In a 1992 publication, Roberts refers to a 1939 text on Number Theory, which tells me that Lamé's remarkable theorem is not very well known, at least among non-specialists. In his Mathematical Gems II, Ross Honsberger refers to the 1924 edition of W. Sierpinski book Elementary Theory of Numbers; the book has been republished twice since, but by now became a bibliographic rarity. It is available online for download as djvu file.
where the supremum includes asymptotic states for bosonic channels. In the following, we prove that these single-letter quantities, and , bound the two-way capacity of basic channels. The first step is the following general result.
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In Supplementary Note 3, we provide various equivalent proofs. The simplest one assumes an exponential growth of the shield system in the target private state27 as proven by ref. 30 and trivially adapted to CVs. Another proof is completely independent from the shield system. Once established the bound , our next step is to simplify it by applying the technique of teleportation stretching, which is in turn based on a suitable simulation of quantum channels.
Assume that the mean photon number of the total register states and is bounded by some large but yet finite value N(n). For instance, we may consider a sequence N(n)=N(0)+nt, where N(0) is the initial photon contribution and t is the channel contribution, which may be negative for energy-decreasing channels (like the thermal-loss channel) or positive for energy-increasing channels (like the quantum amplifier). We then prove
The supremum over all adaptive protocols, which defines disappears in the right hand side of equation (108). The resulting bound applies to both energy-constrained protocols and the limit of energy-unconstrained protocols. The proof of the further condition in equation (15) comes from the subadditivity of the REE over tensor product states. This subadditivity also holds for a tensor product of asymptotic states; it is proven by restricting the minimization on tensor product sequences in the corresponding definition of the REE.
Let us now prove equation (16). The two inequalities in equation (16) are simply obtained by using for a Choi-stretchable channel (where the Choi matrix is intended to be asymptotic for a bosonic channel). Then we show the equality . By restricting the optimization in to an input EPR state Φ, we get the direct part as already noticed in equation (6). For CVs, this means to choose an asymptotic EPR state , so that
Our method can be extended to more complex forms of quantum communication. In fact, our weak converse theorem can be applied to any scenario where two parties produce an output state by means of an adaptive protocol. All the details of the protocol are contained in the LOCCs which are collapsed into by teleportation stretching and then discarded using the REE.
The paper is organized as follows. In Sect. 2, we provide several equivalent statements of the PPT\(^2\) conjecture and also review some useful facts about diagonal unitary covariant maps; all of them are proven in [28]. In Sect. 3, we introduce the notion of factor width for pairwise and triplewise completely positive matrices. These tools are used in Sect. 4 to prove the PPT\(^2\) conjecture for (C)DUC maps. Finally, in Sect. 5 we discuss some open problems and future directions for research.
Next, we state and prove an important proposition, which connects the above composition rules on matrix pairs to the operations of map composition in \(\mathsf DUC_d\) and \(\mathsf CDUC_d\). But first, we need familiarity with the notion of stability under composition. 2ff7e9595c
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