| Summary: | Quantum coherence is known as an important resource in many quantum information tasks, which is a basis-dependent property of quantum states. In this paper, we discuss quantum incoherence based simultaneously on <i>k</i> bases using Matrix Theory Method. First, by defining a correlation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>(</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula> of two orthonormal bases <i>e</i> and <i>f</i>, we investigate the relationships between sets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">I</mi><mo>(</mo><mi>e</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">I</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula> of incoherent states with respect to <i>e</i> and <i>f</i>. We prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">I</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>=</mo><mi mathvariant="script">I</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula> if and only if the rank-one projective measurements generated by <i>e</i> and <i>f</i> are identical. We give a necessary and sufficient condition for the intersection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">I</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>⋂</mo><mi mathvariant="script">I</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula> to include a state except the maximally mixed state. Especially, if two bases <i>e</i> and <i>f</i> are mutually unbiased, then the intersection has only the maximally mixed state. Secondly, we introduce the concepts of strong incoherence and weak coherence of a quantum state with respect to a set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula> of <i>k</i> bases and propose a measure for the weak coherence. In the two-qubit system, we prove that there exists a maximally coherent state with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and it is not the case for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>.
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