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arXiv:2610.11995

Unitary-accessible coherence-erasure distance: Exact qubit solution and a tight qutrit bound

Dong-Ping Xuan, Hua Nan, Zhi-Xi Wang, Shao-Ming Fei

quant-phSubmitted Oct 8, 2026• Announced Oct 9, 2026• 0 citations
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Abstract

We introduce the unitary-accessible coherence-erasure distance, which measures the minimum unitary-control cost required to transform a quantum state into an incoherent state without changing its spectrum. The target states are restricted to incoherent states on the same unitary orbit, leading to a geometry different from conventional state-space proximity. We establish inequalities relating orbit-restricted quantum transport, Bures geometry, and the unitary-control distance. These quantities coincide for pure states but can differ for mixed states. For qubits, we derive an exact expression and show that, in the nondegenerate case, the coherence-erasure cost depends on the eigenbasis orientation rather than on the eigenvalues. For nondegenerate qutrits, we reduce the problem to an optimization over the monomial unitary group. We derive the tight universal bound $\arccos[(2\sqrt2-1)/4]$ and construct an explicit eigenbasis that attains it. The resulting distance also determines the minimum coherence-erasure time under bounded unitary driving, giving it a direct operational meaning.

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20 pages, 4 figures,

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We introduce the unitary-accessible coherence-erasure distance, which measures the minimum unitary-control cost required to transform a quantum state into an incoherent state without changing its spectrum. The target states are restricted to incoherent states on the same unitary orbit, leading to a geometry different from conventional state-space proximity. We establish inequalities relating orbit-restricted quantum transport, Bures geometry, and the unitary-control distance. These quantities coincide for pure states but can differ for mixed states. For qubits, we derive an exact expression and show that, in the nondegenerate case, the coherence-erasure cost depends on the eigenbasis orientation rather than on the eigenvalues. For nondegenerate qutrits, we reduce the problem to an optimization over the monomial unitary group. We derive the tight universal bound arccos⁡[(22−1)/4]\arccos[(2\sqrt2-1)/4]arccos[(22​−1)/4] and construct an explicit eigenbasis that attains it. The resulting distance also determines the minimum coherence-erasure time under bounded unitary driving, giving it a direct operational meaning.
In practical applications of quantum information science, quantum systems can have non-negligible interactions with the environment, and this generally degrades the power of quantum protocols as it introduces noise. Counteracting this by appropriately measuring the environment (and therefore projecting its state) would require access all the necessary degrees of freedom, which in practice can be far too hard to achieve. To better understand one's limitations, we calculate the upper bound of optimal quantum erasure (i.e. the highest recoverable visibility, or "coherence"), when erasure is realistically limited to an accessible subspace of the whole environment. In the particular case of a two-dimensional accessible environment, the bound is given by the sub-fidelity of two particular states of the \emph{inaccessible} environment, which opens a new window into understanding the connection between correlated systems. We also provide an analytical solution for a three-dimensional accessible environment. This result provides also an interesting operational interpretation of sub-fidelity. We end with a statistical analysis of the expected visibility of an optimally erased random state and we find that 1) if one picks a random pure state of 2 qubits, there is an optimal measurement that allows one to distill a 1-qubit state with almost 90\% visibility and 2) if one picks a random pure state of 2 qubits in an inaccessible environment, there is an optimal measurement that allows one to distill a 1-qubit state with almost twice its initial visibility.
In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the ${\ensuremath{\ell}}_{1}$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.
Abstract The Bures-angle distance to a fixed free sector is one-Lipschitz. For QFI-regular paths, a resource change Δ over a control window of width w requires integrated and peak SLD quantum Fisher information (QFI) of at least 4 Δ 2 / w and 4 Δ 2 / w 2 . We develop this standard geometric estimate into a verifiable mixed-state erasure calculation. Lipschitz diagnostics and binary-measurement probabilities certify distance gaps; the latter certificate is exact for isotropic states relative to the full separable set. Their distance is M d ( θ ) = [ arccos ⁡ ( d − 1 / 2 ) − θ ] + and their QFI is 4 on the full-rank resourceful branch in the angular coordinate. For two qubits, the negativity zero set is precisely the separable set, and both the physical depolarizing probability and a fixed-rate depolarizing semigroup give explicit finite-window costs. Affine Bures-arclength traversal of the resourceful isotropic segment saturates all three bounds. A prescribed dimension-growing sequence realizes a finite resource jump and establishes optimal coefficients and width powers without asserting a microscopic transition. For independent local errors an exact flagged-state fidelity gives a two-endpoint constraint on the physical output, refining the QFI-based width bound. Its toric-code specialization remains a necessary constraint conditional on an independently established memory sector and resource drop, not a proof of mixed-state topology.
We compute on all quantum states several measures that characterise asymptotic quantum coherence manipulation under restricted classes of operations. We focus on the distillable coherence, i.e. the maximum rate of production of approximate pure bits of coherence starting from independent copies of an input state ρ, and on the coherence cost, i.e. the minimum rate of consumption of pure coherence bits that is needed to generate many copies of ρ with vanishing error. We obtain the first closed-form expression for the distillable coherence under strictly incoherent operations (SIO), proving that it coincides with that obtained via physically incoherent operations (PIO). This shows that SIO and PIO are equally weak at distilling coherence, sheds light on the recently discovered phenomenon of generic bound coherence, and provides us with an explicit optimal distillation protocol that is amenable to practical implementations. We give a single-letter formula for the coherence cost under PIO, showing that it is finite on a set of states with nonzero volume. Since PIO can be realised in a laboratory with incoherent ancillae, unitaries, and measurements, our result puts fundamental limitations on coherence manipulation in an experimentally relevant setting. We find examples of `abyssally bound' states with vanishing PIO distillable coherence yet infinite PIO coherence cost. Our findings complete the picture of asymptotic coherence manipulation under all the main classes of incoherent operations.
Quantum coherence is a fundamental property that can emerge within any quantum system. Incoherent operations, defined in terms of the Kraus decomposition, play an important role in state transformation. The maximum number of incoherent Kraus operators has been presented in Streltsov A., Rana S., Boes P. and Eisert J., Phys. Rev. Lett. , 119 (2017) 140402. In this work, we show that the number of incoherent Kraus operators for a single qubit can be reduced from 5 to 4 by constructing a proper unitary matrix. For qutrit systems we further obtain 32 incoherent Kraus operators, while the upper bound in the research of Sterltsov gives 39 Kraus operators. Besides, we reduce the number of strictly incoherent Kraus operators from more than 15 to 13. And we consider the state transformation problem for these two types of operations in single qutrit systems.
In a quantum unitary system, an initial state may have different paths to evolve to a target state, and the only requirement is that the initial and target states share the same eigenvalue spectrum. We focus on the evolution between two nondegenerate mixed states in this paper, and investigate the shortest evolution path between them. By minimizing the path distance contained in the unitary operator connecting the initial and target states over a series of phases, the shortest evolution path could be figured out. This minimum path distance has an analytical form in the single-qubit dynamical system, and its solution in the three- or higher-dimensional dynamical system could be obtained numerically. Based on the unitary operator associated with the shortest evolution path, a general form for the Hamiltonian to realize it is presented. Here we present another way to study quantum optimal control, which is based on the path distance between the initial state and its evolution state, rather than the state distance between them.