lot of attention in recent years, two important open questions still remain: copy-protection of point functions in the plain model, which is usually considered as feasibility demonstration, and unclonable encryption with unclonable indistinguishability security in the plain model. A core ingredient of these protocols is the so-called monogamy-of-entanglement property. Such games allow quantifying the correlations between the outcomes of multiple non-communicating parties sharing entanglement in a particular context. Specifically, we define the games between a challenger and three players in which the first player is asked to split and share a quantum state between the two others, who are then simultaneously asked a question and need to output the correct answer. In this work, by relying on previous works of Coladangelo, Liu, Liu, and Zhandry (Crypto'21) and Culf and Vidick (Quantum'22), we establish a new monogamy-of-entanglement property for subspace coset states, which allows us to progress towards the aforementioned goals. However, it is not sufficient on its own, and we present two conjectures that would allow first to show that copy-protection of point functions exists in the plain model, with different challenge distributions (including arguably the most natural ones), and then that unclonable encryption with unclonable indistinguishability security exists in the plain model. We believe that our new monogamy-of-entanglement to be of independent interest, and it could be useful in other applications as well. To highlight this last point, we leverage our new monogamy-of-entanglement property to show the existence of a tokenized signature scheme with a new security definition, called unclonable unforgeability.
In this work, we consider "decision" variants of a monogamy-of-entanglement game by Tomamichel, Fehr, Kaniewski, and Wehner [New Journal of Physics '13]. In its original "search" variant, Alice prepares a (possibly entangled) state on registers ABC; register A, consisting of n qubits, is sent to a Referee, while B and C are sent to Bob and Charlie; the Referee then measures each qubit in the standard or Hadamard basis (chosen uniformly at random). The basis choices are sent to Bob and Charlie, whose goal is to simultaneously guess the Referee's n-bit outcome string x. Tomamichel et al. show that the optimal winning probability is cos2n(8π), following a perfect parallel repetition theorem. We consider the following "decision" variants of this game: - Variant 1, "XOR repetition": Bob and Charlie's goal is to guess the XOR of all the bits of x. Ananth et al. [Asiacrypt '24] conjectured that the optimal advantage over random guessing decays exponentially in n. Surprisingly, we show that this conjecture is false, and, in fact, there is no decay at all: there exists a strategy that wins with probability cos2(8π)≈0.85 for any n. - Variant 2, "Goldreich-Levin": The Referee additionally samples a uniformly random n-bit string r that is sent to Bob and Charlie along with the basis choices. Their goal is to guess the parity of r⋅x. We show that the optimal advantage over random guessing decays exponentially in n for the restricted class of adversaries that do not share entanglement. A similar result was already shown by Champion et al. and Çakan et al.; we give a more direct proof. Additionally, we put forward a reasonably concrete conjecture that is equivalent to exponential decay for general adversaries.
By leveraging the no-cloning principle of quantum mechanics, unclonable cryptography enables us to achieve novel cryptographic protocols that are otherwise impossible classically. Two most notable examples of unclonable cryptography are copy-protection (CP) and unclonable encryption (UE). Most known constructions rely on the QROM (as opposed to the plain model). Despite receiving a lot of attention in recent years, two important open questions still remain: CP for point functions in the plain model, which is usually considered as feasibility demonstration, and UE with unclonable indistinguishability security in the plain model. A core ingredient of these protocols is the so-called monogamy-of-entanglement (MoE) property. Such games allow quantifying the correlations between the outcomes of multiple non-communicating parties sharing entanglement in a particular context. Specifically, we define the games between a challenger and three players in which the first player is asked to split and share a quantum state between the two others, who are then simultaneously asked a question and need to output the correct answer. In this work, by relying on previous works [CLLZ21, CV22], we establish a new MoE property for subspace coset states, which allows us to progress towards the aforementioned goals. However, it is not sufficient on its own, and we present two conjectures that would allow first to show that CP of point functions exists in the plain model, with different challenge distributions, and then that UE with unclonable indistinguishability security exists in the plain model. We believe that our new MoE to be of independent interest, and it could be useful in other applications as well. To highlight this last point, we leverage our new MoE property to show the existence of a tokenized signature scheme with a new security definition, called unclonable unforgeability.
We give a generic compiler for turning a wide class of device-dependent protocols with trusted state preparation into sequential device-independent protocols. In order for our compiler to apply, the trusted-preparation protocol needs to satisfy certain minimal conditions: having an entanglement-based realization in the first place, and the entanglement-based realization corresponding to the optimal strategy of any non-local game that satisfies a fairly weak rigidity condition. We apply our generic compiler to give sequential device-independent protocols for uncloneable encryption, copy-protection of point functions, and blind randomness expansion.
Our technical contribution is a new class of rigidity theorems based on Rényi divergences, with flexible cryptographic applications. They allow our compiler to achieve a constant rate (i.e., the ratio of trusted states prepared to device uses), while tolerating a constant level of noise per device usage. This overcomes a fundamental scaling obstacle faced by previous rigidity theorems for trace distance or fidelity. For the generic sequential case, our proof is based on the sharp-divergence chain rule by Fawzi and Fawzi (2021), which suffices for a constant but fairly low rate. In the special case when the devices retain no memory, we improve the rates by deriving a novel subadditivity property for a modified channel sandwiched divergence, which may be of independent interest. Finally, as a prospect for future improvements, we also prove a chain rule for the iterated-mean divergences from Brown, Fawzi and Fawzi (2021).
We study a generalization of non-local games—which we call extended non-local games —in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on the outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski and Wehner, which they also generalize. We prove that a natural extension of the Navascués–Pironio–Acín hierarchy of semidefinite programmes converges to the optimal commuting measurement value of extended non-local games, and we prove two extensions of results of Tomamichel et al. concerning monogamy-of-entanglement games.