We consider a pair of unstable fermions in a spin-entangled state. After the decay of one fermion, a spin measurement is performed on the surviving partner, with a Stern-Gerlach experiment or similar. The measurement not only projects the spin of the surviving fermion, but is also physically equivalent to a spin projection for the decayed one — even when it no longer exists. This post-selection effect would be experimentally accessible using muon pairs in a maximally-entangled state, produced either in the decay of a scalar particle, or in e+e− collisions at wide angles.We consider a pair of unstable fermions in a spin-entangled state. After the decay of one fermion, a spin measurement is performed on the surviving partner, with a Stern-Gerlach experiment or similar. The measurement not only projects the spin of the surviving fermion, but is also physically equivalent to a spin projection for the decayed one — even when it no longer exists. This post-selection effect would be experimentally accessible using muon pairs in a maximally-entangled state, produced either in the decay of a scalar particle, or in e+e− collisions at wide angles.

Physics Letters BDec 19, 20238 citations

Time, quantum entanglement and particle decay

J. A. Aguilar–Saavedra

We investigate the role of time ordering in entanglement experiments involving unstable particles, focusing on $\mu^+ \mu^-$ pairs produced in a maximally-entangled spin state. We analyse the correlations between measurements performed by two experimenters, Alice (who measures $\mu^-$ spin) and Bob (who measures $\mu^+$ decay products). Remarkably, the observed correlations persist irrespective of whether Bob's muon decays before or after Alice's spin measurement. We further discuss different interpretations of the same empirical results depending on the observer's reference frame. The findings reinforce the viewpoint that the Copenhagen interpretation of measurement is a mathematical tool rather than a literal account of physical reality.We investigate the role of time ordering in entanglement experiments involving unstable particles, focusing on $\mu^+ \mu^-$ pairs produced in a maximally-entangled spin state. We analyse the correlations between measurements performed by two experimenters, Alice (who measures $\mu^-$ spin) and Bob (who measures $\mu^+$ decay products). Remarkably, the observed correlations persist irrespective of whether Bob's muon decays before or after Alice's spin measurement. We further discuss different interpretations of the same empirical results depending on the observer's reference frame. The findings reinforce the viewpoint that the Copenhagen interpretation of measurement is a mathematical tool rather than a literal account of physical reality.

Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)Aug 24, 20260 citations

Spin operator and entanglement in quantum field theory

Kazuo Fujikawa, C. H. Oh, Chengjie Zhang

Entanglement is studied in the framework of Dyson's S-matrix theory in relativistic quantum field theory, which leads to a natural definition of entangled states of a particle-antiparticle pair and the spin operator from a Noether current. As an explicit example, the decay of a massive pseudo-scalar particle into a pair of electron and positron is analyzed. Two spin operators are extracted from the Noether current. The Wigner spin operator characterizes spin states at the rest frame of each fermion and, although not measurable in the laboratory, gives rise to a straightforward generalization of low-energy analysis of entanglement to the ultrarelativistic domain. In contrast, if one adopts a (modified) Dirac spin operator, the entanglement measured by spin correlation becomes maximal near the threshold of the decay, while the entanglement is replaced by the classical correlation for the ultrarelativistic electron-positron pair by analogy to the case of neutrinos, for which a hidden-variables type of description is possible. Chiral symmetry differentiates the spin angular momentum and the magnetic moment. The use of weak interaction that can measure helicity is suggested in the analysis of entanglement at high energies instead of a Stern-Gerlach apparatus, which is useless for the electron. A difference between the electron spin at high energies and the photon linear polarization is also noted. The Standard Model can describe all of the observable properties of leptons.Entanglement is studied in the framework of Dyson's S-matrix theory in relativistic quantum field theory, which leads to a natural definition of entangled states of a particle-antiparticle pair and the spin operator from a Noether current. As an explicit example, the decay of a massive pseudo-scalar particle into a pair of electron and positron is analyzed. Two spin operators are extracted from the Noether current. The Wigner spin operator characterizes spin states at the rest frame of each fermion and, although not measurable in the laboratory, gives rise to a straightforward generalization of low-energy analysis of entanglement to the ultrarelativistic domain. In contrast, if one adopts a (modified) Dirac spin operator, the entanglement measured by spin correlation becomes maximal near the threshold of the decay, while the entanglement is replaced by the classical correlation for the ultrarelativistic electron-positron pair by analogy to the case of neutrinos, for which a hidden-variables type of description is possible. Chiral symmetry differentiates the spin angular momentum and the magnetic moment. The use of weak interaction that can measure helicity is suggested in the analysis of entanglement at high energies instead of a Stern-Gerlach apparatus, which is useless for the electron. A difference between the electron spin at high energies and the photon linear polarization is also noted. The Standard Model can describe all of the observable properties of leptons.

