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Bell Measurements and Observables

D'Ariano, G M, Lo Presti, P, Sacchi, M F

Published Jan 1, 2000• 80 citations
OpenAlex

Abstract

A general matrix approach to study entangled states is presented, based on operator completeness relations. Bases of unitary operators are considered, with focus on irreducible representations of groups. Bell measurements for teleportation are considered, and robustness of teleportation to various kinds of non idealities is shown.

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Quantum Teleportation: Entanglement, Bell Measurement, and Pauli Correction — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum teleportation relies on shared entanglement and classical communication, not instantaneous matter transfer; mathematical core is Bell state measurement and unitary transformation. | MATH: Bell state basis: \(|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)\), \(|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)\). Teleportation protocol: Alice performs Bell measurement on her qubit and entangled pair, sends 2 classical bits to Bob, who applies Pauli correction \(I, X, Z, XZ\) based on outcome. Fidelity = \(|\langle \psi | \rho_{\text{out}} | \psi \rangle|^2\). No faster-than-light signaling: classical bits limit speed to ≤ c. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear. However, Bell states are maximally entangled, corresponding to the four vertices of a regular tetrahedron in the Bloch sphere representation (SO(3) symmetry). The Pauli matrices \(X, Y, Z\) generate the Clifford group, which is the sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 6, 2026•0 citations

From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications

Marek Mozrzymas, Michał Horodecki, Michał Studziński

Abstract In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group $$S(n-1)\subset S(n)$$ S ( n - 1 ) ⊂ S ( n ) . This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.

arXiv:2310.16423•Letters in Mathematical Physics•Apr 16, 2024•1 citation
Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kinds of measurement can be achieved through positive-operator valued measures (POVMs) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.
Bell states and quantum teleportation play crucial roles in quantum information and computation. However, a comprehensive theoretical study of both topics remains to be carried out. This work aims to investigate key algebraic properties of generalized Bell states and explore the topological features of quantum teleportation. First, the basis theorem and the basis group are introduced to show that the extension of a generalized Bell basis by a unitary matrix still forms an orthonormal basis. Then, a twist operator is defined to establish a connection between a generalized multi-qubit Bell state and a tensor product of two-qubit Bell states. In addition, the Temperley--Lieb algebra, the braid group relations, and the Yang--Baxter equation are employed to provide a topological description of generalized Bell states and quantum teleportation. The results demonstrate that our approach not only offers a clear illustration of relevant quantum information protocols but also reveals the topological nature of quantum entanglement and teleportation.
This chapter is concerned with the measure of information contained in qubits . This can be done only through quantum measurement , an operation that has no counterpart in the classical domain. I shall first describe in detail the case of single qubit measurements, which shows under which measurement conditions “classical” bits can be retrieved. Next, we consider the measurements of higher-order or n -qubits. Particular attention is given to the Einstein–Podolsky–Rosen (EPR) or Bell states , which, unlike other joint tensor states, are shown being entangled . The various single-qubit measurement outcomes from the EPR–Bell states illustrate an effect of causality in the information concerning the other qubit. We then focus on the technique of Bell measurement , which makes it possible to know which Bell state is being measured, yielding two classical bits as the outcome. The property of EPR–Bell state entanglement is exploited in the principle of quantum superdense coding , which makes it possible to transmit classical bits at twice the classical rate, namely through the generation and measurement of a single qubit. Another key application concerns quantum teleportation . It consists of the transmission of quantum states over arbitrary distances, by means of a common EPR–Bell state resource shared by the two channel ends. While quantum teleportation of a qubit is instantaneous, owing to the effect of quantum-state collapse, it is shown that its completion does require the communication of two classical bits , which is itself limited by the speed of light.
FINDING: Quantum teleportation relies on shared entanglement and classical communication, not instantaneous matter transfer; mathematical core is Bell state measurement and unitary transformation. | MATH: Bell state basis: ∣Φ±⟩=12(∣00⟩±∣11⟩)|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)∣Φ±⟩=2​1​(∣00⟩±∣11⟩), ∣Ψ±⟩=12(∣01⟩±∣10⟩)|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)∣Ψ±⟩=2​1​(∣01⟩±∣10⟩). Teleportation protocol: Alice performs Bell measurement on her qubit and entangled pair, sends 2 classical bits to Bob, who applies Pauli correction I,X,Z,XZI, X, Z, XZI,X,Z,XZ based on outcome. Fidelity = ∣⟨ψ∣ρout∣ψ⟩∣2|\langle \psi | \rho_{\text{out}} | \psi \rangle|^2∣⟨ψ∣ρout​∣ψ⟩∣2. No faster-than-light signaling: classical bits limit speed to ≤ c. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear. However, Bell states are maximally entangled, corresponding to the four vertices of a regular tetrahedron in the Bloch sphere representation (SO(3) symmetry). The Pauli matrices X,Y,ZX, Y, ZX,Y,Z generate the Clifford group, which is the sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n−1)⊂S(n)S(n-1)\subset S(n)S(n−1)⊂S(n) S ( n - 1 ) ⊂ S ( n ) . This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.