arXiv subjects
Publications and source records attributed to Reinhold A. Bertlmann.
Some personal memories of the Nobel Prize Winners 2022 Alain Aspect, John Clauser, and Anton Zeilinger are summarized and the significance of their works is described in a historical perspective.
My discussions with John Bell about reality in quantum mechanics are recollected. I would like to introduce the reader to Bell's vision of reality which was for him a natural position for a scientist. Bell had a strong aversion against "quantum jumps" and insisted to be clear in phrasing quantum mechanics, his "words to be forbidden" proclaimed with seriousness and wit -- both typical Bell characteristics -- became legendary. I will summarize the Bell-type experiments and what Nature responded, and discuss the implications for the physical quantities considered, the real entities and the nonlocality concept due to Bell's work. Subsequently, I also explain a quite different view of the meaning of a quantum state, this is the information theoretic approach, focussing on the work of Brukner and Zeilinger. Finally, I would like to broaden and contrast the reality discussion with the concept of "virtuality", with the meaning of virtual particle occurring in quantum field theory. With some of my own thoughts I will conclude the paper which is composed more as a historical article than as a philosophical one.
We review the geometric features of negative conditional entropy and the properties of the conditional amplitude operator proposed by Cerf and Adami for two qubit states in comparison with entanglement and nonlocality of the states. We identify the region of negative conditional entropy in the tetrahedron of locally maximally mixed two-qubit states. Within this set of states, negative conditional entropy implies nonlocality and entanglement, but not vice versa, and we show that the Cerf-Adami conditional amplitude operator provides an entanglement witness equivalent to the Peres-Horodecki criterion. Outside of the tetrahedron this equivalence is generally not true.
My collaboration and friendship with John Bell is recollected. I will explain his outstanding contributions in particle physics, in accelerator physics, and his joint work with Mary Bell. Mary's work in accelerator physics is also summarized. I recall our quantum debates, mention some personal reminiscences, and give my personal view on Bell's fundamental work on quantum theory, in particular, on the concept of contextuality and nonlocality of quantum physics. Finally, I describe the huge influence Bell had on my own work, in particular on entanglement and Bell inequalities in particle physics and their experimental verification, and on mathematical physics, where some geometric aspects of the quantum states are illustrated.
I present my encounter with John Bell at CERN, our collaboration and joint work in particle physics. I also will recall our quantum debates and give my personal view on Bell's fundamental work on quantum theory, in particular, on contextuality and nonlocality of quantum physics. Some mathematical and geometric aspects of entanglement are discussed as influence of Bell's Theorem. Finally, I make some historical comments on the experimental side of Bell inequalities.
The familiar Greenberger-Horne-Zeilinger (GHZ) states can be rewritten by entangling the Bell states for two qubits with a state of the third qubit, which is dubbed entangled entanglement. We show that in this way we obtain all 8 independent GHZ states that form the simplex of entangled entanglement, the magic simplex. The construction procedure allows a generalization to higher dimensions both, in the degrees of freedom (considering qudits) as well as in the number of particles (considering n-partite states). Such bases of GHZ-type states exhibit a certain geometry that is relevant for experimental and quantum information theoretic applications. Furthermore, we study the geometry of these particular state spaces, the inherent symmetries, the cyclicity of the phase operations, and the regions of (genuine multi-partite) entanglement and the several classes of separability. We find non-trivial geometrical properties and a conceptually clear procedure to compare state spaces of different dimensions and number of particles.
We trace back the phenomenon of "delayed-choice entanglement swapping" as it was realized in a recent experiment to the commutativity of the projection operators that are involved in the corresponding measurement process. We also propose an experimental set-up which depends on the order of successive measurements corresponding to noncommutative projection operators. In this case entanglement swapping is used to teleport a quantum state from Alice to Bob, where Bob has now the possibility to examine the noncommutativity within the quantum history.
We analyze the operational meaning of the residual entanglement in non-inertial fermionic systems in terms of the achievable violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. We demonstrate that the quantum correlations of fermions, which were previously found to survive in the infinite acceleration limit, cannot be considered to be non-local. The entanglement shared by an inertial and an accelerated observer cannot be utilized for the violation of the CHSH inequality in case of high accelerations. Our results are shown to extend beyond the single mode approximation commonly used in the literature.
