Efficiently Controllable Graphs.

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that, when a controllable quantum network is described by such a graph and the gaps in eigenfrequencies and in transition frequencies are bounded exponentially in the number of vertices, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficiently controllable and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems or qubits.

Additive Classical Capacity of Quantum Channels Assisted by Noisy Entanglement.

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 signal-witness 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.

Superadditivity of the Classical Capacity with Limited Entanglement Assistance.

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.

Supercritical entanglement in local systems: Counterexample to the area law for quantum matter.

Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an "area law": The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system's size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system's size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.

Criticality without frustration for quantum spin-1 chains.

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right brackets separated by empty spaces. Entanglement entropy of one half of the chain scales as 1/2 log n+O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.

Quantum state restoration and single-copy tomography for ground states of Hamiltonians.

Given a single copy of an unknown quantum state, the no-cloning theorem limits the amount of information that can be extracted from it. Given a gapped Hamiltonian, in most situations it is impractical to compute properties of its ground state, even though in principle all the information about the ground state is encoded in the Hamiltonian. We show in this Letter that if you know the Hamiltonian of a system and have a single copy of its ground state, you can use a quantum computer to efficiently compute its local properties. Specifically, in this scenario, we give efficient algorithms that copy small subsystems of the state and estimate the full statistics of any local measurement.

Capacity of quantum erasure channel assisted by backwards classical communication.

We present a communication protocol for the erasure channel assisted by backward classical communication, which achieves a significantly better rate than the best prior result. In addition, we prove an upper bound for the capacity of the channel. The upper bound is smaller than the capacity of the erasure channel when it is assisted by two-way classical communication. Thus, we prove the separation between quantum capacities assisted by backward classical communication and two-way classical communication.

Inequalities and separations among assisted capacities of quantum channels.

We exhibit quantum channels whose classical and quantum capacities, when assisted by classical feedback, exceed their unassisted classical Holevo capacity. These channels are designed to be noisy in a way that can be corrected with the help of the output and a reference system entangled with part of the input. A similar construction yields quantum channels whose classical capacity, when assisted by two-way classical communication independent of the source, exceeds their classical capacity assisted by feedback alone. We give a hierarchy of capacity inequalities and open questions.

Entanglement assisted capacity of the broadband Lossy channel.

We calculate the entanglement assisted capacity of a multimode bosonic channel with loss. As long as the efficiency of the channel is above 50%, the superdense coding effect can be used to transmit more bits than those that can be stored in the message sent down the channel. Bounds for the other capacities of the multimode channel are also provided.

Oblivious remote state preparation.

We characterize the class of remote state preparation (RSP) protocols that use only forward classical communication and entanglement, deterministically prepare an exact copy of a general state, and do so obliviously-without leaking further information about the state to the receiver. We prove that any such protocol can be modified to require from the sender only a single specimen of the state, without increasing the classical communication cost. This implies Lo's conjectured lower bound on the cost for these protocols. We relate our RSP protocols to the private quantum channels and establish a one-to-one correspondence between them.

Superactivation of bound entanglement.

We show that, in a multiparty setting, two nondistillable (bound-entangled) states tensored together can make a distillable state. This is an example of true superadditivity of distillable entanglement. We also show that unlockable bound-entangled states cannot be asymptotically unentangled, providing the first proof that some states are truly bound-entangled in the sense of being both nondistillable and nonseparable asymptotically.