‘Emergence of typical entanglement in random two-party processes’

1. CQC, CambridgeEmergence of typical entanglement in random two-party processes.Oscar C.O. Dahlsten withMartin B. Plenio, Roberto Oliveira and Alessio Serafini www.imperial.ac.uk/quantuminformation

2. ‘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. 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. 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. 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. Most quantum states are maximally entangled • Known result: The entanglement E between NA and NB 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. Typical Entanglement Prob(E) EntanglementE

7. 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. Most stabilizer states are maximally entangled • New Theorem: exact probability distribution of entanglement P(E) of stabilizer states. Distribution Concentrates P(E) E

9. 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. 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. Part II Relating abstract results to random two-party processes

12. 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. 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. The random process Qubit 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. 1 … c U … … V d … N

15. Random process example U2 U1 V4 V2 V3 V1 U3 U4

16. Entanglement after infinite time Qubit • 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. 1 NA … U … … NB V … N

17. Result: efficient generation • Theorem: The average entanglement of the unitarily invariant measure is reached to a fixed arbitrary accuracy ε within O(N3) steps. • In other words the circuit is expected to make the input state maximally entangled in a physical number of steps. 3

18. 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. Efficient generation, proof outline Emax-1<E<Emax • 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 State Space

20. 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. 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. 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. Outlook • There are many waiting results in this area. • Analytical proof of cut-off moment? • Typical entanglement of naturally occurring systems? Alice Bob t=0 t=1

24. 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]