The coherent measurement cost of coherence distillation

Quantum coherence—an indispensable resource for quantum technologies—is known to be distillable from a noisy form using operations that cannot create it. However, distillation exacts a hidden coherent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi><mml:mi>t</mml:mi></mml:math> cost, which has not previously been examined. We devise the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">target effect</mml:mtext></mml:mrow></mml:math> construction to characterize this cost through detailed conditions on the coherence-measuring structure necessary in any process realizing exact (maximal or non-maximal) or approximate distillation. As a corollary, we lower-bound the requisite measurement coherence, as quantified by operationally-relevant measures. We then consider the asymptotic limit of distilling from many copies of a given noisy coherent state, where we offer rigorous arguments to support the conjecture that the (necessary and sufficient) coherent measurement cost scales extensively in the number of copies. We also show that this cost is no smaller than the coherence of measurements saturating the scaling law in the generalized quantum Stein's lemma. Our results and conjectures apply to any task whereof coherence distillation is an incidental outcome (e.g., incoherent randomness extraction). But if pure coherence is the only desired outcome, our conjectures would have the cautionary implication that the measurement cost is often higher than the distilled yield, in which case coherence should rather be prepared afresh than distilled from a noisy input.