Toby S. Cubitt

Undecidability of the spectral gap

The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

Improving Zero-Error Classical Communication with Entanglement

Given one or more uses of a classical channel, only a certain number of messages can be transmitted with zero probability of error. The study of this number and its asymptotic behavior constitutes the field of classical zero-error information theory. We show that, given a single use of certain classical channels, entangled states of a system shared by the sender and receiver can be used to increase the number of (classical) messages which can be sent without error. In particular, we show how to construct such a channel based on any proof of the Kochen-Specker theorem. We investigate the connection to pseudotelepathy games. The use of generalized nonsignaling correlations to assist in this task is also considered. In this case, an elegant theory results and, remarkably, it is sometimes possible to transmit information with zero error using a channel with no unassisted zero-error capacity.

Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel

The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly so that they can be decoded with zero probability of error. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel.

On the dimension of subspaces with bounded Schmidt rank

We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al. [e-print arXiv:quant-ph∕0407049; Commun. Math. Phys., 265, 95 (2006)], which show that in large d×d-dimensional systems there exist random subspaces of dimension almost d2, all of whose states have entropy of entanglement at least logd−O(1). It is also a generalization of results on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in Hayden et al. We also determine the converse: the l...

Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for pxa0>xa01, we demonstrate here by a careful random selection argument that also at pxa0=xa00, and consequently for sufficiently small p, there exist counterexamples.An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≲ 0.11. We conjecture however that violations of additivity exist for all pxa0<xa01.

Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations

The theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly in the presence of various classes of nonsignalling correlations between sender and receiver i.e., shared randomness, shared entanglement and arbitrary nonsignalling correlations. When the channel being simulated is noiseless, this is zero-error coding assisted by correlations. When the resource channel is noiseless, it is the reverse problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, separations between the power of the different classes of assisting correlations are exhibited for finite block lengths. The most striking result here is that entanglement can assist in zero-error communication. In the large block length limit, shared randomness is shown to be just as powerful as arbitrary nonsignalling correlations for exact simulation, but not for asymptotic zero-error coding. For assistance by arbitrary nonsignalling correlations, linear programming formulas for the asymptotic capacity and simulation rates are derived, the former being equal (for channels with nonzero unassisted capacity) to the feedback-assisted zero-error capacity derived by Shannon. Finally, a kind of reversibility between nonsignalling-assisted zero-error capacity and exact simulation is observed, mirroring the usual reverse Shannon theorem.

Unbounded number of channel uses may be required to detect quantum capacity

Transmitting data reliably over noisy communication channels is one of the most important applications of information theory, and is well understood for channels modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities. This is because the formula for the quantum capacity involves maximizing the coherent information over an unbounded number of channel uses. In fact, entanglement across channel uses can even increase the coherent information from zero to non-zero. Here we study the number of channel uses necessary to detect positive coherent information. In all previous known examples, two channel uses already sufficed. It might be that only a finite number of channel uses is always sufficient. We show that this is not the case: for any number of uses, there are channels for which the coherent information is zero, but which nonetheless have capacity.

Stability of Local Quantum Dissipative Systems

Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are potentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time that scales logarithmically with the system size. The proof relies on Lieb–Robinson bounds, which describe a finite group velocity for propagation of information in local systems. As a main example, we prove that classical Glauber dynamics is stable under local perturbations, including perturbations in the transition rates, which may not preserve detailed balance.

An Extreme Form of Superactivation for Quantum Zero-Error Capacities

The zero-error capacity of a channel is the rate at which it can send information perfectly, with zero probability of error, and has long been studied in classical information theory. We show that the zero-error capacity of quantum channels exhibits an extreme form of nonadditivity, one which is not possible for classical channels, or even for the usual capacities of quantum channels. By combining probabilistic arguments with algebraic geometry, we prove that there exist channels and with no zero-error classical capacity whatsoever, , but whose joint zero-error quantum capacity is positive, . This striking effect is an extreme form of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, while being a strictly stronger property of capacities. Superactivation of the quantum zero-error capacity was not previously known.

Extracting Dynamical Equations from Experimental Data is NP Hard

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70xa0years).