Quantum information with top quarks in QCD

Yoav Afik1 and Juan Ramón Muñoz de Nova2

1Experimental Physics Department, CERN, 1211 Geneva, Switzerland
2Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain

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Top quarks represent unique high-energy systems since their spin correlations can be measured, thus allowing to study fundamental aspects of quantum mechanics with qubits at high-energy colliders. We present here the general framework of the quantum state of a top-antitop ($t\bar{t}$) quark pair produced through quantum chromodynamics (QCD) in a high-energy collider. We argue that, in general, the total quantum state that can be probed in a collider is given in terms of the production spin density matrix, which necessarily gives rise to a mixed state. We compute the quantum state of a $t\bar{t}$ pair produced from the most elementary QCD processes, finding the presence of entanglement and CHSH violation in different regions of phase space. We show that any realistic hadronic production of a $t\bar{t}$ pair is a statistical mixture of these elementary QCD processes. We focus on the experimentally relevant cases of proton-proton and proton-antiproton collisions, performed at the LHC and the Tevatron, analyzing the dependence of the quantum state with the energy of the collisions. We provide experimental observables for entanglement and CHSH-violation signatures. At the LHC, these signatures are given by the measurement of a single observable, which in the case of entanglement represents the violation of a Cauchy-Schwarz inequality. We extend the validity of the quantum tomography protocol for the $t\bar{t}$ pair proposed in the literature to more general quantum states, and for any production mechanism. Finally, we argue that a CHSH violation measured in a collider is only a weak form of violation of Bell's theorem, necessarily containing a number of loopholes.

The top quark is the most massive fundamental particle known to exist. This large mass is translated into a lifetime so short that it decays before hadronising, allowing to reconstruct its spin quantum state from its decay products. As a result, spin correlations between top-antitop quarks ($t\bar{t}$) have been intensively studied. However, no link with quantum information theory has been established until very recently.

Here we present the general formalism of the quantum state of a $t\bar{t}$ pair, a unique high-energy realization of a two-qubit state. Remarkably, once the probabilities and density matrices of each $t\bar{t}$ production process are computed by the high-energy theory, we are simply left with a typical problem in quantum information involving the statistical mixture of two-qubit quantum states. This important observation motivates the pedagogical presentation of the article, fully developed within a genuine quantum information approach, aimed at making it easily understandable by the general physics community.

We discuss the experimental study of quantum information concepts such as entanglement, CHSH inequality or quantum tomography with top quarks. Interestingly, both entanglement and CHSH violation can be detected at the Large Hadron Collider (LHC) from the measurement of one single observable, with high-statistical significance in the case of entanglement.

The implementation of these measurements at the LHC paves the way to study quantum information also at high-energy colliders. Due to their genuinely relativistic behavior, the exotic character of the symmetries and interactions involved, as well as their fundamental nature, high-energy colliders are extremely attractive systems for this type of studies. For instance, the proposed detection of entanglement will represent the first detection ever of entanglement between a pair of quarks, and the highest-energy observation of entanglement so far achieved.

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