Geography

# Dr Mark Williamson

Associate Research Fellow

Key publications | Publications by category | Publications by year

### Key publications

**Williamson MS, Lenton TM**(2015). Detection of bifurcations in noisy coupled systems from multiple time series.

*Chaos*,

*25*(3).

## Abstract:

Detection of bifurcations in noisy coupled systems from multiple time series

© 2015 AIP Publishing LLC. We generalize a method of detecting an approaching bifurcation in a time series of a noisy system from the special case of one dynamical variable to multiple dynamical variables. For a system described by a stochastic differential equation consisting of an autonomous deterministic part with one dynamical variable and an additive white noise term, small perturbations away from the system's fixed point will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all such systems exhibit this decay-type behaviour. However, when the deterministic part has multiple coupled dynamical variables, the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case, we find additional indicators to decay rate, such as frequency of oscillation. In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given multiple time series, where the underlying dynamics are not fully known. Our generalisation also allows bifurcation detection to be applied spatially if one treats each spatial location as a new dynamical variable. One may then determine the unstable spatial mode(s). This is also something that has not been possible with the single variable method. The method is applicable to any set of time series regardless of its origin, but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.**Williamson MS, Ericsson M, Johansson M, Sjöqvist E, Sudbery A, Vedral V, Wootters WK**(2011). Geometric local invariants and pure three-qubit states.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*83*(6).

## Abstract:

Geometric local invariants and pure three-qubit states

We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or "gauge" invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of d-level quantum systems. We give an operational interpretation of these invariants in terms of observables. © 2011 American Physical Society.**Williamson MS, Ericsson M, Johansson M, Sjöqvist E, Sudbery A, Vedral V**(2011). Global asymmetry of many-qubit correlations: a lattice-gauge-theory approach.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*84*(3).

## Abstract:

Global asymmetry of many-qubit correlations: a lattice-gauge-theory approach

We introduce a bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the stochastic local operations and classical communication protocols familiar in quantum information. Although the mathematical methods are the same, the meaning of the gauge group is different. The measure we introduce, "twist," is constructed as a Wilson loop from a correlation-induced holonomy. The measure can be understood as the global asymmetry of the bipartite correlations in a loop of three or more qubits; if the holonomy is trivial (the identity matrix), the bipartite correlations can be globally untwisted using general local qubit operations, the gauge group of our theory, which turns out to be the group of Lorentz transformations familiar from special relativity. If it is not possible to globally untwist the bipartite correlations in a state using local operations, the twistedness is given by a nontrivial element of the Lorentz group, the correlation-induced holonomy. We provide several analytical examples of twisted and untwisted states for three qubits, the most elementary nontrivial loop one can imagine. © 2011 American Physical Society.**Modi K, Paterek T, Son W, Vedral V, Williamson M**(2010). Unified View of Quantum and Classical Correlations.

*PHYSICAL REVIEW LETTERS*,

*104*(8). Author URL. DOI.

**Williamson MS, Lenton TM, Shepherd JG, Edwards NR**(2006). An efficient numerical terrestrial scheme (ENTS) for Earth system modelling.

*ECOLOGICAL MODELLING*,

*198*(3-4), 362-374. Author URL. DOI.

**Lenton TM, Williamson MS, Edwards NR, Marsh R, Price AR, Ridgwell AJ, Shepherd JG, Cox SJ, team GENIE**(2006). Millennial timescale carbon cycle and climate change in an efficient Earth system model.

*CLIMATE DYNAMICS*,

*26*(7-8), 687-711. Author URL. DOI.

### Publications by category

### Journal articles

**Bathiany S, Bolt BVD, Williamson MS, Lenton TM, Scheffer M, Nes EV, Notz D**(In Press). Trends in sea-ice variability on the way to an ice-free Arctic.

