Pseudorandom states (PRSs) are state ensembles that cannot be distinguished from Haar random states by any efficient quantum algorithm. However, the definition of PRSs has been limited to pure states and lacks robustness against noise. In this work, we introduce pseudorandom density matrices (PRDMs), ensembles of $n$-qubit states that are computationally indistinguishable from the generalized Hilbert-Schmidt ensemble, which is constructed from $(n+m)$-qubit Haar random states with $m$ qubits traced out. For a mixedness parameter $m=0$, PRDMs are equivalent to PRSs, whereas for $m=\omega(\log n)$, PRDMs are computationally indistinguishable from the maximally mixed state. In contrast to PRSs, PRDMs with $m=\omega(\log n)$ are robust to unital noise channels and a recently introduced $\mathsf{PostBQP}$ attack. Further, we construct pseudomagic and pseudocoherent state ensembles, which possess near-maximal magic and coherence, but are computationally indistinguishable from states with zero magic and coherence. PRDMs can exhibit a pseudoresource gap of $\Theta(n)$ vs $0$, surpassing previously found gaps. We introduce noise-robust EFI pairs, which are state ensembles that are computationally indistinguishable yet statistically far, even when subject to noise. We show that testing entanglement, magic and coherence is not efficient. Further, we prove that black-box resource distillation requires a superpolynomial number of copies. We also establish lower bounds on the purity needed for efficient testing and black-box distillation. Finally, we introduce memoryless PRSs, a noise-robust notion of PRS which are indistinguishable to Haar random states for efficient algorithms without quantum memory. Our work provides a comprehensive framework of pseudorandomness for mixed states, which yields powerful quantum cryptographic primitives and fundamental bounds on quantum resource theories.

翻译：伪随机态（PRS）是指任何高效量子算法都无法将其与哈尔随机态区分开来的态系综。然而，PRS的定义一直局限于纯态，并且缺乏对噪声的鲁棒性。在本工作中，我们引入了伪随机密度矩阵（PRDM），即一类计算上无法与广义希尔伯特-施密特系综区分开的 $n$ 量子比特态系综；该广义系综由 $(n+m)$ 量子比特的哈尔随机态通过迹掉 $m$ 个量子比特而构造得到。当混合度参数 $m=0$ 时，PRDM 等价于 PRS；而当 $m=\omega(\log n)$ 时，PRDM 在计算上无法与最大混合态区分开。与 PRS 不同，具有 $m=\omega(\log n)$ 的 PRDM 对酉噪声信道以及最近提出的 $\mathsf{PostBQP}$ 攻击具有鲁棒性。此外，我们构造了伪魔术态和伪相干态系综，它们具有接近最大的魔术度和相干性，但在计算上无法与魔术度和相干性为零的态区分开。PRDM 可以展现出 $\Theta(n)$ 对 $0$ 的伪资源间隙，超越了先前发现的间隙。我们引入了噪声鲁棒的 EFI 对，即即使在受到噪声影响时，在计算上不可区分但在统计上相距甚远的态系综。我们证明了纠缠、魔术度和相干性的测试不是高效的。此外，我们证明了黑盒资源蒸馏需要超多项式数量的副本。我们还建立了高效测试和黑盒蒸馏所需纯度的下界。最后，我们引入了无记忆 PRS，这是一种噪声鲁棒的 PRS 概念，对于没有量子内存的高效算法而言，它们与哈尔随机态不可区分。我们的工作为混合态的伪随机性提供了一个全面的框架，该框架产生了强大的量子密码学原语以及对量子资源理论的基本限制。