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The quantum magic of top quarks

Chris D. White

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Abstract

In recent years, a growing community of researchers has been applying ideas from Quantum Information theory to high energy physics. Entanglement is one such quantity, but there are by now many others. This contribution will examine the property of magic" (also known as non-stabliserness") which, roughly speaking, measures whether quantum systems have a genuine computational advantage over their classical counterparts. We will review the origin of this concept, before describing recent applications in collider physics, including top quark production.

QCD at the Extremes 2026

Introduction

Quantum Information (QI) theory is by a now a well-established field, with a number of applications. On the one hand, it underlies the design and implementation of quantum computers, and the development of practical algorithms for such devices. On the other hand, QI methods provide a general language for examining quantum systems, ranging from low-energy table top experiments, through to high energy particle physics facilities such as the Large Hadron Collider. A by now well-studied property at colliders is that of quantum entanglement1, with a key example process being that of top quark pair production [7, 8, 9, 10, 11, 12]. Relevant experimental analyses include Refs. [13, 14], and an excellent pedagogical review of this area is that of Ref. [15].

The previous wide interest in entanglement has paved the way, more recently, for the application of other ideas from QI theory to a high energy setting. There are a number of motivations. First, there is the possibility of testing quantum mechanics itself (subject to loopholes and other caveats). Secondly, there is the hope that QI-motivated observables may help in disentangling new physics theories from the current Standard Model of Particle Physics. Thirdly, there is an increasing body of work arguing that QI methods could explain structures within our existing theories, or their possible extensions. More generally, QI theory may give us new ways of thinking about open problems in high energy physics theory or phenomenology.

In this contribution, we will focus on a particular QI property known as magic (more formally, non-stabiliser­ness). This was first introduced in Ref. [16] in the context of quantum computing, where it is of crucial importance for characterising the computational advantage of quantum computers over their classical counterparts, as well as designing fault-tolerant algorithms. Thus, in order to formally define magic, we must first briefly review relevant aspects of quantum computing.

A bit of quantum computing

Classical computers consist of logical bits which can take values 0 or 1. A quantum computer replaces these with qubits2, whose most general state is a superposition of two basis states ∣0⟩|0\rangle and ∣1⟩|1\rangle:

∣ψ⟩=α∣0⟩+β∣1⟩.(1)|\psi\rangle= \alpha|0\rangle+ \beta|1\rangle. \tag*{(1)}

Here α\alpha and β\beta are complex coefficients subject to the normalisation condition

∣α∣2+∣β∣2=1.(2)|\alpha|^2 + |\beta|^2 = 1. \tag*{(2)}

A canonical physical example of a qubit is that of a single spin-1/2 particle, such that the states ∣0⟩|0\rangle and ∣1⟩|1\rangle correspond to the up and down spin states with respect to a chosen direction. A general quantum computer may contain many qubits, in which case a suitable choice of basis states is

∣ψ1ψ2…ψn⟩≡∣ψ1⟩⊗∣ψ2⟩⊗…⊗∣ψn⟩.(3)|\psi_1\psi_2\ldots\psi_n\rangle\equiv|\psi_1\rangle\otimes|\psi_2\rangle\otimes\ldots\otimes|\psi_n\rangle. \tag*{(3)}

Quantum computers then take (superpositions of) such states, and subject them to unitary transformations, where the unitary property follows from the basic postulates of quantum mechanics. It is conventional to draw the action of such transformations using quantum circuit diagrams, an example of which is shown in Figure 1. Each line in the diagram represents a single qubit, where the conventional incoming state is ∣00…0⟩|00\ldots0\rangle. Boxes represent unitary transformations, and the two shown in Figure 1 act on two and three qubits respectively. The transformations are then referred to as quantum gates, by analogy with the well-known logic gates of classical computing. Common gates used for quantum computing have names like Hadamard, phase, CNOT and Pauli. We will not need their precise details for what follows, but it is worth noting that these particular gates tend to be the easiest ones to make in current quantum systems.

An example quantum circuit diagram

Figure 1. An example quantum circuit diagram, where the incoming lines represent the basis state ∣00…0⟩|00\ldots0\rangle, and the boxes are quantum gates.

We are interested in quantum computers because they are expected to vastly outperform their classical counterparts. Naïvely, this is due to the two quantum properties of superposition and entanglement. The former implies that one may have many calculations going on simultaneously, and the latter is a known ingredient in quantum algorithms that are known to be exponentially faster than classical ones for certain problems. However, it is not actually true that entanglement by itself guarantees higher computing power. To see why, one may introduce the notion of a stabiliser state. The starting point is to consider the following set of operators, defined for any number of qubits:

PN=P1⊗P2⊗…⊗PN,Pa∈{σ1(a),σ2(a),σ3(a),I(a)}.(4)\mathcal{P}_{N} = P_{1} \otimes P_{2} \otimes\ldots\otimes P_{N}, \qquad P_{a} \in\{\sigma_{1}^{(a)}, \sigma_{2}^{(a)}, \sigma_{3}^{(a)}, I^{(a)}\}. \tag*{(4)}