## Abstract:

Trends in sea-ice variability on the way to an ice-free Arctic

It has been widely debated whether Arctic sea-ice loss can reach a tippingpoint beyond which a large sea-ice area disappears abruptly. The theory of

dynamical systems predicts a slowing down when a system destabilises towards a

tipping point. In simple stochastic systems this can result in increasing

variance and autocorrelation, potentially yielding an early warning of an

abrupt change. Here we aim to establish whether the loss of Arctic sea ice

would follow these conceptual predictions, and which trends in sea ice

variability can be expected from pre-industrial conditions toward an Arctic

that is ice-free during the whole year. To this end, we apply a model hierarchy

consisting of two box models and one comprehensive Earth system model. We find

a consistent and robust decrease of the ice volume's annual relaxation time

before summer ice is lost because thinner ice can adjust more quickly to

perturbations. Thereafter, the relaxation time increases, mainly because the

system becomes dominated by the ocean water's large heat capacity when the

ice-free season becomes longer. Both trends carry over to the autocorrelation

of sea ice thickness in time series. These changes are robust to the nature and

origin of climate variability in the models and hardly depend on the balance of

feedbacks. Therefore, characteristic trends can be expected in the future. As

these trends are not specific to the existence of abrupt ice loss, the

prospects for early warnings seem very limited. This result also has

implications for statistical indicators in other systems whose effective mass

changes over time, affecting the trend of their relaxation time. However, the

robust relation between state and variability would allow an estimate of

sea-ice variability from only short observations. This could help one to

estimate the likelihood and persistence of extreme events in the future.

**Williamson MS, collins M, drijfhout S, kahana R, mecking J, lenton TM**(2017). Effect of AMOC collapse on ENSO in a high resolution general circulation model.

*Climate Dynamics*Full text. DOI.

**Williamson MS, Bathiany S, M Lenton T**(2016). Early warning signals of tipping points in periodically forced systems.

*Earth System Dynamics*,

*7*(2), 313-326.

## Abstract:

Early warning signals of tipping points in periodically forced systems

© Author(s) 2016. The prospect of finding generic early warning signals of an approaching tipping point in a complex system has generated much interest recently. Existing methods are predicated on a separation of timescales between the system studied and its forcing. However, many systems, including several candidate tipping elements in the climate system, are forced periodically at a timescale comparable to their internal dynamics. Here we use alternative early warning signals of tipping points due to local bifurcations in systems subjected to periodic forcing whose timescale is similar to the period of the forcing. These systems are not in, or close to, a fixed point. Instead their steady state is described by a periodic attractor. For these systems, phase lag and amplification of the system response can provide early warning signals, based on a linear dynamics approximation. Furthermore, the Fourier spectrum of the system's time series reveals harmonics of the forcing period in the system response whose amplitude is related to how nonlinear the system's response is becoming with nonlinear effects becoming more prominent closer to a bifurcation. We apply these indicators as well as a return map analysis to a simple conceptual system and satellite observations of Arctic sea ice area, the latter conjectured to have a bifurcation type tipping point. We find no detectable signal of the Arctic sea ice approaching a local bifurcation.**Bathiany S, Van Der Bolt B, Williamson MS, Lenton TM, Scheffer M, Van Nes EH, Notz D**(2016). Statistical indicators of Arctic sea-ice stability-prospects and limitations.

*Cryosphere*,

*10*(4), 1631-1645.

## Abstract:

Statistical indicators of Arctic sea-ice stability-prospects and limitations

We examine the relationship between the mean and the variability of Arctic sea-ice coverage and volume in a large range of climates from globally ice-covered to globally ice-free conditions. Using a hierarchy of two column models and several comprehensive Earth system models, we consolidate the results of earlier studies and show that mechanisms found in simple models also dominate the interannual variability of Arctic sea ice in complex models. In contrast to predictions based on very idealised dynamical systems, we find a consistent and robust decrease of variance and autocorrelation of sea-ice volume before summer sea ice is lost. We attribute this to the fact that thinner ice can adjust more quickly to perturbations. Thereafter, the autocorrelation increases, mainly because it becomes dominated by the ocean water's large heat capacity when the ice-free season becomes longer. We show that these changes are robust to the nature and origin of climate variability in the models and do not depend on whether Arctic sea-ice loss occurs abruptly or irreversibly. We also show that our climate is changing too rapidly to detect reliable changes in autocorrelation of annual time series. Based on these results, the prospects of detecting statistical early warning signals before an abrupt sea-ice loss at a "tipping point" seem very limited. However, the robust relation between state and variability can be useful to build simple stochastic climate models and to make inferences about past and future sea-ice variability. from only short observations or reconstructions.**Williamson MS, Lenton TM**(2015). Detection of bifurcations in noisy coupled systems from multiple time series.

