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Foundations of Physics

Quantumness of States and Unitary Operations

Joanna Luc

Published Oct 30, 2020• 6 citations
Publisher

Abstract

Abstract This paper investigates various properties that may by possessed by quantum states, which are believed to be specifically “quantum” (entanglement, nonlocality, steerability, negative conditional entropy, non-zero quantum discord, non-zero quantum super discord and contextuality) and their opposites. It also considers their “absolute” counterparts in the following sense: a given state has a given property absolutely if after an arbitrary unitary transformation it still possesses it. The known relations between the listed properties and between their absolute counterparts are summarized. It is proven that the only two-qubit state that has zero quantum discord absolutely is the maximally mixed state. Finally, related conceptual issues concerning the terms “classical” and “quantum” are discussed.

DOI
10.1007/s10701-020-00391-z

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Abstract This paper investigates various properties that may by possessed by quantum states, which are believed to be specifically “quantum” (entanglement, nonlocality, steerability, negative conditional entropy, non-zero quantum discord, non-zero quantum super discord and contextuality) and their opposites. It also considers their “absolute” counterparts in the following sense: a given state has a given property absolutely if after an arbitrary unitary transformation it still possesses it. The known relations between the listed properties and between their absolute counterparts are summarized. It is proven that the only two-qubit state that has zero quantum discord absolutely is the maximally mixed state. Finally, related conceptual issues concerning the terms “classical” and “quantum” are discussed.
The subject of this thesis are various properties of quantum states that make them "non-classical" and their behaviour under unitary operations. In chapter 2 some basic concepts of quantum mechanics and quantum information are reviewed. In chapter 3 properties of quantum states analysed in the literature are identified. I chapter 4 relations between them are analysed: whether possessing of one property by a given state implies possessing another property by the same state. In chapter 5 the central notion of the thesis is introduced, namely possessing a given property in absolute vs. non-absolute way. A property is possessed by a given state in an absolute way if it is preserved under arbitrary unitary operation on this state. This will be applied notion to all properties listed in chapter 3. Two main questions asked here are as follows: Can a given property be possessed both in an absolute way and in a non-absolute way (by different classes of states)? If yes, what are necessary and sufficient conditions for a state possessing a given property in an absolute way?
Measurements of Quantum Systems disturb their states. To quantify this non-classical characteristic, Zurek and Ollivier introduced the quantum discord, a quantum correlation which can be nonzero even when entanglement in the system is zero. Discord has aroused great interest as a resource that is more robust against the effects of decoherence and offers exponential speed up of certain computational algorithms. Here, we study general two-level bipartite systems and give general results on the relationship between discord, entanglement, and linear entropy, and identify the states for which discord takes a maximal value for a given entropy or entanglement, thus placing strong bounds on entanglement-discord and entropy-discord relations. We find out that although discord and entanglement are identical for pure states, they differ when generalized to mixed states as a result of the difference in the method of generalization.
In contrast with entanglement, as measured by concurrence, in general, quantum discord does not possess the property of monogamy; that is, there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party. Here, we show that, as far as monogamy is considered, quantum discord of pure states is equivalent to the entanglement of formation. This result allows one to analytically prove that none of the pure three-qubit states belonging to the subclass of WWW states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.
Due to some ambiguity in the definition of mutual Tsallis entropy in classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discords, called q-discords, are defined in terms of quantum Tsallis entropy. Both q-discords for two-qubit Werner and isotropic states are determined and compared and it is shown that one of them is non-negative, at least for states under investigation, for all q > 0. Finally, an analytical expression for q-discord of certain family of two-qubit X-states is presented. Using this example, we show that both types of q-discords can take negative values for some q > 1, hence their use as correlations measures is rather limited.
We give a pedagogical introduction to quantum discord and discuss the problem of separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy. This allows us to put all correlations on an equal footing. Entanglement and dissonance jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We finally show, using relative entropy, how different notions of quantum correlations are related to each other. This gives a single theory that incorporates all correlations, quantum and classical, and different methods of quantifying them.