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

Technical Summary: Unitary-Accessible Coherence-Erasure Distance

Problem Statement
The paper addresses a limitation in standard quantum coherence measures, which typically quantify coherence by minimizing the distance from a quantum state to the set of all incoherent states. This standard approach assumes that any incoherent state is accessible, ignoring dynamical constraints. Specifically, when evolution is restricted to unitary transformations (closed-system dynamics), the spectrum of the density operator is preserved. Consequently, a state $\rho$ cannot generally be transformed into an arbitrary incoherent state; it can only reach incoherent states that share its spectrum (isospectral states).

The authors introduce the unitary-accessible coherence-erasure distance ($D^U_I(\rho)$) to quantify the minimum unitary-control cost required to transform a state $\rho$ into an incoherent state while strictly preserving its spectrum. This distance represents the operational cost of "erasing" coherence via closed-system unitary dynamics, distinct from dissipative decoherence processes where entropy and purity change.

Methodology
The study employs a resource-theoretic framework combined with geometric quantum information theory:

1. Definitions: The authors define the set of unitary-accessible incoherent states, $I_\rho = \mathcal{I} \cap \mathcal{O}_\rho$, where $\mathcal{I}$ is the set of incoherent states and $\mathcal{O}_\rho$ is the unitary orbit of $\rho$.
2. Distance Metric: The distance is defined as the minimum quantum charging distance ($D_{CD}$) between $\rho$ and any state in $I_\rho$. The quantum charging distance measures the minimum time required to transform one isospectral state to another under a bounded driving Hamiltonian.
3. Eigenbasis Reduction: For nondegenerate states, the authors prove that the optimization problem reduces to finding the minimum unitary rotation required to align the eigenbasis of $\rho$ with the fixed reference basis, up to permutations and phase changes. This is formalized as a distance from the eigenbasis matrix $R$ to the monomial unitary group ($M_d$), which consists of permutation matrices multiplied by diagonal phase matrices.
4. Comparative Analysis: The work establishes inequalities relating $D^U_I(\rho)$ to the orbit-restricted Bures angle and the orbit-restricted revised quantum transport cost.

Key Results

• General Properties:

  • $D^U_I(\rho) = 0$ if and only if $\rho$ is already incoherent.
  • The distance is invariant under monomial unitary transformations.
  • For pure states, the unitary-accessible coherence-erasure distance coincides exactly with the orbit-restricted Bures angle and a rescaled version of the orbit-restricted transport cost.
  • For mixed states, these quantities generally differ, with the Bures angle being strictly smaller than the unitary-control distance for non-degenerate qubits.

• Qubit Solution (Exact):

  • For nondegenerate qubit states ($r > 0$), the authors derive an exact analytical expression:
    $$D^U_I(\rho) = \frac{1}{2} \arccos\left(\frac{|r_z|}{r}\right) = \frac{1}{2} \arcsin\left(\frac{r_\perp}{r}\right)$$
    where $r$ is the Bloch vector magnitude and $r_z$ is its component along the reference axis.
  • Key Insight: The distance depends only on the orientation of the eigenbasis relative to the reference basis, not on the eigenvalues (mixedness) of the state. This contrasts with the Bures angle, which depends on both direction and length.
  • Singularity: At the maximally mixed state ($r=0$), the distance drops discontinuously to zero due to spectral degeneracy, whereas the limit as $r \to 0$ along a fixed direction remains finite.

• Qutrit Solution (Tight Bound):

  • For nondegenerate qutrits, the problem reduces to an optimization over the monomial unitary group $M_3$.
  • The authors derive a tight universal upper bound for the distance:
    $$D^U_I(\rho) \leq \theta^* = \arccos\left(\frac{2\sqrt{2}-1}{4}\right) \approx 1.096 \text{ rad} \approx 62.80^\circ$$
  • An explicit eigenbasis matrix $R^*$ is constructed that attains this bound, demonstrating that the bound is achievable.

Significance and Claims
The paper claims to provide a rigorous operational definition for the cost of coherence erasure under unitary constraints, distinguishing it from standard coherence measures that ignore spectral preservation.

• Operational Meaning: The distance $D^U_I(\rho)$ directly determines the minimum time required to eliminate coherence under bounded unitary driving, providing a concrete physical interpretation in terms of control time.
• Geometric Distinction: The work highlights a fundamental geometric difference between state-space proximity (Bures angle) and unitary accessibility. While they coincide for pure states, they diverge for mixed states, revealing that the "cost" to reach an incoherent state via unitary evolution is strictly higher than the geometric distance suggests for mixed qubits.
• Universality: The derived qutrit bound is independent of the specific eigenvalues of the state, depending solely on the eigenbasis orientation, suggesting a universal limit on the difficulty of unitary coherence erasure for three-level systems.

The authors do not propose specific experimental implementations or future applications beyond the theoretical characterization of these distances and their relation to quantum speed limits and optimal transport costs.