*Chaos*,

*25*(3).

## Abstract:

Detection of bifurcations in noisy coupled systems from multiple time series

© 2015 AIP Publishing LLC. We generalize a method of detecting an approaching bifurcation in a time series of a noisy system from the special case of one dynamical variable to multiple dynamical variables. For a system described by a stochastic differential equation consisting of an autonomous deterministic part with one dynamical variable and an additive white noise term, small perturbations away from the system's fixed point will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all such systems exhibit this decay-type behaviour. However, when the deterministic part has multiple coupled dynamical variables, the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case, we find additional indicators to decay rate, such as frequency of oscillation. In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given multiple time series, where the underlying dynamics are not fully known. Our generalisation also allows bifurcation detection to be applied spatially if one treats each spatial location as a new dynamical variable. One may then determine the unstable spatial mode(s). This is also something that has not been possible with the single variable method. The method is applicable to any set of time series regardless of its origin, but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.**Johansson M, Ericsson M, Singh K, Sjöqvist E, Williamson MS**(2012). Topological phases and multiqubit entanglement.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*85*(3).

## Abstract:

Topological phases and multiqubit entanglement

Global phase factors of topological origin, resulting from cyclic local SU evolution, called topological phases, were first described in, in the case of entangled qubit pairs. In this paper we investigate topological phases in multiqubit systems as the result of cyclic local SU(2) evolution. These phases originate from the topological structure of the local SU(2) orbits and are an attribute of most entangled multiqubit systems. We discuss the relation between topological phases and stochastic local operations and classical communication (SLOCC)-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an n-qubit system is described and a complete list of these phases for up to seven qubits is given. © 2012 American Physical Society.**Johansson M, Ericsson M, Singh K, Sjöqvist E, Williamson MS**(2011). Correlation-induced non-Abelian quantum holonomies.

*Journal of Physics A: Mathematical and Theoretical*,

*44*(14).

## Abstract:

Correlation-induced non-Abelian quantum holonomies

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the Lévay geometric phase (2004 J. Phys. A: Math. Gen. 37 1821) in the case of two-qubit systems undergoing local SU(2) evolutions. © 2011 IOP Publishing Ltd.**Williamson MS, Ericsson M, Johansson M, Sjöqvist E, Sudbery A, Vedral V, Wootters WK**(2011). Geometric local invariants and pure three-qubit states.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*83*(6).

## Abstract:

Geometric local invariants and pure three-qubit states

We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or "gauge" invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of d-level quantum systems. We give an operational interpretation of these invariants in terms of observables. © 2011 American Physical Society.**Williamson MS, Ericsson M, Johansson M, Sjöqvist E, Sudbery A, Vedral V**(2011). Global asymmetry of many-qubit correlations: a lattice-gauge-theory approach.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*84*(3).

## Abstract:

Global asymmetry of many-qubit correlations: a lattice-gauge-theory approach

We introduce a bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the stochastic local operations and classical communication protocols familiar in quantum information. Although the mathematical methods are the same, the meaning of the gauge group is different. The measure we introduce, "twist," is constructed as a Wilson loop from a correlation-induced holonomy. The measure can be understood as the global asymmetry of the bipartite correlations in a loop of three or more qubits; if the holonomy is trivial (the identity matrix), the bipartite correlations can be globally untwisted using general local qubit operations, the gauge group of our theory, which turns out to be the group of Lorentz transformations familiar from special relativity. If it is not possible to globally untwist the bipartite correlations in a state using local operations, the twistedness is given by a nontrivial element of the Lorentz group, the correlation-induced holonomy. We provide several analytical examples of twisted and untwisted states for three qubits, the most elementary nontrivial loop one can imagine. © 2011 American Physical Society.**Modi K, Cable H, Williamson M, Vedral V**(2011). Quantum Correlations in Mixed-State Metrology.

*Physical Review X*,

*1*(2), 1-9.

