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Signal Processing and Communications Applications Conference (SIU)

An analysis of concurrence entanglement measure and quantum fisher information of quantum communication networks of two-qubits

Volkan Erol, Sinan Bugu, Fatih Özaydin, Azmi Ali Altıntaṣ

Published Apr 1, 2014• 13 citations
Publisher

Abstract

Quantum entanglement is one of key concepts in quantum communication engineering. Ordering the quantum systems according to their entanglement measures is a popular problem of the field. For two level (qubit) systems of two particles, state ordering has been studied with respect to well-known entanglement measures such as Concurrence, Negativity and Relative Entropy of Entanglement (REE) [1-5]. In this work, we study the state ordering of the two-qubit systems with respect to Quantum Fisher Information vs. Concurrence. In particular, constructing 1K random states and calculating their Concurrences and Negativities, we obtain the orderings of the states by comparing these results with Quantum Fisher Information values and present our results which are interesting when compared to that of two-level systems.

DOI
10.1109/siu.2014.6830229

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Quantum entanglement is one of key concepts in quantum communication engineering. Ordering the quantum systems according to their entanglement measures is a popular problem of the field. For two level (qubit) systems of two particles, state ordering has been studied with respect to well-known entanglement measures such as Concurrence, Negativity and Relative Entropy of Entanglement (REE) [1-5]. In this work, we study the state ordering of the two-qubit systems with respect to Quantum Fisher Information vs. Concurrence. In particular, constructing 1K random states and calculating their Concurrences and Negativities, we obtain the orderings of the states by comparing these results with Quantum Fisher Information values and present our results which are interesting when compared to that of two-level systems.
Entanglement has been studied extensively for unveiling the mysteries of non-classical correlations between quantum systems. In the bipartite case, there are well known measures for quantifying entanglement such as concurrence, relative entropy of entanglement (REE) and negativity, which cannot be increased via local operations. It was found that for sets of non-maximally entangled states of two qubits, comparing these entanglement measures may lead to different entanglement orderings of the states. On the other hand, although it is not an entanglement measure and not monotonic under local operations, due to its ability of detecting multipartite entanglement, quantum Fisher information (QFI) has recently received an intense attraction generally with entanglement in the focus. In this work, we revisit the state ordering problem of general two qubit states. Generating a thousand random quantum states and performing an optimization based on local general rotations of each qubit, we calculate the maximal QFI for each state. We analyze the maximized QFI in comparison with concurrence, REE and negativity and obtain new state orderings. We show that there are pairs of states having equal maximized QFI but different values for concurrence, REE and negativity and vice versa.
Quantum Computers are provisioned as very high performance computers using quantum mechanical aspects for information processing. Quantum Information Theory and Quantum Computing topics are popular topics in academia aiming the construction of theoretical background of Quantum Computers. The information processing units for Quantum Computers are defined as qubits but for some problems three level (trinary) systems may be applied as well. We call these three level systems as qutrits. The scope of this work is the analysis of well-defined entanglement measures Negativity and Relative Entropy of Entanglement (REE) for two qutrit (3- level/trinary) quantum systems. In this manner, for randomly generated 1000 two qutrit and 1000 two qubit states the mentioned measures are calculated and these values are compared. These comparisons are analyzed and for quantum state ordering problem, some interesting results are reported.
Quantum Computers are provisioned as very high performance computers using quantum mechanical aspects for information processing. Quantum Information Theory and Quantum Computing topics are really popular topics in academia aiming the construction of theoretical background of Quantum Computers. The information processing units for Quantum Computers are defined as qubits but for some problems three level (trinary) systems may be applied as well. We call these three level systems as qutrits. The scope of this work is the analysis of well-defined entanglement measures Negativity and Relative Entropy of Entanglement (REE) [1-5] for qubit-qutrit quantum systems. In this manner, for randomly generated 1000 qubit-qutrit and 1000 two qubit states the mentioned measures are calculated and these values are compared. These comparisons are analyzed and for quantum state ordering problem, very interesting results are reported.
Entanglement measures such as concurrence, negativity and REE are well-known tools for quantifying non-classical correlations in quantum systems. However, these measures can lead to different state orderings for non-maximally entangled states. On the other hand, Quantum Fisher Information (QFI), while not an entanglement measure, provides a framework for analyzing a state's metrological potential. In this study, we numerically analyze the relationship between these entanglement measures and QFI for a large ensemble of random two-qubit states. We specifically focus on the maximized QFI (MQFI) obtained through local unitary rotations. Our findings demonstrate a strong correlation between entanglement and a state's metrological capacity, confirming that entanglement is a valuable resource. We show that while a state's QFI with a fixed generator can vary widely for a given entanglement value, local optimization to find the MQFI leads to a tighter, more predictable relationship. Our results also reinforce the principle that the metrological performance of all mixed states is fundamentally bounded by the theoretical limit of pure states. The polynomial fit equations for the upper and lower bounds of our data provide a quantitative description of these complex relationships.
Entanglement measures such as concurrence, negativity and REE are well-known tools for quantifying non-classical correlations in quantum systems. However, these measures can lead to different state orderings for non-maximally entangled states. On the other hand, Quantum Fisher Information (QFI), while not an entanglement measure, provides a framework for analyzing a state's metrological potential. In this study, we numerically analyze the relationship between these entanglement measures and QFI for a large ensemble of random two-qubit states. We specifically focus on the maximized QFI (MQFI) obtained through local unitary rotations. Our findings demonstrate a strong correlation between entanglement and a state's metrological capacity, confirming that entanglement is a valuable resource. We show that while a state's QFI with a fixed generator can vary widely for a given entanglement value, local optimization to find the MQFI leads to a tighter, more predictable relationship. Our results also reinforce the principle that the metrological performance of all mixed states is fundamentally bounded by the theoretical limit of pure states. The polynomial fit equations for the upper and lower bounds of our data provide a quantitative description of these complex relationships.
Studies about Quantum Information Theory continue actively in many research institutions. Very recently, pratical setups of large scale quantum computers are widely studied e.g. quantum repeaters, memories and processors. Entanglement provides us a computational advantage in realization of quantum algorithms. Some ways to quantifiying entanglement were defined. The best formal way to quantify it, is the methods that we call Entanglement Measures or Entanglement Monotones. In this research area, State Ordering Problem is defined and still an open problem especially for multiparticle entangled states. Fisher Information provides a good background for the solution of some actual Information Theory problems. Quantum Fisher Information (QFI) is a value that could be used in situations where phase sensitivity is important and this concept is expected to be the quantum version of the mentioned inftrastructure for Quantum Systems. QFI cannot be defined as an Entaglement Measure or Monotone. In the scope of this thesis, a new optimization technique is proposed for optimizing QFI and thanks to this optimization method, the ordering relation between QFI and entanglement measures is studied. Based on the studies made in the area of quantum state ordering, new and interesting analysis results are found and reported. The main important and interesting result achieved is that for two qubit quantum states, QFI maximized under Local Operation and Classical Communication has an interesting ordering relation with entanglement measures especially with Relative Entropy of Entanglement. Our study is extended for the qubit-qutrit systems and ordering relations and classification results are presented. For some quantum systems, the changes in QFI under decoherence channels are also considered.