We establish fundamental upper bounds on the amount of secret key that can be extracted from quantum Gaussian states by using only local Gaussian operations, local classical processing, and public communication. For one-way public communication, or when two-way public communication is allowed but Alice and Bob first perform destructive local Gaussian measurements, we prove that the key is bounded by the R\'enyi-$2$ Gaussian entanglement of formation $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$. Since the inequality is saturated for pure Gaussian states, this yields an operational interpretation of the R\'enyi-$2$ entropy of entanglement as the secret key rate of pure Gaussian states that is accessible with Gaussian operations and one-way communication. In the general setting of two-way communication and arbitrary interactive protocols, we argue that $2 E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ is still an upper bound on the extractable key. We conjecture that the factor of $2$ is spurious, which would imply that $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ coincides with the secret key rate of Gaussian states under Gaussian measurements and two-way public communication. We use these results to prove a gap between the secret key rates obtainable with arbitrary versus Gaussian operations. Such a gap is observed for states produced by sending one half of a two-mode squeezed vacuum through a pure loss channel, in the regime of sufficiently low squeezing or sufficiently high transmissivity. Finally, for a wide class of Gaussian states that includes all two-mode states, we prove a recently proposed conjecture on the equality between $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ and the Gaussian intrinsic entanglement. The unified entanglement quantifier emerging from such an equality is then endowed with a direct operational interpretation as the value of a quantum teleportation game.

翻译：我们建立了通过仅使用局部高斯操作、局部经典处理及公共通信从量子高斯态中提取秘密密钥量的基本上界。对于单向公共通信，或允许双向公共通信但Alice和Bob首先执行破坏性局部高斯测量的情况，我们证明了密钥量受限于Rényi-2高斯纠缠形成$E_{F,2}^{\mathrm{\scriptscriptstyle G}}$。由于该不等式对于纯高斯态是紧的，这为纯高斯态的Rényi-2纠缠熵提供了操作性的解释，即其可通过高斯操作和单向通信达到的秘密密钥率。在双向通信及任意交互协议的一般设置下，我们论证$2 E_{F,2}^{\mathrm{\scriptscriptstyle G}}$仍是可提取密钥量的上界。我们推测因子$2$是多余的，这意味着$E_{F,2}^{\mathrm{\scriptscriptstyle G}}$与高斯态在测量及双向公共通信下的秘密密钥率一致。我们利用这些结果证明了任意操作与高斯操作可获得的秘密密钥率之间存在差距。这种差距在通过纯损耗信道传输双模压缩真空态的一半所产生的态中观察到，出现在足够低压缩或足够高透射率的区域。最后，对于包含所有双模态的广泛高斯态类，我们证明了最近提出的关于$E_{F,2}^{\mathrm{\scriptscriptstyle G}}$与高斯内禀纠缠等价的猜想。由此产生的统一纠缠量度被赋予了直接的操作性解释，即作为量子隐形传态博弈的价值。