Undecidability of the Spectral Gap (full version)

We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion paper arXiv:1502.04135 (same title and authors) for a short version omitting technical details. v2: Small but important fix to wording of abstract. v3: Simplified and shortened some parts of the proof; minor fixes to other parts. Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to introductio

Undecidability of the Spectral Gap in One Dimension

The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum spin systems in two (or more) spatial dimensions: there exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional spin systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG - and even provably polynomial-time ones - for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.Comment: 7 figure

Bounds on entanglement assisted source-channel coding via the Lovasz theta number and its variants

We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs $G$ and $H$. Such vectors exist if and only if $\vartheta(\overline{G}) \le \vartheta(\overline{H})$ where $\vartheta$ represents the Lov\'asz number. We also obtain similar inequalities for the related Schrijver $\vartheta^-$ and Szegedy $\vartheta^+$ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: $\alpha^*(G) \le \vartheta^-(G)$. Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lov\'asz number. Beigi introduced a quantity $\beta$ as an upper bound on $\alpha^*$ and posed the question of whether $\beta(G) = \lfloor \vartheta(G) \rfloor$. We answer this in the affirmative and show that a related quantity is equal to $\lceil \vartheta(G) \rceil$. We show that a quantity $\chi_{\textrm{vect}}(G)$ recently introduced in the context of Tsirelson's conjecture is equal to $\lceil \vartheta^+(\overline{G}) \rceil$. In an appendix we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.Comment: Fixed proof of multiplicativity; more connections to prior work in conclusion; many changes in expositio

Lord Cochrane and the Chilean Navy, 1818-1823, with an inventory of the Dundonald papers relating to his service with the Chilean Navy

In the late 18th century and the first two decades of the 19th, Spanish seapower in the Pacific was in a state of decline, though it remained strong enough to contribute to the overthrow of the first attempt of the Chileans to liberate their colony from Spain, in 1814. By the time of the second, successful, emancipation of Chile in 1817, the patriots had realised the need for seapower. In that year they took into their service Lord Cochrane, a noted British naval officer then unemployed. Lord Cochrane arrived in Chile at the end of 1818. The squadron at that time is described. With this squadron Lord Cochrane made his first cruise, a reconnaissance in force of the royalist-held Peruvian coast during which Callao was attacked without success. Arising from this reconnaissance, the physical environment of the Mar del Sur is reviewed, together with the state of navigational knowledge. The intention of Lord Cochrane's second cruise, which began in September 1819, was to stage a major attack on Callao. This object was not achieved because of the' squadron's inadequate means and the viceroy's defensive measures, so in December 1819 Lord Cochrane sailed to Valdivia, a fortified city in the south of Chile still in Spanish hands, and captured it by assault in February 1820. There has been same debate about his intentions when he sailed for Valdivia. By early 1820 some of the basic social characteristics of the Chilean navy had emerged and these are examined, firstly from the point of view of the manning of the ships and secondly from the point of view of the problems of discipline and morale that arose. At the same time, the system of naval administration should be examined as its defects and malfunctioning had serious effects on the operating of the squadron, and its efficiency. This data forms the background to the squadron's participation in the liberation of Peru. Initially it played a significant role, firstly by shipping the expedition to Peru and secondly by boarding and taking out of Callao harbour the principal Spanish warship there. These successes were in 1820; in 1821 the squadron's role became less important as the relations between Lord Cochrane and San Martin, the commander-in-chief, deteriorated as a result of the refusal or inability of the latter to pay the squadron. In September 1821 Lord Cochrane seized the Peruvian public funds, allegedly to indemnify the expenses of the squadron, and left Peru. His last cruise, from October 1821 to May 1822, had the object of hunting down the remaining Spanish warships in the Pacific. This cruise here receives its first full account. The cruise completed, though not as successfully as he had hoped, Lord Cochrane returned to Chile. His brief remaining stay in that country was disturbed by difficulties in paying off the ships, disputes with San Martin, and the deteriorating political position of the government. When he received an invitation in November 1822 to take command of the Brazilian navy he accepted, resigned from the Chilean service, and left the country at the beginning of 1823. The dissertation is supplemented by the inventory of the papers in the Dundonald collection which relate to the period of Lord Cochrane's service with Chile. These amount to 2286 items

Area law for fixed points of rapidly mixing dissipative quantum systems

We prove an area law with a logarithmic correction for the mutual information for fixed points of local dissipative quantum system satisfying a rapid mixing condition, under either of the following assumptions: the fixed point is pure, or the system is frustration free.Comment: 17 pages, 1 figure. Final versio

Stability of Local Quantum Dissipative Systems

This is the author accepted manuscript. The final version is available from Springer at http://link.springer.com/article/10.1007%2Fs00220-015-2355-3.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 which 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

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

This is the author accepted manuscript. The final version is available from NPG at http://www.nature.com/ncomms/2015/150331/ncomms7739/full/ncomms7739.html.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.DE and DP acknowledge nancial support from the European CHIST-ERA project CQC (funded partially by MINECO grant PRI-PIMCHI-2011-1071) and from Comunidad de Madrid (grant QUITEMAD+-CM, ref. S2013/ICE-2801). TSC is supported by the Royal Society. MO acknowledges nancial support from European Union under project QALGO (Grant Agreement No. 600700). SS acknowledges the support of Sidney Sussex College. This work was made possible through the support of grant #48322 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily re ect the views of the John Templeton Foundation