Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that "hides" initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom.

The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This Letter investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Rényi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Rényi entanglement entropies and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Rényi entanglement entropies to the complexity of scrambling and quantum chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.

We give a capacity formula for the classical information transmission over a noisy quantum channel, with separable encoding by the sender and limited resources provided by the receiver's preshared ancilla. Instead of a pure state, we consider the signal-ancilla pair in a mixed state, purified by a "witness." Thus, the signalwitness correlation limits the resource available from the signal-ancilla correlation. Our formula characterizes the utility of different forms of resources, including noisy or limited entanglement assistance, for classical communication. With separable encoding, the sender's signals across multiple channel uses are still allowed to be entangled, yet our capacity formula is additive. In particular, for generalized covariant channels, our capacity formula has a simple closed form. Moreover, our additive capacity formula upper bounds the general coherent attack's information gain in various two-way quantum key distribution protocols. For Gaussian protocols, the additivity of the formula indicates that the collective Gaussian attack is the most powerful. DOI: 10.1103/PhysRevLett.118.200503 Communication channels model the physical medium for information transmission between the sender (Alice) and the receiver (Bob). Classical information theory [1,2] says that a channel is essentially characterized by a single quantity-the (classical) channel capacity, i.e., its maximum (classical) information transmission rate. However, quantum channels [3] can transmit information beyond the classical. Formally, a (memoryless) quantum channel is a time-invariant completely positive trace preserving (CPTP) linear map between quantum states. Various types of information lead to various capacities, e.g., classical capacity C [4,5] for classical information transmission encoded in quantum states, and quantum capacity Q [6-8] for quantum information transmission. For both cases, implicit constraints on the input Hilbert space, e.g., fixed dimension or energy, quantify the resources. Resources can also be in the form of assistance: given unlimited entanglement, one has the entanglement-assisted classical capacity C E [9]. References [10,11] provide a capacity formula for the trade-off of classical and quantum information transmission and entanglement generation (or consumption).With the trade-off capacity formula in hand, it appears that the picture of communication over quantum channels is complete. However, our understanding about the trade-off is plagued by the "nonadditivity" issue [3], best illustrated by the example of C. The Holevo-Schumacher-Westmoreland (HSW) theorem [4,5] gives the one-shot capacity C ð1Þ ðΨÞ of channel Ψ, which assumes product-state input in multiple channel uses. Consider An exception is the (unlimited) entanglement-assisted classical capacity C E [9]. Since it has the form of quantum mutual information [14,15], C E is additive [9,16]. One immediately hopes that the additivity can be extended to classical communication assisted by imperfect entanglement since entanglement is fragile...

Finding the optimal encoding strategies can be challenging for communication using quantum channels, as classical and quantum capacities may be superadditive. Entanglement assistance can often simplify this task, as the entanglement-assisted classical capacity for any channel is additive, making entanglement across channel uses unnecessary. If the entanglement assistance is limited, the picture is much more unclear. Suppose the classical capacity is superadditive, then the classical capacity with limited entanglement assistance could retain superadditivity by continuity arguments. If the classical capacity is additive, it is unknown if superadditivity can still be developed with limited entanglement assistance. We show this is possible, by providing an example. We construct a channel for which the classical capacity is additive, but that with limited entanglement assistance can be superadditive. This shows entanglement plays a weird role in communication, and we still understand very little about it. DOI: 10.1103/PhysRevLett.119.040503 In Shannon's classical information theory [1], a classical (memoryless) channel is a probabilistic map from input states to output states. This has been extended to the quantum world. A (memoryless) quantum channel is a time-invariant completely positive trace preserving (CPTP) linear map from input quantum states to output quantum states [2]. A classical channel can only transmit classical information, and the maximum communication rate is fully characterized by its capacity. A quantum channel can be used to transmit not only classical information but also quantum information. Hence, there are different types of capacity, such as classical capacity C for classical communication [3,4] and quantum capacity Q for quantum communication [5][6][7].Since quantum channels transmit quantum states, and quantum states can be entangled with other parties, it is natural to ask if entanglement can assist the communication. This was first considered by Bennett et al., who showed that unlimited preshared entanglement could improve the classical capacity of a noisy channel [8,9]. Shor examined the case where only finite preshared entanglement is available and obtained a trade-off curve that illustrates how the optimal rate of classical communication depends on the amount of entanglement assistance (CE trade-off) [10]. One can also consider how entanglement (E), classical communication (C), or quantum communication (Q) can trade-off against each other as resources. The trade-off capacity of almost any two resources was studied by Devetak et al. [11,12], such as entanglement-assisted quantum capacity (QE trade-off). Subsequently, the triple resource (CQE) trade-off capacity was also characterized [13][14][15].However, almost all the capacity formulas above are given by regularized expressions. They are difficult to evaluate because they require an optimization over an infinite number of channel uses, which is typically intractable. The existence of this regularization is because entanglement acr...

Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas.While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this work, we design quantum machine learning algorithms to model copulas. We show that any copula can be naturally mapped to a multipartite maximally entangled state. A variational ansatz we christen as a 'qopula' creates arbitrary correlations between variables while maintaining the copula structure starting from a set of Bell pairs for two variables, or GHZ states for multiple variables. As an application, we train a Quantum Generative Adversarial Network (QGAN) and a Quantum Circuit Born Machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables for historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to 8 qubits and show that our results outperform those obtained through equivalent classical generative learning. Further, we present theoretical arguments for exponential advantage in our model's expressivity over classical models based on communication and computational complexity arguments.