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.
We prove that learning an unknown mixed fermionic Gaussian state on m modes to trace distance ε requires Ω(m3/ε2) copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds for 0<ε≤1/3600 and separates this model from the known collective rate Θ(m2/ε2). It follows from a uniform single-copy Fisher-information budget and a dimension-independent comparison between trace distance and Gaussian parameters. On covariance matrices of operator norm at most 1−c, we prove a Frobenius-to-trace-norm continuity bound with constant [2c(2−c)]−1/2; matchgate shadows then attain O(m3/(cε2)) copies. This also gives explicit error certificates for thermal free-fermion states. For a class of mixed passive bosonic Gaussian states with total mean photon number at most one, a reduction to bounded-block qudit tomography gives a single-copy lower bound Ω(m3/(ε2log(m/ε))), versus collective complexity Θ(m2/ε2). These separations answer the mixed-state measurement-resource question posed by Chen et al. A two-mode example has optimal two-copy Fisher trace 5, versus the separable ceiling 4. An IBM replication gave a pre-registered witness lower bound 4.12 at one-sided 95% shot-noise confidence; its interpretation assumes the prescribed preparations and a common measurement channel and does not bound preparation systematics.
We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, X states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between l1-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both lp-norm- and Schatten-p-norm--based measures violate the (strong) monotonicity for all $p\ensuremath{\in}(1,\ensuremath{\infty})$.