1. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 CQC, Cambridge Emergence of typical entanglement in random two-party processes. Oscar C.O. Dahlsten with Martin B. Plenio, Roberto Oliveira and Alessio Serafini www.imperial.ac.uk/quantuminformation

2. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 ‘Emergence of typical entanglement in random two-party processes’ By entanglement we mean, unless otherwise stated, that taken between two parties sharing a pure state. By typical/generic entanglement we mean the entanglement average over pure states picked from the uniform (Haar) distribution. [Hayden, Leung, Winter, Comm. Math.Phys. 2006] By two-party process we mean that there are only two-body (and thus local) interactions. By random two-party process we mean that the local interaction is picked at random (using classical probabilities).

3. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Aim of talk This talk aims to explain key points of a series of related papers: Efficient Generation of Generic Entanglement. [Oliveira, Dahlsten and Plenio, quant-ph/0605126] Entanglement probability distribution of bipartite randomised stabilizer states. [Dahlsten and Plenio, Quant. Inf. Comp., 6 no.6 (2006)] Thermodynamical state space measure and typical entanglement of pure Gaussian states. [Serafini, Dahlsten and Plenio, quant-ph/0610090] Emergence of typical entanglement in two-party random processes. [Dahlsten, Oliveira and Plenio, to appear on quant-ph]

4. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Talk Structure Part 1: Abstract mathematical results on typical entanglement. Known Theorems:Typical/Generic entanglement of general pure states. New Theorem: Typical entanglement of pure stabilizer states. New Theorem: Typical entanglement of pure Gaussian states. Part 2: Relating abstract results to random two-party process. New Theorem : Generic entanglement is generated efficiently New Numerical Observation: Generic entanglement is achieved at a particular instant. Conclusion and Outlook

5. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Part I: Typical entanglement- abstract view Restricting entanglement types to those that are typical/generic could give a simplified entanglement theory. [Hayden, Leung, Winter, Comm. Math.Phys. 2006] ‘Typical’ has been defined relative to a flat distribution on pure states, the unitarily invariant measure, where. For a single qubit this can be visualised as an even density on the Bloch sphere. If the associated entanglement probability distribution is concentrated around a value->typical entanglement.

6. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Most quantum states are maximally entangled Known result: The entanglement E between N A and N B qubits is expected to be nearly maximal. [Lubkin,,J. Math. Phys.1978][Lloyd, Pagels, Ann. of Phys. 1988][Page, PRL, 1993] [Foong, Kanno PRL, 1994] [Hayden, Leung, Winter, Comm. Math.Phys. 2006] The distribution concentrates around the average with increasing N-> the average Is the typical value. Entanglement E Prob(E ) Typical Entanglement

7. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Stabilizer, Gaussian states Can ask what the typical entanglement is of interesting subsets of states. We determined the typical entanglement of stabilizer states[Dahlsten&Plenio, QIC] and Gaussian states[Serafini, Dahlsten&Plenio]. -see also [Smith&Leung quant-ph/0510232] Stabilizer states are a finite subset of general states, including the EPR and GHZ states. Gaussian states have analogous importance in the continuous variable setting. Here one considers entanglement between ‘modes’. We now briefly mention results on the typical entanglement of stabilizer states and Gaussian states.

8. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Most stabilizer states are maximally entangled New Theorem: exact probability distribution of entanglement P(E) of stabilizer states. E P(E) Distribution Concentrates

9. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Typical entanglement of Gaussian states Called ‘Gaussian’ since they are uniquely specified by (i) first moments (expectations)of pairs(modes) of canonical X, P operators (ii) a matrix,, giving the second moments of the operators’ expectations. The quantum correlations between modes depend only on. New Result: we construct a “Thermodynamical state space measure of Gaussian states” to pick in an unbiased manner. [Matlab code available at www.imperial.ac.uk/quantuminformation] New Result: the typical entanglement is when and

10. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Part 1 conclusion Known result: Most (relative to flat distribution on pure states) pure states are maximally entangled New result: Most (relative to flat distribution) stabilizer states are maximally entangled. New result: Most(relative to a distribution we invented) Gaussian states have a typical entanglement. But are statements relative to the (flat) unitarily invariant measure physically relevant?

11. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Relating abstract results to random two- party processes Part II

12. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Part II: Motivation and Aim From part I: Entanglement is typically maximal. ‘Typical’ has been defined relative to a flat distribution on pure states. However exp(N=system size) two-qubit gates are necessary to get that flat distribution on states, so it seems not physically relevant. Aim: to investigate what entanglement emerges in poly(N) applications of two-qubit gates.

13. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Part 2: Overview Setting: The two-party random process(es). New Result (Theorem) : Generic entanglement is generated efficiently. New Result (Numerics): Generic entanglement is achieved at a particular instant.

14. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 The random process Consider random two-party interactions modelled as two- qubit gates: 1. Pick two single qubit unitaries, U and V, uniformly from the Bloch Sphere. 2. Choose a pair of qubits {c,d} without bias. 3. Apply U to c and V to d. 4. Apply a CNOT on c and d. U V … … … … Qubit 1 c d N

15. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Random process example U1U1 V1V1 U2U2 V2V2 V3V3 U3U3 V4V4 U4U4

16. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Entanglement after infinite time After infinite time(steps), the entanglement E is expected to be nearly maximal since get uniform distribution, [Lubkin,,J. Math. Phys.1978][Lloyd, Pagels, Ann. of Phys. 1988][Page, PRL, 1993][Foong, Kanno PRL, 1994] [Hayden, Leung, Winter, Comm. Math.Phys. 2006][Emerson, Livine, Lloyd, PRA 2005] But this average is only physical if it is reached in poly(N) steps. NANA NBNB U V … … … … Qubit 1 N

17. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Result: efficient generation Theorem: The average entanglement of the unitarily invariant measure is reached to a fixed arbitrary accuracy ε within O(N 3 ) steps. In other words the circuit is expected to make the input state maximally entangled in a physical number of steps. 3

18. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Efficient generation, exact statement Theorem: Let some arbitrary ε ∈ (0, 1) be given. Then for a number n of gates in the random circuit satisfying n ≥ 9N(N − 1)[(4 ln 2)N + ln ε −1 ]/4 we have and, for maximally entangled

19. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Efficient generation, proof outline The random circuit does a random walk on a massive state space. One could consider mapping the random walk onto an associated, faster converging, random walk on the entanglement state space. It is a bit more complicated though. In fact we map it onto a random walk relating to the purity. We then use known Markov Chain methods to bound the rate of convergence of this smaller walk. 0<E<1 E max -1<E<E max State Space

20. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Result: moment it becomes typical Numerical observation: can associate a specific time with achievement of generic entanglement. This figure shows the total variation, TV, distance to the asymptotic entanglement probability distribution. It tends to a step function with increasing N. For larger N we used stabilizer states and tools for efficient evaluation of their entanglement. [Gottesmann PhD] [Audenaert,Plenio, NJP 2005] We term this the variation cut-off moment after [Diaconis, Cut-off effect in Markov chains]

21. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Part II: Conclusion Result: Proof that generic entanglement is physical as it can be generated using poly(N) two-qubit gates. Implication: arguments and protocols assuming generic entanglement gain relevance. [Abeyesinghe,Hayden,Smith, Winter, quant-ph/0407061][Harrow, Hayden, Leung, PRL 2004] Result: Numerical observation that generic entanglement is achieved at a particular instant.

22. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Summary Part 1: Abstract mathematical results on typical entanglement. Known Result (Theorem):Typical/Generic entanglement of general pure states is near maximal. New Result (Theorem): Typical entanglement of pure Stabilizer states is near maximal. New Result (Theorem): There is a typical entanglement of pure Gaussian states. Part 2: Relating abstract results to random two-party processes. New Result (Theorem) : Generic entanglement is generated efficiently New Result (Numerics): Generic entanglement is achieved at a particular instant.

23. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Outlook There are many waiting results in this area. Analytical proof of cut-off moment? Typical entanglement of naturally occurring systems? t=0t=1 Alice Bob

24. Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep 2006 Acknowledgements, References We acknowledge initial discussions with Jonathan Oppenheim, and later discussions with numerous people. Funding by The Leverhulme Trust, EPSRC QIP-IRC, EU Integrated Project QAP, EU Marie-Curie, the Royal Society, the NSA, the ARDA, Imperial’s Institute for Mathematical Sciences. Papers discussed here: Efficient Generation of Generic Entanglement. [Oliveira, Dahlsten and Plenio, quant-ph/0605126] Entanglement probability distribution of bipartite randomised stabilizer states. [Dahlsten and Plenio, Quant. Inf. Comp., 6 no.6 (2006)] Thermodynamical state space measure and typical entanglement of pure Gaussian states. [Serafini, Dahlsten and Plenio, quant-ph/0610090] Emergence of typical entanglement in a two-party random process. [Dahlsten, Oliveira and Plenio, to appear on quant-ph]