Quantum entanglement in the 21 st century. John Preskill The Quantum Century: 100 Years of the Bohr Atom 3 October 2013

Transcription

1 Quantum entanglement in the 21 st century John Preskill The Quantum Century: 100 Years of the Bohr Atom 3 October 2013

2 My well-worn copy, bought in 1966 when I was 13.

3 George Gamow, recalling Bohr s Theoretical Physics Institute : Bohr s Institute quickly became the world center of quantum physics, and to paraphrase the old Romans, all roads led to Blegdamsvej 17 The popularity of the institute was due both to the genius of its director and his kind, one might say fatherly, heart Almost every country in the world has physicists who proudly say: I used to work with Bohr. Thirty Years That Shook Physics, 1966, p. 51.

4 George Gamow, recalling Bohr s Theoretical Physics Institute : Bohr, Fru Bohr, Casimir, and I were returning home from the farewell dinner for Oscar Klein on the occasion of his election as a university professor in his native Sweden. At that late hour the streets of the city were empty. On the way home we passed a bank building with walls of large cement blocks. At the corner of the building the crevices between the courses of the blocks were deep enough to give a toehold to a good alpinist. Casimir, an expert climber, scrambled up almost to the third floor. When Cas came down, Bohr, inexperienced as he was, went up to match the deed. When he was hanging precariously on the second-floor level, and Fru Bohr, Casimir, and I were anxiously watching his progress, two Copenhagen policeman approached from behind with their hands on their gun holsters. One of them looked up and told the other, Oh, it is only Professor Bohr! and they went quietly off to hunt for more dangerous bank robbers. Thirty Years That Shook Physics, 1966, p. 57.

5 Werner Heisenberg on Schrödinger s 1926 visit to Coperhagen: Bohr s discussions with Schrödinger began at the railway station and continued daily from early morning until late at night. Schrödinger stayed at Bohr s house so that nothing would interrupt the conversations After a few days, Schrödinger fell ill, perhaps as a result of his enormous effort; in any case he was forced to keep to his bed with a feverish cold. While Mrs. Bohr nursed him and brought in tea and cake, Niels Bohr kept sitting on the edge of the bed talking at Schrödinger: But surely you must admit that No real understanding could be expected since, at that time, neither side was able to offer a complete and coherent interpretation of quantum mechanics. Physics and Beyond, 1971, p

6 Classical Correlations

7 Classical Correlations Quantum Correlations Aren t boxes like soxes?

8 Einstein s 1935 paper, with Podolsky and Rosen (EPR), launched the theory of quantum entanglement. To Einstein, quantum entanglement was so unsettling as to indicate that something is missing from our current understanding of the quantum description of Nature.

9 If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

10 there is no question of a mechanical disturbance of the system under investigation during the critical last stage of the measuring procedure. But even at this stage there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system.

11 Quantum entanglement. This Page Blank This Page Blank This Page Blank This Page Blank This Page Blank. Nearly all the information in a typical entangled quantum book is encoded in the correlations among the pages. You can't access the information if you read the book one page at a time.

12 To describe 300 qubits, we would need more numbers than the number of atoms in the visible universe!

13 We can t even hope to describe the state of a few hundred qubits in terms of classical bits. Might a computer that operates on qubits rather than bits (a quantum computer) be able to perform tasks that are beyond the capability of any conceivable classical computer?

14 Peter Shor

15 Problems Quantumly Hard Quantumly Easy Classically Easy

16 Problems Quantumly Hard Quantumly Easy Classically Easy What s in here?

17 Three Questions About Quantum Computers 1. Why build one? How will we use it, and what will we learn from it? A quantum computer may be able to simulate efficiently any process that occurs in Nature! 2. Can we build one? Are there obstacles that will prevent us from building quantum computers as a matter of principle? Using quantum error correction, we can overcome the damaging effects of noise at a reasonable overhead cost. 3. How will we build one? What kind of quantum hardware is potentially scalable to large systems?

18 Quantum entanglement in the 21 st century Algorithms Error Correction Matter Spacetime

19 arxiv papers with entanglement in the title quant-ph

20 arxiv papers with entanglement in the title cond-mat hep-th gr-qc

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23 Classical correlations are polygamous Betty Adam Charlie

24 Quantum correlations are monogamous Betty fully entangled unentangled Adam Charlie

25 Quantum correlations are monogamous Betty unentangled fully entangled Adam Charlie

26 Monogamy is frustrating! Betty fully entangled unentangled Adam cryptography quantum matter black holes Charlie

27 Information Puzzle: Is a black hole a quantum cloner? Suppose that the collapsing body s quantum information is encoded in the emitted Hawking radiation; the information is thermalized, not destroyed. The green time slice crosses both the collapsing body behind the horizon and nearly all of the radiation outside the horizon. Thus the same (quantum) information is in two places at the same time. A quantum cloning machine has operated, which is not allowed by the linearity of quantum mechanics. We re stuck: either information is destroyed or cloning occurs. Either way, quantum physics needs revision. outgoing radiation singularity time slice time (outside horizon) collapsing body event horizon

