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Signal Processing and Communications Applications Conference (SIU)

Analysis of Negativity and Relative Entropy of Entanglement measures for qubit-qutrit Quantum Communication systems

Volkan Erol, Fatih Özaydin, Azmi Ali Altıntaṣ

Published May 1, 2015• 11 citations
Publisher

Abstract

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.

DOI
10.1109/siu.2015.7129800

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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.
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 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.
We review some counterintuitive properties of standard measures describing quantum entanglement and violation of Bell's inequality (often referred to as “nonlocality”) in two-qubit systems. By comparing the nonlocality, negativity, concurrence, and relative entropy of entanglement, we show: (i) ambiguity in ordering states with the entanglement measures, (ii) ambiguity of robustness of entanglement in lossy systems and (iii) existence of two-qubit mixed states more entangled than pure states having the same negativity or nonlocality. To support our conclusions, we performed a Monte Carlo simulation of 106two-qubit states and calculated all the entanglement measures for them. Our demonstration of the relativity of entanglement measures implies also how desirable is to properly use an operationally-defined entanglement measure rather than to apply formally-defined standard measures. In fact, the problem of estimating the degree of entanglement of a bipartite system cannot be analyzed separately from the measurement process that changes the system and from the intended application of the generated entanglement.
It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.
The problem of ordering of two-qubit states imposed by the relative entropy of entanglement ( E ) in comparison with the concurrence ( C ) and negativity ( N ) is studied. Analytical examples of states consistently and inconsistently ordered by the entanglement measures are given. In particular, the states for which any of the three measures imposes order opposite to that given by the other two measures are described. Moreover, examples are given of pairs of the states for which (i) and but is different from , (ii) and but differs from , (iii) , and ; and (iv) states having the same E , C and N but still violating the Bell–Clauser–Horne–Shimony–Holt inequality to different degrees.
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.