Entanglement cost and quantum channel simulation

This paper proposes a revised definition for the entanglement cost of a quantum channel $\mathcal{N}$. In particular, it is defined here to be the smallest rate at which entanglement is required, in addition to free classical communication, in order to simulate $n$ calls to $\mathcal{N}$, such that the most general discriminator cannot distinguish the $n$ calls to $\mathcal{N}$ from the simulation. The most general discriminator is one who tests the channels in a sequential manner, one after the other, and this discriminator is known as a quantum tester [Chiribella et al., Phys. Rev. Lett., 101, 060401 (2008)] or one who is implementing a quantum co-strategy [Gutoski et al., Symp. Th. Comp., 565 (2007)]. As such, the proposed revised definition of entanglement cost of a quantum channel leads to a rate that cannot be smaller than the previous notion of a channel's entanglement cost [Berta et al., IEEE Trans. Inf. Theory, 59, 6779 (2013)], in which the discriminator is limited to distinguishing parallel uses of the channel from the simulation. Under this revised notion, I prove that the entanglement cost of certain teleportation-simulable channels is equal to the entanglement cost of their underlying resource states. Then I find single-letter formulas for the entanglement cost of some fundamental channel models, including dephasing, erasure, three-dimensional Werner--Holevo channels, epolarizing channels (complements of depolarizing channels), as well as single-mode pure-loss and pure-amplifier bosonic Gaussian channels. These examples demonstrate that the resource theory of entanglement for quantum channels is not reversible. Finally, I discuss how to generalize the basic notions to arbitrary resource theories.Comment: 28 pages, 7 figure

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Quantization of universal Teichm\"uller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite braid group $B_\infty$, and by abelianizing the kernel $B_\infty$ one constructs central extensions $T^*_{ab}$, $T^\sharp_{ab}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show $\hat{T}^{Kash}\cong T^\sharp_{ab}$. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension $\hat{T}^{CF}$ of $T$ resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to $12$ times the Euler class and that $\hat{T}^{CF} \cong T^*_{ab}$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.Comment: 43 pages, 15 figures. v2: substantially revised from the first version, and the author affiliation changed. // v3: Groups M and T are shown to be anti-isomorphic (new Prop.2.32), which makes the whole construction more natural. And some minor changes // v4: reflects all changes made for journal publication (to appear in Adv. Math.