## Abstract:

Quantum Correlations in Mixed-State Metrology

We analyze the effects of quantum correlations, such as entanglement and discord, on the efficiency of phase estimation by studying four quantum circuits that can be readily implemented using NMR techniques. These circuits define a standard strategy of repeated single-qubit measurements, a classical strategy where only classical correlations are allowed, and two quantum strategies where nonclassical correlations are allowed. In addition to counting space (number of qubits) and time (number of gates) requirements, we introduce mixedness as a key constraint of the experiment.We compare the efficiency of the four strategies as a function of the mixedness parameter. We find that the quantum strategy gives ffiffiffiffi N p enhancement over the standard strategy for the same amount of mixedness. This result applies even for highly mixed states that have nonclassical correlations but no entanglement.**Modi K, Paterek T, Son W, Vedral V, Williamson M**(2010). Unified View of Quantum and Classical Correlations.

*PHYSICAL REVIEW LETTERS*,

*104*(8). Author URL. DOI.

**Williamson MS, Heaney L, Son W**(2010). Violation of multipartite Bell inequalities with classical subsystems via operationally local transformations.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*82*(3).

## Abstract:

Violation of multipartite Bell inequalities with classical subsystems via operationally local transformations

Recently, it was demonstrated by Son, Phys. Rev. Lett.PRLTAO0031-900710. 1103/PhysRevLett.102.110404 102, 110404 (2009), that a separable bipartite continuous-variable quantum system can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality via operationally local transformations. Operationally local transformations are parametrized only by local variables; however, in order to allow violation of the CHSH inequality, a maximally entangled ancilla was necessary. The use of the entangled ancilla in this scheme caused the state under test to become dependent on the measurement choice one uses to calculate the CHSH inequality, thus violating one of the assumptions used in deriving a Bell inequality, namely, the free will or statistical independence assumption. The novelty in this scheme however is that the measurement settings can be external free parameters. In this paper, we generalize these operationally local transformations for multipartite Bell inequalities (with dichotomic observables) and provide necessary and sufficient conditions for violation within this scheme. Namely, a violation of a multipartite Bell inequality in this setting is contingent on whether an ancillary system admits any realistic local hidden variable model (i.e. whether the ancilla violates the given Bell inequality). These results indicate that violation of a Bell inequality performed on a system does not necessarily imply that the system is nonlocal. In fact, the system under test may be completely classical. However, nonlocality must have resided somewhere, this may have been in the environment, the physical variables used to manipulate the system or the detectors themselves provided the measurement settings are external free variables. © 2010 the American Physical Society.**Williamson MS, Vedral V**(2009). Effect of entanglement on geometric phase for multi-qubit states.

*Open Systems and Information Dynamics*,

*16*(2-3), 305-323.

## Abstract:

Effect of entanglement on geometric phase for multi-qubit states

When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state projective Hilbert space. A correction term to this geometric phase, in addition to the local subsystem phases, may appear from correlations between the subsystems. We find that this correction term can be characterized completely either by the entanglement or by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,)-invariant called twist. © 2009 World Scientific Publishing Company.**Williamson MS, Vedral V**(2007). Composite geometric phase for multipartite entangled states.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*76*(3).

## Abstract:

Composite geometric phase for multipartite entangled states

When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show that the dynamical phase acquired by an entangled state in such a scenario can always be understood as the sum of the dynamical phases of its subsystems. In contrast, the equivalent statement for the geometric phase is not generally true unless the state is separable. For an entangled state an additional term is present, the mutual geometric phase, that measures the change the additional correlations present in the entangled state make to the geometry of the state space. For N qubit states we find that this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for W states. © 2007 the American Physical Society.**Lenton TM, Marsh R, Price AR, Lunt DJ, Aksenov Y, Annan JD, Cooper-Chadwick T, Cox SJ, Edwards NR, Goswami S, et al**(2007). Effects of atmospheric dynamics and ocean resolution on bi-stability of the thermohaline circulation examined using the Grid ENabled Integrated Earth system modelling (GENIE) framework.

*CLIMATE DYNAMICS*,

*29*(6), 591-613. Author URL. DOI.

**Williamson MS, Lenton TM, Shepherd JG, Edwards NR**(2006). An efficient numerical terrestrial scheme (ENTS) for Earth system modelling.

*ECOLOGICAL MODELLING*,

*198*(3-4), 362-374. Author URL. DOI.

