Abstract
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.
Highlights
The linearity of quantum mechanics gives rise to the concept of superposition of quantum states of a quantum system and is one of the characteristic properties that makes a clear distinction between the ways a classical and a quantum system can behave
We explore how the coherence of a given quantum state in one reference basis is restricted by the coherence of the same quantum state in other reference bases for single and bipartite quantum systems
We provide a non trivial upper bound on the absolute value of the differences between the relative entropy of coherence obtained in two different bases
Summary
The linearity of quantum mechanics gives rise to the concept of superposition of quantum states of a quantum system and is one of the characteristic properties that makes a clear distinction between the ways a classical and a quantum system can behave. [2], provides a set of conditions on a real valued function of quantum states for it to be a bona fide quantifier of quantum coherence This resource theory is based on the set of incoherent operations as the free operations and the set of incoherent states as the set of free states. We provide lower and upper bounds on the sum of the relative entropies of coherence of a given state with respect to two or more incompatible bases. The lower bounds on the sum of the relative entropies of coherence for single and bipartite quantum systems are facilitated through the use of entropic uncertainty relations with and without memory effects. (the superscript i is used to denote the reference basis) is said to be a valid measure of quantum coherence of the state ρ if it satisfies following conditions: (1) C (i) (ρ) = 0 iff ρ ∈ I .
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