Interference and entanglement: an intrinsic approach

We introduce the notion of bipartite entanglement for pure states in finite-dimensional quantum mechanical systems and classify the states into two types depending on whether they contain non-trivial entanglement entropy that quantifies the amount of quantum entanglement. A set of inequalities of entanglement entropy which play crucial rules will be stated without proofs. I. BIPARTITE ENTANGLEMENT Given a lattice model or QFT, suppose the system is in a pure ground state |ψ that is the density matrix for the Hilbert space H tot is given by ρ tot = |ψψ|. We normalize the ground state as ψ|ψ = 1 so that tr tot (ρ tot) = 1. We then divide the total system into two subsystems A and B = A complementary to each other. In the spin chain example, we cut off the chain in between the sites and divide the lattice points into two groups. Note that the procedure of cutting is an imaginary process without changing the system at all. In what follows, the total Hilbert space will be assumed to take a direct product form of two Hilbert spaces pf the subsystems H tot = H A ⊗ H B. Let {|i B , i = 1, 2, ...} be an orthonormal basis in H B and define the reduced density matrix ρ A of the system A by taking the partial trace over the system B ρ A ≡ tr B (ρ tot) ≡ i B i|ρ tot |i B. Note that this definition depends on the choice of the subsystem but does not depend on the choice of the orthonormal basis. For example if ρ tot is the tensor product of the density matrices ρ A and ρ B describing the subsystems that is ρ tot = ρ A ⊗ ρ B then the partial trace simply recovers them tr B (ρ tot) = ρ A , tr A (ρ tot) = ρ B. The total density matrix needs not be pure but one may ask once the complete information ρ A about a system A is given if it is possible to find a pure density matrix in an enlarged Hilbert space of H A whose partial trace recovers ρ A. Indeed one can always construct such an enlarged Hilbert space and the density matrix in the following way. The most general density matrix is of the form ρ A = i p i |i AA i| where {|i A } is an orthonormal basis of H A and the coefficients p i ≥ 0 satisfy i p i = 1. We then copy H A into another Hilbert space H A with the basis given by {|i A } and define the pure density matrix ρ by ρ = |χχ|, |χ ≡ i √ p i |i A ⊗ |i A in the enlarged Hilbert space H = H A ⊗ H A. It is easy to check that this construction correctly reproduces ρ A under the partial trace over A.The operation is called entanglement purification as it constructs

2019

This work is a collection of papers on quantum entanglement. It is intended as a glimpse for the younger colleagues of the author at Ericsson Hungary. Our intention was to introduce as many features as possible, within a readable extent. Our selection criteria are very wide, the work spans the gap from university lecture through research paper to magazine article. Our goal is to illustrate that the topic is very interesting and it covers several unsolved problems. It can be a good basis for research.

Physical Review Letters, 2008

We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is directly related to the Meyer-Wallach multipartite entanglement measure. Using channels that depolarize across different partitions, we obtain related multipartite entanglement measures. We show that this measure is the sum of expectation values of local observables on two copies of the state.

International Journal of Modern Physics B, 2013

The aspects of many particle systems as far as their entanglement is concerned is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.

2003

The significance of the quantum feature of entanglement between physical systems is investigated in the context of quantum measurements. It is shown that, while there are measurement couplings that leave the object and probe systems nonentangled, no information transfer from object to probe can take place unless there is at least some intermittent period where the two systems are entangled.

2005

International Journal of Theoretical Physics, 2018

We investigate entanglement and separability criteria of multipartite (n-partite) state by examining ranks of its reduced density matrices. Firstly, we construct the general formula to determine the criterion. A rank of origin density matrix always equals one, meanwhile ranks of reduced matrices have various ranks. Next, separability and entanglement criterion of multipartite is determined by calculating ranks of reduced density matrices. In this article we diversify multipartite state criteria into completely entangled state, completely separable state, and compound state, i.e. sub-entangled state and subentangledseparable state. Furthermore, we also shorten the calculation proposed by the previous research to determine separability of multipartite state and expand the methods to be able to differ multipartite state based on criteria above.

2017

We show that the two notions of entanglement: the maximum of the geometric measure of entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-density tensor is separable if and only if its nuclear norm is $1$. We suggest an alternating method for computing the nuclear norm of tensors. We apply the above results to symmetric tensors. We give many numerical examples.

Academia Quantum, 2024

Complementary relationships exist among interference properties of particles such as pattern visibility, predictability, and distinguishability. Additionally relationships between average information gain G ̄ and measurement disturbance F for entangled spin pairs are well established. This article examines whether a similar complementary relationship exists between entanglement and measurement. For qubit systems, both measurements on a single system and measurements on a bipartite system are considered in regard to entanglement. It is proven that E ̄ + D ≤ 1 holds, where E ̄ is the average entanglement after a measurement is made and D is a measure of the measurement disturbance of a single measurement. Assuming measurements on a bipartite system shared by Alice and Bob, it is shown that E ̄ + G ̄ ≤ 1, where G ̄ is the maximum average information gain that Bob can obtain regarding Alice’s result. These results are generalized to arbitrary initial mixed states and non-Hermitian operators. In the case of maximally entangled initial states, it is found that D ≤ EL and G ̄ ≤ EL, where EL is the loss of entanglement due to measurement by Alice. We conclude that the amount of disturbance and average information gain one can achieve is strictly limited by entanglement.

We propose two necessary sufficient (NS) criteria to decide the separability of quantum states. They follow from two independent ideas: i) the Bloch-sphere-like-representation of states and ii) the proportionality of lines (rows, columns etc.) of certain multimatrix [1] associated with states. The second criterion proposes a natural way to determine the possible partial (or total, when possible) factorization of given multipartite state and in a sense can be used to determine the structure of the entanglement. We also introduce three entanglement measures based on the proposed new characterizations of entanglement. At last we discuss the second criterion mentioned above in the language of density matrix which is an inevitable language especially for mixed states.

2003

Entanglement is perhaps the most important new feature of the quantum world. It is expressed in quantum theory by the joint measurement formula. We prove the formula for projection valued observables from a plausible assumption, which for spacelike separated measurements is a consequence of causality. State reduction is simply a way to express the joint measurement formula after one measurement has been made, and its result known.

2012

Quantum entanglement is a huge and active research field these days. Not only the philosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting, but also the possibilities to make use of it in our everyday life is thrilling. In the last few years many possible applications, mostly within the ’Quantum Information’ field, have been developed. Of course to make use of this feature one demands tools to control entanglement in a certain sense. How can one define entanglement? How can one identify an entangled quantum system? Can entanglement be measured? These are questions one desires an answer for and indeed many answers have been found. However today entanglement is not yet fully in control by mathematics; many problems are still not solved. This paper aims to provide a theoretical introduction to get a feeling for the mathematical problems concerning entanglement and presents approaches to handle entanglement identification or entanglement measures for s...

Physical Review A, 2011

Achieving a satisfactory understanding of the nature and structure of quantum correlations is of paramount importance in quantum-information theory [1] as well as in the study of complex quantum systems [2]; very recent speculations even hint at possible fundamental roles played by quantum entanglement in biological systems and processes with enhanced properties of quantum coherence [3]. For bipartite quantum systems prepared in globally pure states, there is universal consensus on the fact that quantum correlations identify with ...

Physical Review A, 2006

Arxiv preprint arXiv:0706.2019, 2007

Abstract: We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is ...