28 Black hole complementarity singularity Perhaps the lesson is that, for mysterious reasons that should be elucidated by a complete theory of quantum gravity, it is wrong to think of the outside and inside portions of the time slice as two separate subsystems of a composite system. H H in H out Rather, the inside and outside are merely complementary descriptions of the same system. Which description is appropriate depends on whether the observer enters the black hole or stays outside (Susskind, 1993). time slice outgoing radiation time (outside horizon) collapsing body event horizon

29 No-cloning lower bound on the information retention time Let s demand that verifiable cloning does not occur. Then the proper time during which Alice can send her qubits to Bob cannot be larger than O(1) in Planck units: singularity Bob τ r exp t / r O(1) r (Alice) proper and therefore ( ) S S S t O r log r ( ) S S S Planck Alice (where r S is measured in Planck units ). If Alice s quantum information were revealed in the Hawking radiation faster than this, then Alice and Bob would be able to verify that Alice s quantum information is in two places at once, in violation of the no-cloning principle.

30 Alice throws k qubits (maximally entangled with reference system N) into an old black hole. As radiation R escapes, the correlation of N with B decays. Eventually, N is nearly uncorrelated with B and nearly maximally entangled with a subsystem of ER --- at that stage, Bob can decode Alice s quantum message with high fidelity (Hayden-Preskill, 2007). Haar Black holes as mirrors Bob s decoder E R B' radiation black hole ( dv ) ρ V ρ ρmax 1 V B black hole maximal entanglement Alice s qubits reference system k N 2 ( ) = = 2 k + c R 2 B NB B N B c time Bob can decode with high fidelity after receiving only k+c qubits of Hawking radiation, where c is a constant, if the mixing unitary V B is Haar random, or even if it is a typical unitary realized by a small quantum circuit (depth ~log r s ). N

31 Black hole complementarity challenged Three reasonable beliefs, not all true! [Almheri, Marolf, Polchinski, Sully (AMPS) 2012]: (1) The black hole scrambles information, but does not destroy it. (2) An observer who falls through the black hole horizon sees nothing unusual (at least for a while). (3) An observer who stays outside the black hole sees nothing unusual. Conservative resolution: A firewall at the horizon.

32 Complementarity Challenged singularity (1) For an old black hole, recently emitted radiation (B) is highly entangled with radiation emitted earlier (R) by the time it reaches Robert. Robert R (2) If freely falling observer sees vacuum at the horizon, then the recently emitted radiation (B) is highly entangled with modes behind the horizon (A). (3) If B is entangled with R by the time it reaches Robert, it was already entangled with R at the time of emission from the black hole. Monogamy of entanglement violated! outgoing radiation time (outside horizon) B Betty A Adam event horizon

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34 What s inside a black hole? black hole Bob Alice

35 A. An unlimited amount of stuff. singularity forward light cone There is all that stuff that fell in and it crashed into the singularity and that s it. Bye-bye. Bill Unruh But -- Why S = Area / 4? time -- What about AdS/CFT duality? collapsing matter

36 B. Nothing at all. singularity It is time to constrain and construct the dynamics of firewalls. Raphael Bousso time But -- Curtains for the equivalence principle? (Braunstein, 2009) collapsing matter

37 C. A huge but finite amount of stuff, which is also outside the black hole. A black hole wormhole-connected to the Hawking radiation it has emitted (Maldacena and Susskind). B (recent radiation) can be entangled with both A (behind the horizon) and R (early radiation), because A and R are two descriptions of the same system. Complementarity rescued, perhaps by identifying nontraversable wormholes with entanglement (ER = EPR). But -- R could be far, far away from the black hole.

38 What s inside a black hole? A. An unlimited amount of stuff. B. Nothing at all. C. A huge but finite amount of stuff, which is also outside the black hole. D. None of the above.

