Unitarity provides mathematical and physical constraints on quantum information systems. e.g., in entanglement swapping, unitarity requires the same von Neumann entanglement entropy generation for either a particle interaction or an act of measurement. For the first time, the language of non-relativistic quantum mechanics is presented to derive the density matrix for hard scattering. We show that unitarity allows for finding the latter's cross section without using the scattering amplitude or the Lippmann-Schwinger equation plus Green's function. We also show the language of relativistic quantum mechanics can be used to derive the momentum entropy or Sackur-Tetrode equation for the inelastic scattering of an electron from a proton. The latter entropy derives from a Shannon entropy and an additional entropy that evokes the uncertainty principle. This article's presentation allows particle physicists to readily begin calculating quantum information metrics such as correlations and mutual information for any particle interaction.
Unitarity provides mathematical and physical constraints on quantum information systems. e.g., in entanglement swapping, unitarity requires the same von Neumann entanglement entropy generation for either a particle interaction or an act of measurement. For the first time, the language of non-relativistic quantum mechanics is presented to derive the density matrix for hard scattering. We show that unitarity allows for finding the latter's cross section without using the scattering amplitude or the Lippmann-Schwinger equation plus Green's function. We also show the language of relativistic quantum mechanics can be used to derive the momentum entropy or Sackur-Tetrode equation for the inelastic scattering of an electron from a proton. The latter entropy derives from a Shannon entropy and an additional entropy that evokes the uncertainty principle. This article's presentation allows particle physicists to readily begin calculating quantum information metrics such as correlations and mutual information for any particle interaction.
Regularized quantum information metrics are calculated for the scattering process e − e + → γ , Z → μ − μ + that has a witness photon entangled with the initial electron-positron state. Unitarity implies the correct regularization of divergences that appear in both the final density matrix and von Neumann entanglement entropies. The entropies are found to quantify uncertainty or randomness. The variation of information, entanglement entropy, and correlation between the muon’s and witness photon’s helicities are found to convey equivalent information. The magnitude of the muon’s expected helicity rises (falls) as the helicity entropy falls (rises). Area, or the scattering cross section, is a source of entropy for the muon’s helicity entropy and momentum entropy. The muon’s differential angular entropy distribution is similar to the differential angular cross section distribution, capturing the forward-backward asymmetry at high center-of-mass energies. Published by the American Physical Society 2024
We study relativistic scattering when one only has access to a subset of the particles, using the language of quantum measurement theory. We give an exact, non-perturbative formula for the von Neumann entanglement entropy of an apparatus particle scattered off an arbitrary set of system particles, in either the elastic or inelastic regime, and show how to evaluate it perturbatively. We give general formulas for the late-time expectation values of apparatus observables. Some simple example applications are included: in particular, a protocol to verify preparation of coherent superpositions of spatially localized system states using position-space information in the outgoing apparatus state, at lowest order in perturbation theory in a weak apparatus-system coupling.
Unitarity and the optical theorem are used to derive the reduced density matrices of Compton scattering in the presence of a witness particle. Two photons are initially entangled wherein one photon participates in Compton scattering, while the other is a witness, i.e., does not interact with the electron. Unitarity is shown to require that the entanglement entropy of the witness photon does not change after its entangled partner undergoes scattering. The final mutual information of the electron's and witness particle’s polarizations is shown to be nonzero for low-energy Compton scattering. This indicates that the two particles became correlated in spite of no direct interaction. Assuming an initial maximally entangled state, the change in entanglement entropy of the scattered photon’s polarization is calculated in terms of Stokes parameters. A common ratio of areas occurs in the final reduced density matrix elements, von Neumann entropies, Stokes parameter, and mutual information. This common ratio consists of the Thomson scattering cross-section and an accessible regularized scattering area.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
Classical information theory, built on Shannon entropy, and quantum information theory, built on von Neumann entropy, quantify uncertainty and compression properties of probability distributions and density operators. Both are powerful, but both treat information as a property of states of individual systems or ensembles. In this manuscript we argue that entanglement—the pattern of non-classical correlations between systems—is more fundamental than either Shannon or von Neumann information. We motivate this claim with a condensate thought experiment in which many bosons occupy the same single-particle state: Shannon and von Neumann entropies assign them identical information content, yet they can be distinguished by their distinct entanglement histories with external systems. Formally, states with identical marginal density operators but different purifications represent different global information, accessible only through their entanglement structure. Entropic quantities are then seen as coarse-grained summaries of this underlying relational information. We connect this perspective to an entanglement-geometric view of physics, including Entanglement Manifold Cosmology (EMC), in which spacetime and physical “truth” emerge from the geometry of correlations. The goal is an information-theoretic narrative that is rigorous enough for physics and information theory while readable to a non-specialist who is comfortable with basic linear algebra and quantum mechanics.
The theory of majorization is a powerful mathematical tool which naturally arises in the quantum theory as a consequence of its fundamental connection with unitarity. In this work, we look for majorization relations based on entropic inequalities. We begin by introducing the theory of majorization. We then give some notions of quantum mechanics, like the density matrix and some algebra, before presenting the concept of entropy which is closely related to the theory of majorization. We define in the process the Shannon, von Neumann and Renyi entropies, along with some of their properties and applications. We then define quantum entanglement and give some criteria for separability, which turn out to be applications of the concepts of majorization and entropy in the quantum theory. We present some new majorization relations similar to already existing entropic inequalities, and exhibit their advantages and limitations. The rest of the report is dedicated to the study of the field of Gaussian quantum information. We present the theory, before introducing other majorization relations, which turn out to be useful because they allow one to know if it is possible to transform one quantum state into another when there exists majorization relations between them.