Abstract This paper studies the order relations among various quantum coherence measures in a single-qubit system. Ten representative coherence measures are selected from three categories: distance-based, fidelity-based, and relative entropy-based measures. The analytical expressions for the coherence values are derived in the Bloch sphere representation of the single-qubit state, and the order relations within the same category and across categories are systematically analyzed. The study shows that coherence measures within the same category generally maintain consistent order relations in the single-qubit system. However, there is typically no fixed order relation between fidelity-based coherence measures (represented by geometric coherence) and relative entropy-based coherence measures (represented by α -affinity coherence measures). Yet, in pure single-qubit states and certain restricted sets of mixed states, these two types of coherence measures exhibit the same order relation. The results indicate that the order relations between coherence measures depend not only on their construction but also on the quantum state space structure and the constraints on parameters.
Abstract This paper studies the order relations among various quantum coherence measures in a single-qubit system. Ten representative coherence measures are selected from three categories: distance-based, fidelity-based, and relative entropy-based measures. The analytical expressions for the coherence values are derived in the Bloch sphere representation of the single-qubit state, and the order relations within the same category and across categories are systematically analyzed. The study shows that coherence measures within the same category generally maintain consistent order relations in the single-qubit system. However, there is typically no fixed order relation between fidelity-based coherence measures (represented by geometric coherence) and relative entropy-based coherence measures (represented by α -affinity coherence measures). Yet, in pure single-qubit states and certain restricted sets of mixed states, these two types of coherence measures exhibit the same order relation. The results indicate that the order relations between coherence measures depend not only on their construction but also on the quantum state space structure and the constraints on parameters.
Ordering states of Tsallis relative α-entropies of coherence
In this paper, we study the ordering states with Tsallis relative α-entropies of coherence and l1 norm of coherence for single-qubit states. We show that any Tsallis relative α-entropies of coherence and l1 norm of coherence give the same ordering for single-qubit pure states. However, they don't generate the same ordering for some high dimensional pure states, even though these states are pure. We also consider three special Tsallis relative α-entropies of coherence, such as C1 ,C2 and C21, and show any one of these three measures and Cl1 will not generate the same ordering for single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states.
In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and l1 norm of coherence. Firstly, we show that these measures give the same ordering for single-qubit states with a fixed mixedness or a fixed length along the direction σz. Secondly, we consider some special cases of high dimensional states, we show that these measures generate the same ordering for the set of high dimensional pure states if any two states of the set satisfy majorization relation. Moreover, these three measures generate the same ordering for all X states with a fixed mixedness. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find phase damping channel don't change the coherence-induced state ordering for some single-qubit states with fixed mixedness, instead amplitude damping channel change the coherence-induced ordering even though for single-qubit states with fixed mixedness.
The uncertainty relation reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, the quantum uncertainty relations of the sandwiched R\'enyi relative entropy of coherence (SRREOC) and unified $(\ensuremath{\alpha},\ensuremath{\beta})$-relative entropy of coherence (UREOC) are studied. First, the uncertainty relation of the SRREOC for pure states, with extension to mixed states by convex-roof construction, is given in a d-dimensional quantum system. Second, the analytical formula of the UREOC is presented; it is shown that the UREOC is a good candidate for a quantum coherence measure. Furthermore, the quantum uncertainty relation of the UREOC for pure states is obtained in a d-dimensional quantum system, then it is generalized to mixed states. In addition, our results cover the quantum uncertainty relations of the other three coherence measures. Third, two methods are given to compare the lower bounds of the SRREOC and UREOC for single-qubit pure states. By comparison, we find that the results in this paper not only can be applied to single-qubit states but also can be extended to a d-dimensional quantum system.
Abstract In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and l 1 norm of coherence, and give the sufficient conditions of the same state order induced by above coherence measures. First, we show that the above measures give the same ordering for single-qubit states in some special cases. Second, we consider some special states in a d -dimensional quantum system. We show that the above measures generate the same ordering for these special states. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find amplitude damping channel changes the coherence-induced ordering even though for single-qubit states with fixed mixedness.
In this paper, we study single-qubit coherence with the central question: when and why do different measures disagree enough to reverse channel orderings? First, we compile closed-form expressions and analyze behavior on pure and mixed states across widely used quantifiers. Next, we compare noisy evolutions using two resource-theoretic measures the [Formula: see text] norm [Formula: see text] and the relative entropy of coherence [Formula: see text] — on a common basis, enabling unbiased cross-model comparison via an iso-measure protocol (fix one measure, read off the other). Finally, we delineate conditions for ranking reversal: we establish a no-reversal regime for basis-consistent unital, phase-covariant evolutions on equatorial inputs and identify mechanisms that permit reversals nonunital population bias and axis mixing that couples transverse and longitudinal components. The framework yields actionable, side-by-side rankings and clarifies when conclusions depend on the chosen quantifier.
Characterizing noise in quantum circuits is fundamentally limited by gauge degrees of freedom; certain parameters, such as the individual contributions of state preparation and measurement (SPAM) errors, are in principle unlearnable from any experiment within the gate set. Here, we show that the physical structure of realistic noise processes imposes approximate symmetry constraints on the Pauli fidelities of gate noise channels. These symmetries relate the fidelity of a Pauli P and its gate-conjugate UgPUg†, and can be used to fix the gauge using only knowledge of the error type and not its magnitude. Using Lindbladian perturbation theory, we analyze a broad class of Clifford gates, including ZZπ/2, CZ, CNOT, iSWAP, and SWAP, and demonstrate that coherent errors do not induce first-order asymmetry, while only a restricted set of predominantly off-diagonal dissipative errors can break the symmetry at first order, for which we derive simple selection rules. Notably, common single-qubit noise sources such as T1-relaxation and T2φ-pure-dephasing can only cause asymmetry at second order. Leveraging these symmetries to fix the gauge enables systematic identification of SPAM errors, simplifying error characterization and mitigation. We validate our results numerically and experimentally on IBM Kingston.