39 Holographic entanglement entropy minimal bulk surface bulk boundary To compute entropy of region A in the boundary field theory, find minimal area of the bulk surface with the same boundary: 1 S( A) = min m= A area(m ) + 4 G N Ryu and Takayanagi, 2006 Recover, for example, in 1+1 dimensional conformal field theory: c S( A( L)) = log( L / a) + 3

40 Strong subadditivity from holography minimal bulk surface S(A) + S(B) S(A»B) + S(A B) Headrick and Takayanagi, 2007 bulk bulk boundary boundary Tripartite Info: I(A;B) + I(A;C) I(A;BC) 0 ( extensivity of mutual information). True for holographic theories, not in general. Hayden, Headrick, Maloney, 2011

41 Building spacetime from quantum entanglement i e β E i /2 i β E e /2 i E E i i A connected geometry is constructed as a superposition of disconnected geometries. The entangled state becomes a product state as the neck pinches off and the geometry becomes disconnected. (Van Raamsdonk, 2010).

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43 Love in a wormhole throat singularity Alice Bob time Alice and Bob are in different galaxies, but each lives near a black hole, and their black holes are connected by a wormhole. If both jump into their black holes, they can enjoy each other s company for a while before meeting a tragic end.

44 C. A huge but finite amount of stuff, which is also outside the black hole. A black hole wormhole-connected to the Hawking radiation it has emitted (Maldacena and Susskind). B (recent radiation) can be entangled with both A (behind the horizon) and R (early radiation), because A and R are two descriptions of the same system. Complementarity rescued, perhaps by identifying nontraversable wormholes with entanglement (ER = EPR). But -- R could be far, far away from the black hole.

45 Horowitz-Maldacena Proposal (2003) singularity singularity S in out out = M time M time Quantum information escapes from a black hole via postselected teleportation. The black hole S-matrix is unitary if the Unruh vacuum at the horizon is maximally entangled and the postselected final state at the horizon is also maximally entangled. Monogamy of entanglement and no-cloning are (temporarily) violated, allowing smoothness of the horizon to be reconciled with unitarity. (Lloyd and Preskill, 2013).

46 Horowitz-Maldacena Proposal (2003) out 2 out 1 singularity S in 2 out in 1 = M 1 M 2 M time Quantum information escapes from a black hole via postselected teleportation. The black hole S-matrix is unitary if the Unruh vacuum at the horizon is maximally entangled and the postselected final state at the horizon is also maximally entangled. Monogamy of entanglement and no-cloning are (temporarily) violated, allowing smoothness of the horizon to be reconciled with unitarity. (Lloyd and Preskill, 2013).

47 Generic final state N out Considering dividing the infalling matter into a relatively small subsystem M 1 (matter that collapses quickly) and a larger subsystem M 2 (which collapses slowly). M 1 U M 2 0 in If M 2 is initially in a fixed (vacuum) state, then a generic final state boundary condition, will project onto a very nearly maximally entangled state of M 1 and the outgoing radiation; hence the black hole S- matrix will be very nearly unitary. H L1 norm deviation from unitarity: M1 3/2 exp ( SBH / 2 + O( m )) Hin Such a small violation of unitarity may be an artifact of the semiclassical framework used in the analysis, as nonperturbative quantum gravity corrections of that order are expected. 1/2

48 Entanglement Renormalization and Holography In AdS/CFT, the emergent dimension of space can be regarded as a renomalization scale. Entanglement renorm., run backwards, prepares a region of length L in circuit depth O(log L). View the bulk space as a prescription for building up the boundary state (Swingle, 2009). Think of a growing tensor network as a model of an evolving bulk spatial slice. The slice expands, corresponding to adding additional layers to the network.

49 Niels Bohr to Wolfgang Pauli, 1958: We are all agreed that your theory is crazy. The question that divides us is whether it is crazy enough to have a chance of being correct. All the proposed resolutions of the black hole firewall puzzle are crazy, but are any of them crazy enough? Bohr probably said something like this on multiple ocassions. Quoted by Freeman Dyson, Scientific American, September Another eyewitness account: Jeremy Bernstein, The life it brings, 1987, p. 139

50 Frontiers of Physics short distance long distance complexity Higgs boson Neutrino masses Supersymmetry Quantum gravity String theory Large scale structure Cosmic microwave background Dark matter Dark energy More is different Many-body entanglement Phases of quantum matter Quantum computing

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52 Freeman Dyson on discussion with Bohr in San Diego, It was his habit to walk and talk. All his life he had been walking and talking, usually with a single listener who could concentrate his full attention upon Bohr s convoluted sentences and indistinct voice. That evening he wanted to talk about the future of atomic energy. He signaled for me to come with him, and we walked together up and down the beach. I was delighted to be so honored I clutched at every word as best I could. But Bohr s voice was at the best of times barely audible. There on the beach, each time he came to a particularly crucial point of his confrontations with Churchill and Roosevelt, his voice seemed to sink lower and lower until it was utterly lost in the ebb and flow of the waves. Disturbing the Universe, 1979, p. 102.

53 Niels Theoretical Physicist Tweets Stop telling God what to do! Niels If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.