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Physical Review A

Quantum coherence distribution and high-dimensional complementarity

Paula Machado, Carlos Henrique Monken, Sebastião Pádua

Published Jan 16, 2024• 6 citations
Publisher

Abstract

In the last few years, the ${l}_{1}$ norm has been used as a quantum coherence measure by many authors. In this work, we introduce a variable that mediates the complementarity relation between the ${l}_{1}$ norm of coherence, linear entropy, and the predictability of a quantum state of dimension greater than two. We show that this variable indicates the coherence distribution among the pairs of base states expanding the Hilbert space. We also show that a uniform quantum coherence distribution creates a direct complementarity between the ${l}_{1}$ norm, predictability, and linear entropy.

DOI
10.1103/physreva.109.012210

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In the last few years, the l1{l}_{1}l1​ norm has been used as a quantum coherence measure by many authors. In this work, we introduce a variable that mediates the complementarity relation between the l1{l}_{1}l1​ norm of coherence, linear entropy, and the predictability of a quantum state of dimension greater than two. We show that this variable indicates the coherence distribution among the pairs of base states expanding the Hilbert space. We also show that a uniform quantum coherence distribution creates a direct complementarity between the l1{l}_{1}l1​ norm, predictability, and linear entropy.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.
Abstract Quantum coherence is a relevant resource for various quantum information processing tasks, but it is fragile since it is generally affected by environmental noise. This is reflected in the loss of purity of the system, which in turn limits the amount of quantum coherence of it. As a consequence, a complementarity relation between coherence and mixedness arises. Previous works characterize this complementarity through inequalities between the ℓ 1-norm of coherence and linear entropy, and between the relative entropy of coherence and von Neumann entropy. However, coherence–mixedness complementarity is expected to be a general feature of quantum systems, regardless of the measures used. Here, an alternative approach to coherence–mixedness complementarity, based on majorization theory, is proposed. Vectorial quantifiers of coherence and mixedness, namely the coherence vector and the spectrum, respectively, are used, instead of scalar measures. A majorization relation for the tensor product of both vectorial quantifiers is obtained, capturing general aspects of the trade-off between coherence and mixedness. The optimal bound for qubit systems and numerical bounds for qutrit systems are analyzed. Finally, coherence–mixedness complementarity relations are derived for a family of symmetric, concave and additive functions. These results provide a deeper insight into the relation between quantum coherence and mixedness.
Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constraints on the accuracy of two noncommuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation using coherence between two measurement noncommuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1{l}_{1}l1​ norm of coherence.
Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constrains on the accuracy of two non-commuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation of coherence between two measurement non-commuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1l_1l1​ norm of coherence.
Establishing the correspondence of two dimensional paraxial and three dimensional non-paraxial optical beams with the qubit and qutrit systems respectively, we derive a complementary relation between Hilbert-Schmidt coherence, generalized predictability and linear entropy. The linear entropy, a measure of mixedness is shown to saturate the complementarity relation for mixed bi-partite states. For pure two qubit and qutrit systems, it quantifies the global entanglement and reduces the comple-mentarity relation to the triality relation between coherence, predictability and entanglement. We analyze these relations in wedge-product formalism in order to investigate the innate geometry of the complex vector space. The derived complementary relations offer insights into our ability to manipulate and utilize quantum properties for practical advancements.
Lower Bound of l1l_{1}l1​ Norm of Coherence of Bipartite Qubit-Qudit System and its Application in the Detection of Entangled Tripartite Qudit-Qubit-Qudit System
Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound LLL of l1l_{1}l1​ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound UUU of l1l_{1}l1​ norm of coherence of separable bipartite qubit-qudit system using the separability criterion. Thus, we find that if any l1l_{1}l1​ norm of coherence of bipartite qubit-qudit system is greater than the upper bound UUU then the given qubit-qudit state is entangled. Finally, we obtained the upper bound U1U_{1}U1​ of l1l_{1}l1​ norm of coherence of separable tripartite state lies either in 2⊗d⊗d2 \otimes d \otimes d2⊗d⊗d or d⊗2⊗dd \otimes 2 \otimes dd⊗2⊗d or d⊗d⊗2d \otimes d \otimes 2d⊗d⊗2 dimensional Hilbert space using the upper bound UUU. We have shown that if the l1l_{1}l1​ norm of coherence of any tripartite qubit−qudit−quditqubit-qudit-quditqubit−qudit−qudit or qudit−qubit−quditqudit-qubit-quditqudit−qubit−qudit or qudit−qudit−qubitqudit-qudit-qubitqudit−qudit−qubit system is greater than the derived upper bound U1U_{1}U1​ then the given tripartite system represent an entangled state.