These are called Pauli strings, and consist of a series of transformations acting simultaneously on each individual qubit, and where the operator acting on the atha^{\mathrm{th}} qubit is either an identity matrix or a Pauli matrix. There are then 4N4^{N} such operators for NN qubits. Then, given a state ∣ψ⟩|\psi\rangle, its Pauli spectrum consists of the set of expectation values for every Pauli string:

spec⁡(∣ψ⟩)={⟨ψ∣P∣ψ⟩,P∈PN}.(5)\operatorname{spec}(|\psi\rangle) = \{\langle\psi|P|\psi\rangle,\quad P \in\mathcal{P}_{N}\}. \tag*{(5)}

Stabiliser states are such that 2N2^{N} expectation values are +1+1 or −1-1, with the rest being zero. Non-stabiliser states will have a mixture of different values in the Pauli spectrum. Despite this very abstract definition, stabiliser states can be interpreted physically as those states one makes by using the Hadamard, phase, CNOT and Pauli gates mentioned earlier. Their importance stems from a result called the Gottesman-Knill theorem, which one may paraphrase as follows: For every quantum computer containing stabiliser states only, there is a classical computer that is just as efficient.

Put more bluntly: there is no computational advantage to be gained by building an expensive quantum computer, if all it contains are the very gates that are easiest to make in the lab! Unsurprisingly, this tells us that quantum computing is a rather difficult endeavour. The result can also be puzzling when first encountered, as it turns out that stabiliser states include maximally entangled ones (e.g. the well-known Bell states for two qubits). A priori, one might think that the more entanglement one has, the more powerful one’s quantum computer is. The Gottesman-Knill theorem tells us that this is not true at all, and it follows that something other than entanglement is needed for computational advantage. It is this resource that has been called magic in the QI literature. As a term, it is synonymous with non-stabiliserness of a quantum state, which is what it ultimately measures.

Various definitions of magic exist in the literature, motivated by ease of computation or measurement in particular scenarios. A popular definition is the set of so-called Stabiliser Rényi entropies, first presented in Ref. [17]:

Mq=11−qlog⁡2(ζq),ζq≡∑P∈PN⟨ψ∣P∣ψ⟩2q2N.(6)M_q = \frac{1}{1-q}\log_2(\zeta_q), \qquad\zeta_q \equiv\sum_{P\in\mathcal{P}_N}\frac{\langle\psi|P|\psi\rangle^{2q}}{2^N}. \tag*{(6)}

There is one such quantity for each value of the integer q≥2q \ge2, and we see that each involves the Pauli spectrum (through expectation values of Pauli strings PP), as it should given that deviations from a simple Pauli spectrum constitute non-stabiliserness. One may check that the magic defined in this way indeed vanishes for stabiliser states, and it is also additive when multiqubit systems are combined. For what follows, choosing q=2q = 2 will suffice, and the quantity M2M_2 is then usually referred to as the Second Stabiliser Rényi Entropy (SSRE).

The magic of top quarks

There is by now a well-established and vibrant global community of researchers interested in many-body quantum magic, with even an entire annual interdisciplinary conference devoted to the subject 3. As noted above, magic plays a key role in generating computational advantage and fault tolerance of quantum computers. It has also become increasingly clear that magic is an important tool for characterising interesting behaviours such as phase transitions and confinement (see e.g. ref. [18] for a recent review). People are thus interested in any physical system where magic can be produced and / or enhanced, so that new insights can be obtained, which may in turn be portable to other quantum systems. High energy scattering processes provide one such playground, and the Large Hadron collider excels in particular at generating final states containing two qubits (spin-1/2 particles). A key example is that of top quark pair production, already mentioned above in the context of entanglement studies. At leading order in the Standard Model, top quarks are produced form either a quark-antiquark or gluon-gluon initial state, and the final state will consist of different possible pure quantum states, weighted by different probabilities as dictated by the rules of quantum field theory. This is a so-called mixed state, and one must generalise the above measure of magic accordingly. First, one calculates a (spin) density matrix for the final state, schematically given by

ραβ,α′β′∝⟨Aαβ[Aα′β′]∗⟩,(7)\rho_{\alpha\beta,\alpha'\beta'} \propto\left\langle\mathcal{A}_{\alpha\beta}\left[\mathcal{A}_{\alpha'\beta'}\right]^* \right\rangle, \tag*{(7)}

where Aαβ\mathcal{A}_{\alpha\beta} is the scattering amplitude with final-state spin indices α,β\alpha,\beta, and the angle brackets denote averaging / projection over non-spin quantum numbers (see Refs. [7, 8] for further details). Armed with the density matrix, one may calculate the mixed-state SSRE via

