En av flere [[Kvanteressurser]]e, og sammen med [[Kvantesuperposisjon]] den (populært sett) definerende kvanteeffekten (dog [[Kvantemagi]] kan vise seg å være enda viktigere). Kvantiseres via f.eks hvor mange Bell Pairs tilstanden kan gjøres om til via LOCC (Local Operators and Classical Communication) operasjoner (entanglement estimation/distillation), eller *entanglement entropy*, se [Horodecki x4's Entanglement Review](https://arxiv.org/pdf/quant-ph/0702225.pdf) (110 sider som ser veldig verdt ut å lese!). Generelt vanskelig å kvantisere (no pun intended), men (fra [Magic-induced computational separation in entanglement theory](https://arxiv.org/abs/2403.19610)): > A previous work showed that entanglement could be calculated exactly for stabilizer states (FCY+04). ... altså de forberedt av [[Clifford-kretser]]. På én måte gir dette intuitivt mening, gitt at dette er nøyaktig de tilstandene som lar seg klassisk simulere. Det overraskende, da, er at for en stor klasse problemer (nemlig de hvor [[Kvantemagi]] dominerer [[Sammenfiltring]], se paperet lenket til over), så er det bevisbart vanskelig (med mindre [[Kompleksitetsteori]]-landskapet skulle endre seg signifikant i nærmere fremtid) å estimere mengden [[Sammenfiltring]] *selv for kvantedatamaskiner*: > That is, for a number of entanglement detection and manipulation problems, there are sample- and time-efficient quantum algorithms that solve these problems for ED (entanglement-dominated) states. Conversely, we show that these entanglement manipulation tasks are provably computationally intractable for the MD (magic-dominated) phase. Se [[Entanglement vs Magic]] for mer. En interessant ting jeg lærer etter hvert som jeg leser, er at to oppgaver som jeg tidligere tar trodd var den samme er faktisk separate, nemlig *entanglement estimation* (hvor mange Bell-pairs kunne i prinsippet blitt generert av mengden [[Sammenfiltring]] i tilstanden (filtringsestimat?)) og *entanglement distillation* (å faktisk generere disse Bell pairsene av den tilgjengelige sammenfiltringen (filtringsdistillering?)). Motsatt vei har vi også *entanglement dilution*, altså oppgaven å benytte Bell pairs som en ressurs i forberedelsen av en gitt sammenfiltret tilstand (filtrings...utnyttelse?), kjent som tilstandens *entanglement cost*. De skriver: > Our first step in understanding the transition ED-MD is in the setting of entanglement estimability. We find that, in the ED phase, we can efficiently estimate entanglement entropy with an asymptotically (in n) vanishing error, even for volume-law states. Conversely, we show that in the MD phase, entanglement estimation is inefficient beyond logarithmic entanglement (below this, the swap test allows for efficient estimation). Finding clear evidence of an ED-MD computational separation for entanglement estimation, we then study entanglement distillation. We show that within the class of ED states, we can always efficiently find a polynomial-depth circuit that distills almost all of the entanglement into Bell pairs. We also prove the converse, which says that in the MD phase, it is impossible to find such efficient and optimal distillation protocols. Similarly, in the context of entanglement dilution, we demonstrate that we can always identify an efficient dilution protocol that utilizes an optimal number of Bell pairs to prepare ED states. Conversely, for MD states, we rule out the possibility of an efficient dilution protocol that consumes anything close to an optimal number of Bell pairs. De oppsummerer: > Within the ED phase, entanglement is always structured in such a way that it can be manipulated efficiently and (almost) reversibly. In contrast, for general states in the MD phase, there is almost no structure in the entanglement, making entanglement manipulation inefficient and irreversible. Man har også entanglement *witnessing*: > While precisely measuring entanglement can be a challenging and noisesensitive task, the less ambitious goal of merely witnessing entanglement can be easier and more noise-resilient (...) The purpose of a witness is to experimentally validate the presence of genuine entanglement in an imperfectly prepared version of the target state. Et relevant spørsmål åpenbarer seg: Hvor stor del av dette ED-landskapet, hvor [[Sammenfiltring]] er enklet å manipulere, lar seg *ikke* forberede av [[Clifford-kretser]], og er dermed *ikke* lett å klassisk simulere? > To start, the majority of states within the Hilbert space are magic-dominated. However, it is well-known that most states in Hilbert space also cannot be prepared in polynomial time, hence are irrelevant in practice. In contrast, there is a large class of well-known unitary operations which almost always produce ED states, namely circuits dominated by Clifford gates. In particular, Clifford-dominated circuits with t = o(n) non-Clifford gates (importantly, this far exceeds the classical simulability limit O(log n)) produces ED states with overwhelming probability. (...) However, we emphasize that Clifford-dominated circuits only generate a tiny fraction of all ED states; in fact, most ED states are constructed from far more than just O(n) T-gates. Svar: Mye. Heldigvis. Okei, neste spørsmål: Hvor stor del av MD-landskapet er tilstnander som *uansett ikke lar seg effektivt forberede av en kvantealgoritme* (og er dermed ikke relevant i praksis)?