This paper establishes single-letter formulas for the exact entanglement cost of simulating quantum channels under free quantum operations that completely preserve positivity of the partial transpose (PPT). First, we introduce the $\kappa$-entanglement measure for point-to-point quantum channels, based on the idea of the $\kappa$-entanglement of bipartite states, and we establish several fundamental properties for it, including amortization collapse, monotonicity under PPT superchannels, additivity, normalization, faithfulness, and non-convexity. Second, we introduce and solve the exact entanglement cost for simulating quantum channels in both the parallel and sequential settings, along with the assistance of free PPT-preserving operations. In particular, we establish that the entanglement cost in both cases is given by the same single-letter formula, the $\kappa$-entanglement measure of a quantum channel. We further show that this cost is equal to the largest $\kappa$-entanglement that can be shared or generated by the sender and receiver of the channel. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum channels. Noting that the sequential regime is more powerful than the parallel regime, another notable implication of our result is that both regimes have the same power for exact quantum channel simulation, when PPT superchannels are free. For several basic Gaussian quantum channels, we show that the exact entanglement cost is given by the Holevo--Werner formula [Holevo and Werner, Phys. Rev. A 63, 032312 (2001)], giving an operational meaning of the Holevo-Werner quantity for these channels.

翻译：本文建立了在完全保持部分转置正性（PPT）的自由量子操作下模拟量子信道的精确纠缠代价的单字母公式。首先，我们基于两体量子态的κ-纠缠概念，引入点对点量子信道的κ-纠缠度量，并建立其若干基本性质，包括摊销坍塌、PPT超信道下的单调性、可加性、归一化、保真性和非凸性。其次，我们引入并求解了在并行和顺序设置下，借助自由PPT保持操作模拟量子信道的精确纠缠代价。特别地，我们证明这两种情况下的纠缠代价均由同一个单字母公式给出，即量子信道的κ-纠缠度量。我们进一步证明该代价等于信道发送方和接收方可以共享或生成的最大κ-纠缠。该公式可通过半定规划计算，从而为一般量子信道提供高效可解方案。注意到顺序设置比并行设置更强大，我们结果的另一个显著推论是：当PPT超信道自由时，这两种设置在精确量子信道模拟中具有相同的能力。对于几个基本的高斯量子信道，我们证明精确纠缠代价由Holevo-Werner公式[Holevo and Werner, Phys. Rev. A 63, 032312 (2001)]给出，从而为这些信道的Holevo-Werner量赋予操作意义。