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Register a new userMaximally nonlocal theories cannot be maximally random
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Correlations that violate a Bell Inequality are said to be nonlocal, i.e. they do not admit a local and deterministic explanation. Great effort has been devoted to study how the amount of nonlocality (as measured by a Bell inequality violation) serves to quantify the amount of randomness present in observed correlations. In this work we reverse this research program and ask what do the randomness certification capabilities of a theory tell us about the nonlocality of that theory. We find that, contrary to initial intuition, maximally nonlocal theories cannot allow maximal randomness certification. We go on and show that quantum theory, in contrast, permits certification of maximal randomness in all dichotomic scenarios. We hence pose the question of whether quantum theory is optimal for randomness, i.e. is it the most nonlocal theory that allows maximal randomness certification? We answer this question in the negative by identifying a larger-than-quantum set of correlations capable of this feat. Not only are these results relevant to understanding quantum mechanics fundamental features, but also put fundamental restrictions on device-independent protocols based on the no-signaling principle.
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This paper considers a special class of nonlocal games $(G,psi)$, where $G$ is a two-player one-round game, and $psi$ is a bipartite state independent of $G$. In the game $(G,psi)$, the players are allowed to share arbitrarily many copies of $psi$. The value of the game $(G,psi)$, denoted by $omega^*(G,psi)$, is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states $psi$. For a noisy maximally entangled state $psi$, a two-player one-round game $G$ and an arbitrarily small precision $epsilon>0$, this paper proves an upper bound on the number of copies of $psi$ for the players to win the game with a probability $epsilon$ close to $omega^*(G,psi)$. Hence, it is feasible to approximately compute $omega^*(G,psi)$ to an arbitrarily precision. Recently, a breakthrough result by Ji, Natarajan, Vidick, Wright and Yuen showed that it is undecidable to approximate the values of nonlocal games to a constant precision when the players preshare arbitrarily many copies of perfect maximally entangled states, which implies that $mathrm{MIP}^*=mathrm{RE}$. In contrast, our result implies the hardness of approximating nonlocal games collapses when the preshared maximally entangled states are noisy. The paper develops a theory of Fourier analysis on matrix spaces by extending a number of techniques in Boolean analysis and Hermitian analysis to matrix spaces. We establish a series of new techniques, such as a quantum invariance principle and a hypercontractive inequality for random operators, which we believe have further applications.
Discrete-spin systems with maximally random nearest-neighbor interactions that can be symmetric or asymmetric, ferromagnetic or antiferromagnetic, including off-diagonal disorder, are studied, for the number of states $q=3,4$ in $d$ dimensions. We use renormalization-group theory that is exact for hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic lattices. For all d>1 and all non-infinite temperatures, the system eventually renormalizes to a random single state, thus signaling qxq degenerate ordering. Note that this is the maximally degenerate ordering. For high-temperature initial conditions, the system crosses over to this highly degenerate ordering only after spending many renormalization-group iterations near the disordered (infinite-temperature) fixed point. Thus, a temperature range of short-range disorder in the presence of long-range order is identified, as previously seen in underfrustrated Ising spin-glass systems. The entropy is calculated for all temperatures, behaves similarly for ferromagnetic and antiferromagnetic interactions, and shows a derivative maximum at the short-range disordering temperature. With a sharp immediate contrast of infinitesimally higher dimension 1+epsilon, the system is as expected disordered at all temperatures for d=1.
Integral invariants in maximally supersymmetric Yang-Mills theories are discussed in spacetime dimensions $4leq Dleq 10$ for $SU(k)$ gauge groups. It is shown that, in addition to the action, there are three special invariants in all dimensions. Two of these, the single- and double-trace $F^4$ invariants, are of Chern-Simons type in $D=9,10$ and BPS type in $Dleq 8$, while the third, the double-trace of two derivatives acting on $F^4$, can be expressed in terms of a gauge-invariant super-$D$-form in all dimensions. We show that the super-ten-forms for $D=10$ $F^4$ invariants have interesting cohomological properties and we also discuss some features of other invariants, including the single-trace $d^2 F^4$, which has a special form in $D=10$. The implications of these results for ultra-violet divergences are discussed in the framework of algebraic renormalisation.
Maximally Natural Supersymmetry, an unusual weak-scale supersymmetric extension of the Standard Model based upon the inherently higher-dimensional mechanism of Scherk-Schwarz supersymmetry breaking (SSSB), possesses remarkably good fine tuning given present LHC limits. Here we construct a version with precision $SU(2)_{rm L} times U(1)_{rm Y} $ unification: $sin^2 theta_W(M_Z) simeq 0.231$ is predicted to $pm 2%$ by unifying $SU(2)_{rm L} times U(1)_{rm Y} $ into a 5D $SU(3)_{rm EW}$ theory at a Kaluza-Klein scale of $1/R_5 sim 4.4,{rm TeV}$, where SSSB is simultaneously realised. Full unification with $SU(3)_{rm C}$ is accommodated by extending the 5D theory to a $N=4$ supersymmetric $SU(6)$ gauge theory on a 6D rectangular orbifold at $1/R_6 sim 40 ,{rm TeV}$. TeV-scale states beyond the SM include exotic charged fermions implied by $SU(3)_{rm EW}$ with masses lighter than $sim 1.2,{rm TeV}$, and squarks in the mass range $1.4,{rm TeV} - 2.3,{rm TeV}$, providing distinct signatures and discovery opportunities for LHC run II.
By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of $d$ maximally entangled states in a $dotimes d$ system are systematically constructed for all $dge 4$ as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of $k$ locally indistinguishable maximally entangled states with the ratio $k/d$ approaching 3/4. In particular, in a $dotimes d$ system with even $dge 6$, there always exist $d-1$ maximally entangled states that are locally indistinguishable by separable measurements.
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