**Lunt DJ, Williamson MS, Valdes PJ, Lenton TM, Marsh R**(2006). Comparing transient, accelerated, and equilibrium simulations of the last 30 000 years with the GENIE-1 model.

*CLIMATE OF THE PAST*,

*2*(2), 221-235. Author URL. DOI.

**Lunt DJ, Williamson MS, Valdes PJ, Lenton TM**(2006). Comparing transient, accelerated, and equilibrium simulations of the last 30 000 years with the GENIE-1 model.

*Climate of the Past Discussions*,

*2*(3), 267-283.

**Lenton TM, Williamson MS, Edwards NR, Marsh R, Price AR, Ridgwell AJ, Shepherd JG, Cox SJ, team GENIE**(2006). Millennial timescale carbon cycle and climate change in an efficient Earth system model.

*CLIMATE DYNAMICS*,

*26*(7-8), 687-711. Author URL. DOI.

**Marsh R, Smith MPLM, Rohling EJ, Lunt DJ, Lenton TM, Williamson MS, Yool A**(2006). Modelling ocean circulation, climate and oxygen isotopes in the ocean over the last 120000 years.

*Climate of the Past Discussions*,

*2*(5), 657-709.

**Williamson M, Vedral V**(2003). Eavesdropping on practical quantum cryptography.

*Journal of Modern Optics*,

*50*(13), 1989-2011.

## Abstract:

Eavesdropping on practical quantum cryptography

Practical implementations of quantum cryptography use attenuated laser pulses as the signal source rather than single photons. The channels used to transmit are also lossy. Here we give a simple derivation of two beam-splitting attacks on quantum cryptographic systems using laser pulses, either coherent or mixed states with any mean photon number. We also give a simple derivation of a photon-number splitting attack, the most advanced, both in terms of performance and technology required. We find bounds on the maximum disturbance for a given mean photon number and observed channel transmission efficiency for which a secret key can be distilled. We start by reviewing two incoherent attacks that can be used on single photon quantum cryptographic systems. These results are then adapted to systems that use laser pulses and lossy channels.### Publications by year

### In Press

**Bathiany S, Bolt BVD, Williamson MS, Lenton TM, Scheffer M, Nes EV, Notz D**(In Press). Trends in sea-ice variability on the way to an ice-free Arctic.

## Abstract:

Trends in sea-ice variability on the way to an ice-free Arctic

It has been widely debated whether Arctic sea-ice loss can reach a tippingpoint beyond which a large sea-ice area disappears abruptly. The theory of

dynamical systems predicts a slowing down when a system destabilises towards a

tipping point. In simple stochastic systems this can result in increasing

variance and autocorrelation, potentially yielding an early warning of an

abrupt change. Here we aim to establish whether the loss of Arctic sea ice

would follow these conceptual predictions, and which trends in sea ice

variability can be expected from pre-industrial conditions toward an Arctic

that is ice-free during the whole year. To this end, we apply a model hierarchy

consisting of two box models and one comprehensive Earth system model. We find

a consistent and robust decrease of the ice volume's annual relaxation time

before summer ice is lost because thinner ice can adjust more quickly to

perturbations. Thereafter, the relaxation time increases, mainly because the

system becomes dominated by the ocean water's large heat capacity when the

ice-free season becomes longer. Both trends carry over to the autocorrelation

of sea ice thickness in time series. These changes are robust to the nature and

origin of climate variability in the models and hardly depend on the balance of

feedbacks. Therefore, characteristic trends can be expected in the future. As

these trends are not specific to the existence of abrupt ice loss, the

prospects for early warnings seem very limited. This result also has

implications for statistical indicators in other systems whose effective mass

changes over time, affecting the trend of their relaxation time. However, the

robust relation between state and variability would allow an estimate of

sea-ice variability from only short observations. This could help one to

estimate the likelihood and persistence of extreme events in the future.

### 2017

**Williamson MS, collins M, drijfhout S, kahana R, mecking J, lenton TM**(2017). Effect of AMOC collapse on ENSO in a high resolution general circulation model.

*Climate Dynamics*Full text. DOI.

### 2016

**Williamson MS, Bathiany S, M Lenton T**(2016). Early warning signals of tipping points in periodically forced systems.

*Earth System Dynamics*,

*7*(2), 313-326.