We discuss the concept of how entanglement changes with respect to different factorizations of the total algebra which describes the quantum states. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this free choice of factorization, which is naturally fixed for an experimentalist. For theorists it offers an extension of the interpretations and is adequate to generalizations, as we point out in the examples of quantum teleportation and entanglement swapping.
We describe the spin and momentum degrees of freedom of a system of two massive spin--$\tfrac{1}{2}$ particles as a 4 qubit system. Then we explicitly show how the entanglement changes between different partitions of the qubits, when considered by different inertial observers. Although the two particle entanglement corresponding to a partition into Alice's and Bob's subsystems is, as often stated in the literature, invariant under Lorentz boosts, the entanglement with respect to other partitions of the Hilbert space on the other hand, is not. It certainly does depend on the chosen inertial frame and on the initial state considered. The change of entanglement arises, because a Lorentz boost on the momenta of the particles causes a Wigner rotation of the spin, which in certain cases entangles the spin- with the momentum states. We systematically investigate the situation for different classes of initial spin states and different partitions of the 4 qubit space. Furthermore, we study the behavior of Bell inequalities for different observers and demonstrate how the maximally possible degree of violation, using the Pauli-Lubanski spin observable, can be recovered by any inertial observer.
We establish a relation between entanglement of a many-body system and its diffractive properties, where the link is given by structure factors. Based on these, we provide a general analytical construction of multi-qubit entanglement witnesses. The proposed witnesses contain two-point correlations. They could be either measured in a scattering experiment or via local measurements, depending on the underlying physical system. For some explicit examples of witnesses we analyze the properties of the states that are detected by them. We further study the robustness of these witnesses with respect to noise.
We construct entanglement witnesses with regard to the geometric structure of the Hilbert--Schmidt space and investigate the geometry of entanglement. In particular, for a two--parameter family of two--qutrit states that are part of the magic simplex we calculate the Hilbert--Schmidt measure of entanglement. We present a method to detect bound entanglement which is illustrated for a three--parameter family of states. In this way we discover new regions of bound entangled states. Furthermore we outline how to use our method to distinguish entangled from separable states.
We investigate the entanglement properties of a three--parameter family of states that are part of the magic simplex of two qutrits, which is a simplex of states that are mixtures of maximally entangled two--qutrit Bell states. Using entanglement witnesses we reveal large regions of bound entangled and separable states.
We present three different matrix bases that can be used to decompose density matrices of $d$--dimensional quantum systems, so-called qudits: the \emph{generalized Gell-Mann matrix basis}, the \emph{polarization operator basis}, and the \emph{Weyl operator basis}. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so--called standard matrices, consider the important case of an isotropic two--qudit state and decompose it according to each basis. In case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements, which is appropriate for an experimental realization.
We study entanglement witnesses that can be constructed with regard to the geometrical structure of the Hilbert-Schmidt space, i.e. we present how to use these witnesses in the context of quantifying entanglement and the detection of bound entangled states. We give examples for a particular three-parameter family of states that are part of the \emph{magic simplex} of two-qutrit states.
We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it according to each basis. Investigating the geometry of entanglement of special parameterized two--qubit and two--qutrit states, in particular we calculate the Hilbert--Schmidt measure of entanglement, we find that the Weyl operator basis is the optimal choice since it is closely connected to the entanglement of the considered states.
The nonlocal property of quantum mechanics can be nicely tested in high energy physics; in particular, the neutral kaon pairs as produced at DA$Φ$NE, Frascati, are very well suited. The analogies of kaons as compared to polarized photons or spin--1/2 particles --the kaonic qubit feature-- are reviewed. However, there are also fundamental differences which occur due to the kaon time evolution and due to internal symmetries; in particular, the violation of CP symmetry is related to the violation of Bell inequalities. Two type of Bell inequalities for kaons are presented, one for the variation of the ``quasi--spin'' and the other for different detection times of the kaon.
We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-hermitian Hamiltonian H_{eff}, in accord with the Wigner--Weisskopf approximation, and an additional term of the Lindblad form, the socalled dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-hermitian part of H_{eff} --the ``particle decay''-- can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the new formulation of the time evolution on the enlarged space has a hermitian Hamiltonian and is probability conserving. The equivalence of the new formulation with the original one demonstrates that the time evolution which is governed by a non-hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with hermitian Hamiltonians.