M~2(ρ)=−log⁡2(∑P∈PnTr⁡4(ρP)∑P∈PnTr⁡2(ρP)).(8)\widetilde{M}_2(\rho)=-\log_2\left(\frac{\sum_{P\in\mathcal{P}_n}\operatorname{Tr}^4(\rho P)}{\sum_{P\in\mathcal{P}_n}\operatorname{Tr}^2(\rho P)}\right). \tag*{(8)}

The density matrix itself depends on the basis chosen for identifying spin directions with physical directions in space. A common choice is the helicity basis (see e.g. Ref. [19]), which aligns the spin direction with the outgoing top quark, as shown in Figure 2. One may then express the amount of magic (SSRE) as a function of the cosine of the scattering angle and the 4-velocity of the top quark (in units of the speed of light). These are given respectively by

Helicity basis showing the top quark spin directions and scattering angle

Figure 2. In the helicity basis, the top quark spin directions are mapped to spatial directions parallel or orthogonal to the top quark itself. The scattering angle θ\theta is defined with respect to the first incoming beam.

z=cos⁡θ,β=1−4mt2s^,(9)z=\cos\theta,\qquad\beta=\sqrt{1-\frac{4m_t^2}{\hat{s}}}, \tag*{(9)}

where mtm_t is the top quark mass, and s^\hat{s} the partonic centre of mass energy. As explained in more detail in Ref. [20], this now provides all the ingredients needed to compute the magic of top quark final states at the LHC. Results are presented in Figure 3 at leading order in the Standard Model. One sees that nature chooses to produce magic top quarks over the majority of the final state phase space where, unsurprisingly, the profile of magic is different for different initial states. For both the qqˉq\bar{q} and gggg channels, zero magic is observed in certain kinematic regions, and one may indeed verify analytically that the corresponding states are stabiliser [20].

Three heatmap panels showing the magic of a mixed top-antitop final state

Figure 3. The magic of a mixed top-antitop final state in: (a) the qqˉq\bar{q} channel; (b) the gggg channel; (c) for initial state protons.

Reference [21] studied whether magic – and other quantum information quantities – can be used to disentangle new physics from the Standard Model. This was investigated in the specific context of top quark production in Ref. [22]. The authors used the well-known Standard Model Effective Theory (SMEFT) to add additional corrections to the Standard Model, and examined how the magic profile changed when such contributions are included. An

example is shown in Figure 4, showing that the magic profile indeed changes when new physics is present. Reference [22] also looked at whether magic or alternative QI measures (e.g. so-called trace distance and fidelity) are more or less sensitive as probes of new physics, including compared to simply using spin correlation coefficients by themselves. Depending on the kinematic region or specific new physics type, any of the examined measures can end up being most sensitive, corroborating similar conclusions in different scattering processes from [21]. Such results are also of interest to the general many body magic community, who often study magic in systems that can be deformed by varying one or more parameters. Related studies of magic or similar quantities in a variety of quantum systems include Refs. [11, 17, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

Two different ways of measuring the difference in magic for a new physics theory

Figure 4. Two different ways of measuring the difference in magic for a new physics theory (corresponding to a single additional SMEFT operator added to the Standard Model). Figure taken from [22].

In parallel with theoretical studies of magic in scattering processes, experimental collaborations have also become interested. For example, Ref. [42] presents a differential measurement of magic in top pair final states by the CMS collaboration, in bins of invariant mass and scattering angle. The ATLAS experiment has produced a general document outlining its plan for QI measurements, which explicitly lists magic as an interesting quantity for further study [5]. A more comprehensive study is that of Ref. [43], an input to a recent update of the European strategy for particle physics. Two particular themees that emerge are the use of collider experiments to probe quantum measurements that can be difficult to reconstruct in traditional table top experiments; and the development of better analysis tools for new physics.

Summary and outlook

Magic (also known as non-stabiliser­ness) is a well-studied concept in quantum information theory, due to its crucial importance for developing efficient and fault-tolerant quantum computers. Recent years have seen widespread study of magic in various quantum systems, where it seems to be an efficient marker for novel behaviours including phase transitions and confinement. In this contribution, we have examined recent work demonstrating that magic is naturally produced in top quark final states at the LHC, and can also be a useful probe of new physics. This itself forms part of a wider dialogue between the fields of QI theory and particle physics, that promises to be a fruitful activity in the coming years. High energy physics offers new playgrounds (both theoretical and experimental) for QI theorists to sharpen their tools and insights. What is often invisible to pure particle physics audiences is the high level of interest for high energy physics scenarios that has arisen amongst the global community of people working on many body quantum magic. Likewise, both high energy theorists and experimenters can benefit from knowing more about the language of quantum information theory. The coming years are likely to see rapid developments in this area, and increased interdisciplinary collaboration.

Acknowledgements

This research was supported by the UK Science and Technology Facilities Council (STFC) Consolidated Grant ST/P000754/1 “String theory, gauge theory and duality”.

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