## Abstract:

Early warning signals of tipping points in periodically forced systems

© Author(s) 2016. The prospect of finding generic early warning signals of an approaching tipping point in a complex system has generated much interest recently. Existing methods are predicated on a separation of timescales between the system studied and its forcing. However, many systems, including several candidate tipping elements in the climate system, are forced periodically at a timescale comparable to their internal dynamics. Here we use alternative early warning signals of tipping points due to local bifurcations in systems subjected to periodic forcing whose timescale is similar to the period of the forcing. These systems are not in, or close to, a fixed point. Instead their steady state is described by a periodic attractor. For these systems, phase lag and amplification of the system response can provide early warning signals, based on a linear dynamics approximation. Furthermore, the Fourier spectrum of the system's time series reveals harmonics of the forcing period in the system response whose amplitude is related to how nonlinear the system's response is becoming with nonlinear effects becoming more prominent closer to a bifurcation. We apply these indicators as well as a return map analysis to a simple conceptual system and satellite observations of Arctic sea ice area, the latter conjectured to have a bifurcation type tipping point. We find no detectable signal of the Arctic sea ice approaching a local bifurcation.**Bathiany S, Van Der Bolt B, Williamson MS, Lenton TM, Scheffer M, Van Nes EH, Notz D**(2016). Statistical indicators of Arctic sea-ice stability-prospects and limitations.

*Cryosphere*,

*10*(4), 1631-1645.

## Abstract:

Statistical indicators of Arctic sea-ice stability-prospects and limitations

We examine the relationship between the mean and the variability of Arctic sea-ice coverage and volume in a large range of climates from globally ice-covered to globally ice-free conditions. Using a hierarchy of two column models and several comprehensive Earth system models, we consolidate the results of earlier studies and show that mechanisms found in simple models also dominate the interannual variability of Arctic sea ice in complex models. In contrast to predictions based on very idealised dynamical systems, we find a consistent and robust decrease of variance and autocorrelation of sea-ice volume before summer sea ice is lost. We attribute this to the fact that thinner ice can adjust more quickly to perturbations. Thereafter, the autocorrelation increases, mainly because it becomes dominated by the ocean water's large heat capacity when the ice-free season becomes longer. We show that these changes are robust to the nature and origin of climate variability in the models and do not depend on whether Arctic sea-ice loss occurs abruptly or irreversibly. We also show that our climate is changing too rapidly to detect reliable changes in autocorrelation of annual time series. Based on these results, the prospects of detecting statistical early warning signals before an abrupt sea-ice loss at a "tipping point" seem very limited. However, the robust relation between state and variability can be useful to build simple stochastic climate models and to make inferences about past and future sea-ice variability. from only short observations or reconstructions.### 2015

**Williamson MS, Lenton TM**(2015). Detection of bifurcations in noisy coupled systems from multiple time series.

*Chaos*,

*25*(3).

## Abstract:

Detection of bifurcations in noisy coupled systems from multiple time series

© 2015 AIP Publishing LLC. We generalize a method of detecting an approaching bifurcation in a time series of a noisy system from the special case of one dynamical variable to multiple dynamical variables. For a system described by a stochastic differential equation consisting of an autonomous deterministic part with one dynamical variable and an additive white noise term, small perturbations away from the system's fixed point will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all such systems exhibit this decay-type behaviour. However, when the deterministic part has multiple coupled dynamical variables, the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case, we find additional indicators to decay rate, such as frequency of oscillation. In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given multiple time series, where the underlying dynamics are not fully known. Our generalisation also allows bifurcation detection to be applied spatially if one treats each spatial location as a new dynamical variable. One may then determine the unstable spatial mode(s). This is also something that has not been possible with the single variable method. The method is applicable to any set of time series regardless of its origin, but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.### 2012

**Johansson M, Ericsson M, Singh K, Sjöqvist E, Williamson MS**(2012). Topological phases and multiqubit entanglement.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*85*(3).

