‘ Typical entanglement and random states in the continuous variable regime ’

1. Randomization workshop, IQC WaterlooTypical entanglement and random states in the continuous variable regime Oscar C.O. Dahlsten withAlessio Serafini, Martin B. Plenio and David Gross.www.imperial.ac.uk/quantuminformation

2. ‘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]

3. 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/0701051] • Teleportation fidelities of squeezed states from thermodynamical state space measures. [Serafini, Dahlsten and Plenio, quant-ph/0610090, Phys. Rev. Lett. 98, 170501 (2007)] Aims of the above work: -Define how to pick Gaussian continuous variable states at random. -Study typical entanglement of such states.

4. 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

5. 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.

6. 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(n2) 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.

7. 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.

8. Random Gaussian states-splitting s • For pure Gaussian states, s=STS where S is a real symplectic matrix. Furthermore • where O is an orthogonal symplectic transform • and Z=diag(z1, z2…zn ) . • Key point (not a new result though): • wherein the compact part, O, has been ‘split’ from the non-compact part, Z.

9. 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.

10. Random Gaussian states-non-compact part • The energy of each mode j is • Let • For some finite energy capping Emax., 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. 2006 ]: “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.

11. 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: • Is there a typical entanglement? • For how many modes does it become typical? • Is the typical entanglement maximal? Alice Bob 1 2 3n

12. Typical entanglement-it exists • Key point: Recall that the measure complies with the general canonical principle, so sA 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)

13. Typical entanglement-when does it exist?

14. 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: E1=E/2, • but the measure implies that typically • E1=E/n.

15. 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.

16. 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. Alice Bob t=0 t=1

17. 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.

18. 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/0701051] • Teleportation fidelities of squeezed states from thermodynamical state space measures. [Serafini, Dahlsten and Plenio, quant-ph/0610090, Phys. Rev. Lett. 98, 170501 (2007)] • The Matlab code spitting out random Gaussian states can be downloaded at www.imperial.ac.uk/quantuminformation