ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Randomization workshop, IQC Waterloo Typical entanglement and random states in the continuous variable regime Oscar C.O. Dahlsten with Alessio Serafini, Martin B. Plenio and David Gross.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 ‘Typical entanglement and random states in the continuous variable regime’ By the continuous variable regime we mean states of infinite-level systems, so- called Gaussian states in particular. By random states we mean states picked according to a probability distribution we define. By entanglement we mean, unless otherwise stated, that taken between two parties sharing a pure state (i.e. Von Neumann entropy of one party). By typical/generic entanglement we mean the peak of the entanglement probability distribution associated with the random states. [Hayden, Leung, Winter, Comm. Math.Phys. 2006]

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Aim of talk This talk aims to explain key points of: Canonical and micro-canonical typical entanglement of continuous variable systems. [Serafini, Dahlsten, Gross and Plenio, quant-ph/ ] Teleportation fidelities of squeezed states from thermodynamical state space measures. [Serafini, Dahlsten and Plenio, quant-ph/ , Phys. Rev. Lett. 98, (2007)] Aims of the above work: -Define how to pick Gaussian continuous variable states at random. -Study typical entanglement of such states.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Talk Structure ABC of Gaussian continuous variable states Random Gaussian states Result 1: Microcanonical measure on Gaussian states Their entanglement probability distribution Result 2: The typical entanglement of Gaussian states Summary Outlook

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 ABC of Gaussian continuous variable states Gaussian states are defined by having Gaussian Wigner functions. They are continuous variable states because x and p have continuous spectra. Coherent states, thermal states and many ground states are Gaussian states. Although they are continuous variable states they can be conveniently parameterised.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 ABC of Gaussian continuous variable states cont’d Let, where n is the number of modes. Their Gaussian nature means they are completely specified by first moments R and second moments s, with O(n 2 ) variables. We will here refer to the ‘energy’ of a Gaussian state as. This is important since we will bound the energy later to tame the non-compactness. Key point at this stage: alone carries the entanglement information, so we will focus on it.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Random Gaussian states-strategy To talk about properties of random gaussian states we need a method of picking them at random, i.e. a measure on the state space. We now define and justify such a measure. The problem is not trivial because a priori there is non-compactness and divergence. We focus on s as that is where the quantum correlations are encoded. The strategy is to split sigma into compact and non-compact parts. A Haar measure is used for the compact part and the energy is bound to deal with the other.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Random Gaussian states -splitting s For pure Gaussian states, s=S T S where S is a real symplectic matrix. Furthermore where O is an orthogonal symplectic transform and Z=diag(z 1, z 2… z n ). Key point (not a new result though): wherein the compact part, O, has been ‘split’ from the non-compact part, Z.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Random Gaussian states -compact part Before taming the non-compactness we deal with the compact part, O in O is a member of the orthogonal and symplectic group K(n) It turns out K(n) is isomorphic to U(n). O must have form And U=X+iY can be shown to be an isomorphism We thus get the invariant measure on O by picking U from the unitarily invariant measure. We define this to be the way of picking it.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Random Gaussian states -non-compact part The energy of each mode j is Let For some finite energy capping E max., Define a measure on the energies Justification: -Energy is always finite in reality. -Lack of knowledge of local energies is maximised. -The resulting measure is compliant with the ‘general canonical principle’. [Popescu, Short, Winter, Nature Phys ]: “Given a sufficiently small subsystem of the universe, almost every pure state of the universe is such that the subsystem is approximately in the canonical state” which is here simply a thermal state.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Alice Bob Typical entanglement The question of typical/generic entanglement has been extensively studied in the finite dimensional setting. [Aspects of Generic Entanglement, Hayden, Winter, Leung] Equipped with the ‘microcanonical’ measure just described, we now ask what the typical entanglement of Gaussian states is. Three key questions are: 1.Is there a typical entanglement? 2.For how many modes does it become typical? 3.Is the typical entanglement maximal? n n

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Typical entanglement -it exists Key point: Recall that the measure complies with the general canonical principle, so s A concentrates around a thermal state, where it is simply diag(1+T/2, 1+T/2... ) using temperature T=(E-2n)/n. Thus: 1.There is a typical entanglement in the thermodynamical limit, it is the entropy of a thermal state. In formulae: Where (forgive me for exposing you to this)

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Typical entanglement -when does it exist?

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Typical entanglement -why it is not in general maximal The typical entanglement is not in general maximal given the total energy restriction. The energy in the first mode effectively controls how many levels are available to entangle. The construction of the measure implies that the energy is typically shared out equally between all modes (equipartition of energy). Consider e.g. the entanglement between 1 mode and another n-1 modes. For maximal entanglement the first mode should have half the energy: E 1 =E/2, but the measure implies that typically E 1 = E/n.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Summary Gaussian states are uniquely specified by first moments R, and second moments s. We define a method of picking Gaussian states at random, where we put a cap on the energy to tame the non-compactness: this is called a microcanonical measure on Gaussian states. This is justified as systems will have finite energy. Furthermore our method of capping the energy is consistent with the general canonical principle. Using this measure we determine that there is a typical entanglement of Gaussian states. It is non-maximal given the energy restriction because there is equipartition of energy, due to the thermal nature of the measure.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Outlook 1: Dynamics The measure presented corresponds to the asymptotic time limit of some ‘thermalization’ process, what would this be? E.g. we could track the x,p in Briegel’s spin gases (suggested by J.Eisert), or find the analogy of random circuits in this setting. t=0t=1 Alice Bob

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Outlook 2: Experiments & general states One can teleport with the information stored in the second moments. Can then use the measure to evaluate teleportation fidelity F of a single mode as a function of E, the maximum energy of the states. Red dots presumes no shared entanglement that can be used as a resource, whereas blue and green use entanglement. We are looking for experimenters to test this. Another hope is to use Gaussian states instead of averaging over all continuous variable states.

ABCRandom GaussiansConclusion&OutlookIntroductionTypical entanglement IQC 27 June 2007 Acknowledgements, References We thank R. Oliveira, J.Eisert and K.Audenaert for discussions. Funding by The Leverhulme Trust, EPSRC QIP-IRC, EU Integrated Project QAP, EU Marie-Curie, the Royal Society, Imperial’s Institute for Mathematical Sciences. Papers discussed here: Canonical and micro-canonical typical entanglement of continuous variable systems. [Serafini, Dahlsten, Gross and Plenio, quant-ph/ ] Teleportation fidelities of squeezed states from thermodynamical state space measures. [Serafini, Dahlsten and Plenio, quant-ph/ , Phys. Rev. Lett. 98, (2007)] The Matlab code spitting out random Gaussian states can be downloaded at