## Abstract:

Topological phases and multiqubit entanglement

Global phase factors of topological origin, resulting from cyclic local SU evolution, called topological phases, were first described in, in the case of entangled qubit pairs. In this paper we investigate topological phases in multiqubit systems as the result of cyclic local SU(2) evolution. These phases originate from the topological structure of the local SU(2) orbits and are an attribute of most entangled multiqubit systems. We discuss the relation between topological phases and stochastic local operations and classical communication (SLOCC)-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an n-qubit system is described and a complete list of these phases for up to seven qubits is given. © 2012 American Physical Society.### 2011

**Johansson M, Ericsson M, Singh K, Sjöqvist E, Williamson MS**(2011). Correlation-induced non-Abelian quantum holonomies.

*Journal of Physics A: Mathematical and Theoretical*,

*44*(14).

## Abstract:

Correlation-induced non-Abelian quantum holonomies

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the Lévay geometric phase (2004 J. Phys. A: Math. Gen. 37 1821) in the case of two-qubit systems undergoing local SU(2) evolutions. © 2011 IOP Publishing Ltd.**Williamson MS, Ericsson M, Johansson M, Sjöqvist E, Sudbery A, Vedral V, Wootters WK**(2011). Geometric local invariants and pure three-qubit states.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*83*(6).

## Abstract:

Geometric local invariants and pure three-qubit states

We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or "gauge" invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of d-level quantum systems. We give an operational interpretation of these invariants in terms of observables. © 2011 American Physical Society.**Williamson MS, Ericsson M, Johansson M, Sjöqvist E, Sudbery A, Vedral V**(2011). Global asymmetry of many-qubit correlations: a lattice-gauge-theory approach.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*84*(3).

## Abstract:

Global asymmetry of many-qubit correlations: a lattice-gauge-theory approach

We introduce a bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the stochastic local operations and classical communication protocols familiar in quantum information. Although the mathematical methods are the same, the meaning of the gauge group is different. The measure we introduce, "twist," is constructed as a Wilson loop from a correlation-induced holonomy. The measure can be understood as the global asymmetry of the bipartite correlations in a loop of three or more qubits; if the holonomy is trivial (the identity matrix), the bipartite correlations can be globally untwisted using general local qubit operations, the gauge group of our theory, which turns out to be the group of Lorentz transformations familiar from special relativity. If it is not possible to globally untwist the bipartite correlations in a state using local operations, the twistedness is given by a nontrivial element of the Lorentz group, the correlation-induced holonomy. We provide several analytical examples of twisted and untwisted states for three qubits, the most elementary nontrivial loop one can imagine. © 2011 American Physical Society.**Modi K, Cable H, Williamson M, Vedral V**(2011). Quantum Correlations in Mixed-State Metrology.

*Physical Review X*,

*1*(2), 1-9.

## Abstract:

Quantum Correlations in Mixed-State Metrology

We analyze the effects of quantum correlations, such as entanglement and discord, on the efficiency of phase estimation by studying four quantum circuits that can be readily implemented using NMR techniques. These circuits define a standard strategy of repeated single-qubit measurements, a classical strategy where only classical correlations are allowed, and two quantum strategies where nonclassical correlations are allowed. In addition to counting space (number of qubits) and time (number of gates) requirements, we introduce mixedness as a key constraint of the experiment.We compare the efficiency of the four strategies as a function of the mixedness parameter. We find that the quantum strategy gives ffiffiffiffi N p enhancement over the standard strategy for the same amount of mixedness. This result applies even for highly mixed states that have nonclassical correlations but no entanglement.### 2010

**Modi K, Paterek T, Son W, Vedral V, Williamson M**(2010). Unified View of Quantum and Classical Correlations.

*PHYSICAL REVIEW LETTERS*,

*104*(8). Author URL. DOI.

**Williamson MS, Heaney L, Son W**(2010). Violation of multipartite Bell inequalities with classical subsystems via operationally local transformations.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*82*(3).