arXiv:1312.0326Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJul 22, 201411 citations

Universal and Maximal Entanglement Swapping in General Fermionic Gaussian States

Jiyuan Fang, Qicheng Tang, Xueda Wen

Exploring universal entanglement structure in many-body systems is both fundamental and challenging, particularly when the system undergoes non-unitary operations. In this work, we uncover a universal mechanism for realizing maximal entanglement swapping in fermionic Gaussian states subjected to projective Bell measurements. We consider two initially decoupled, half-filled copies of a free-fermion system in arbitrary dimensions and perform post-selective Bell measurements on half of the corresponding sites across the two copies. Remarkably, the post-measurement state factorizes into a product of Bell pairs, establishing maximal interlayer entanglement entirely independent of the initial Gaussian state. We derive this post-measurement state exactly for general particle-number-conserving fermionic Gaussian states, establishing both the validity and universality of the mechanism, with numerical simulations serving as consistency checks. This phenomenon arises from a robust interplay between fermionic statistics and Gaussianity, revealing a distinct fermionic route to measurement-induced maximal entanglement.Exploring universal entanglement structure in many-body systems is both fundamental and challenging, particularly when the system undergoes non-unitary operations. In this work, we uncover a universal mechanism for realizing maximal entanglement swapping in fermionic Gaussian states subjected to projective Bell measurements. We consider two initially decoupled, half-filled copies of a free-fermion system in arbitrary dimensions and perform post-selective Bell measurements on half of the corresponding sites across the two copies. Remarkably, the post-measurement state factorizes into a product of Bell pairs, establishing maximal interlayer entanglement entirely independent of the initial Gaussian state. We derive this post-measurement state exactly for general particle-number-conserving fermionic Gaussian states, establishing both the validity and universality of the mechanism, with numerical simulations serving as consistency checks. This phenomenon arises from a robust interplay between fermionic statistics and Gaussianity, revealing a distinct fermionic route to measurement-induced maximal entanglement.

quant-pharXiv:2512.15890Dec 19, 20250 citations

S-Matrix Theory and Entanglement

Kazuo Fujikawa

But no good and simple criterion of entanglement for more complicated cases is known. This situation is rather strange since everybody talks about entanglement. How to produce and how to measure the entanglement are also interesting issues. When one measures the spin of the electron, for example, there are three major operators one can think of [1] a) Wigner spin, b) Dirac spin, c) Magnetic moment. The notion of chirality is important to distinguish these operators. Incidentally, many people talk about the Stern-Gerlach apparatus to measure the spin polarization of the electron. This is also strange since N. Bohr and N. F. Mott and other prominent people have shown many years ago that one cannot measure the electron spin polarization by a Stern-Gerlach apparatus, since the inhomogeneous magnetic field disturbs the motion of the electron so much and it is impossible to detect the spin orientation. The Stern-Gerlach particles are electrically neutral. One may ask if novel features appear in high energy physics? Chiral symmetry (including the use of weak interaction) can become important such as in a neutrino-antineutrino pair in Z decayBut no good and simple criterion of entanglement for more complicated cases is known. This situation is rather strange since everybody talks about entanglement. How to produce and how to measure the entanglement are also interesting issues. When one measures the spin of the electron, for example, there are three major operators one can think of [1] a) Wigner spin, b) Dirac spin, c) Magnetic moment. The notion of chirality is important to distinguish these operators. Incidentally, many people talk about the Stern-Gerlach apparatus to measure the spin polarization of the electron. This is also strange since N. Bohr and N. F. Mott and other prominent people have shown many years ago that one cannot measure the electron spin polarization by a Stern-Gerlach apparatus, since the inhomogeneous magnetic field disturbs the motion of the electron so much and it is impossible to detect the spin orientation. The Stern-Gerlach particles are electrically neutral. One may ask if novel features appear in high energy physics? Chiral symmetry (including the use of weak interaction) can become important such as in a neutrino-antineutrino pair in Z decay

Jun 29, 20150 citations

Zig-zag dynamics in a Stern–Gerlach spin measurement

Simon Krekels, Christian Maes, Kasper Meerts +1 more

The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.

arXiv:2311.13406Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesMar 1, 20243 citations

e^+ e^-

collisions at wide angles.

We investigate the role of time ordering in entanglement experiments involving unstable particles, focusing on

\mu^+ \mu^-

pairs produced in a maximally-entangled spin state. We analyse the correlations between measurements performed by two experimenters, Alice (who measures

\mu^-

spin) and Bob (who measures

\mu^+

decay products). Remarkably, the observed correlations persist irrespective of whether Bob's muon decays before or after Alice's spin measurement. We further discuss different interpretations of the same empirical results depending on the observer's reference frame. The findings reinforce the viewpoint that the Copenhagen interpretation of measurement is a mathematical tool rather than a literal account of physical reality.