## Abstract:

Violation of multipartite Bell inequalities with classical subsystems via operationally local transformations

Recently, it was demonstrated by Son, Phys. Rev. Lett.PRLTAO0031-900710. 1103/PhysRevLett.102.110404 102, 110404 (2009), that a separable bipartite continuous-variable quantum system can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality via operationally local transformations. Operationally local transformations are parametrized only by local variables; however, in order to allow violation of the CHSH inequality, a maximally entangled ancilla was necessary. The use of the entangled ancilla in this scheme caused the state under test to become dependent on the measurement choice one uses to calculate the CHSH inequality, thus violating one of the assumptions used in deriving a Bell inequality, namely, the free will or statistical independence assumption. The novelty in this scheme however is that the measurement settings can be external free parameters. In this paper, we generalize these operationally local transformations for multipartite Bell inequalities (with dichotomic observables) and provide necessary and sufficient conditions for violation within this scheme. Namely, a violation of a multipartite Bell inequality in this setting is contingent on whether an ancillary system admits any realistic local hidden variable model (i.e. whether the ancilla violates the given Bell inequality). These results indicate that violation of a Bell inequality performed on a system does not necessarily imply that the system is nonlocal. In fact, the system under test may be completely classical. However, nonlocality must have resided somewhere, this may have been in the environment, the physical variables used to manipulate the system or the detectors themselves provided the measurement settings are external free variables. © 2010 the American Physical Society.### 2009

**Williamson MS, Vedral V**(2009). Effect of entanglement on geometric phase for multi-qubit states.

*Open Systems and Information Dynamics*,

*16*(2-3), 305-323.

## Abstract:

Effect of entanglement on geometric phase for multi-qubit states

When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state projective Hilbert space. A correction term to this geometric phase, in addition to the local subsystem phases, may appear from correlations between the subsystems. We find that this correction term can be characterized completely either by the entanglement or by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,)-invariant called twist. © 2009 World Scientific Publishing Company.### 2007

**Williamson MS, Vedral V**(2007). Composite geometric phase for multipartite entangled states.

*Physical Review a - Atomic, Molecular, and Optical Physics*,

*76*(3).

## Abstract:

Composite geometric phase for multipartite entangled states

When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show that the dynamical phase acquired by an entangled state in such a scenario can always be understood as the sum of the dynamical phases of its subsystems. In contrast, the equivalent statement for the geometric phase is not generally true unless the state is separable. For an entangled state an additional term is present, the mutual geometric phase, that measures the change the additional correlations present in the entangled state make to the geometry of the state space. For N qubit states we find that this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for W states. © 2007 the American Physical Society.**Lenton TM, Marsh R, Price AR, Lunt DJ, Aksenov Y, Annan JD, Cooper-Chadwick T, Cox SJ, Edwards NR, Goswami S, et al**(2007). Effects of atmospheric dynamics and ocean resolution on bi-stability of the thermohaline circulation examined using the Grid ENabled Integrated Earth system modelling (GENIE) framework.

*CLIMATE DYNAMICS*,

*29*(6), 591-613. Author URL. DOI.

### 2006

**Williamson MS, Lenton TM, Shepherd JG, Edwards NR**(2006). An efficient numerical terrestrial scheme (ENTS) for Earth system modelling.

*ECOLOGICAL MODELLING*,

*198*(3-4), 362-374. Author URL. DOI.

**Lunt DJ, Williamson MS, Valdes PJ, Lenton TM, Marsh R**(2006). Comparing transient, accelerated, and equilibrium simulations of the last 30 000 years with the GENIE-1 model.

*CLIMATE OF THE PAST*,

*2*(2), 221-235. Author URL. DOI.

**Lunt DJ, Williamson MS, Valdes PJ, Lenton TM**(2006). Comparing transient, accelerated, and equilibrium simulations of the last 30 000 years with the GENIE-1 model.

*Climate of the Past Discussions*,

*2*(3), 267-283.

**Lenton TM, Williamson MS, Edwards NR, Marsh R, Price AR, Ridgwell AJ, Shepherd JG, Cox SJ, team GENIE**(2006). Millennial timescale carbon cycle and climate change in an efficient Earth system model.

*CLIMATE DYNAMICS*,

*26*(7-8), 687-711. Author URL. DOI.

**Marsh R, Smith MPLM, Rohling EJ, Lunt DJ, Lenton TM, Williamson MS, Yool A**(2006). Modelling ocean circulation, climate and oxygen isotopes in the ocean over the last 120000 years.

*Climate of the Past Discussions*,

*2*(5), 657-709.

### 2003

**Williamson M, Vedral V**(2003). Eavesdropping on practical quantum cryptography.

*Journal of Modern Optics*,

*50*(13), 1989-2011.