Quantum Entanglement in the 21St Century

Quantum entanglement in the 21 st century

John Preskill The Quantum Century: 100 Years of the Bohr Atom 3 October 2013 My well-worn copy, bought in 1966 when I was 13. George Gamow, recalling Bohr’s Theoretical Physics Institute 1928-31:

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. George Gamow, recalling Bohr’s Theoretical Physics Institute 1928-31:

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. 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. 73-76. Classical Correlations Classical Correlations Quantum Correlations

Aren’t boxes like soxes? 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. “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.” “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.” Quantum entanglement

…. This This This This This …. Page Page Page Page Page Blank Blank Blank Blank 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. To describe 300 qubits, we would need more numbers than the number of atoms in the visible universe! 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? Peter Shor Problems

Quantumly Hard

Quantumly Easy

Classically Easy Problems

Quantumly Hard

Quantumly Easy

Classically Easy

What’s in here? 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? Quantum entanglement in the 21 st century

Algorithms Error Correction

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0 2005 2006 2007 2008 2009 2010 2011 2012 2013 cond-mat hep-th gr-qc

Classical correlations are polygamous

Betty

Adam Charlie Quantum correlations are monogamous

Betty

fully entangled unentangled

Adam Charlie Quantum correlations are monogamous

Betty

fully unentangled entangled

Adam Charlie Monogamy is frustrating !

Betty

fully entangled unentangled

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

Perhaps the lesson is that, for “time slice” 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 outgoing subsystems of a composite system. radiation H H H ≠in ⊗ out event Rather, the inside and outside are horizon merely complementary descriptions of the same system. Which description is appropriate depends on whether the observer enters the time black hole or stays outside (outside (Susskind, 1993). horizon) collapsing body “No-cloning” lower bound on the information retention time

Let’s demand that verifiable cloning singularity 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: Bob (Alice) r trO r τ proper ≈Sexp( −∆ S / S ) ≤ (1) × Planck and therefore Alice ∆tS ≥ Or( Slog r S )

(where rS 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. “Black holes as mirrors” Alice throws k qubits (maximally Bob’s decoder entangled with reference ER B' N system N) into an “old” black hole. As radiation R escapes, black reference reference the correlation of N with B′ hole system decays. Eventually, N is nearly uncorrelated with B′ and nearly V B maximally entangled with a subsystem of ER --- at that radiation black Alice’s stage, Bob can decode Alice’s hole qubits quantum message with high fidelity (Hayden-Preskill, 2007). maximal entanglement time N k dVBNBBNB′ V ′ 2 − c () ρ ρ ρ ( )−⊗max ≤=k c = 2 ∫Haar 1 R 2 + Bob can decode with high fidelity after receiving only k+c qubits of Hawking radiation, where c is a constant, if the mixing unitary VB is Haar random, or

even if it is a typical unitary realized by a small quantum circuit (depth ~log rs). 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. Complementarity Challenged singularity

(1) For an old black hole, recently emitted radiation (B) is highly entangled with radiation Robert emitted earlier (R) by the time it R reaches Robert.

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

What’s inside a black hole?

black hole

Bob

Alice 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 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 C. A huge but finite amount of stuff, which is also outside the black hole.

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). A black hole wormhole-connected to the Hawking radiation it has emitted But … (Maldacena and Susskind). -- R could be far, far away from the black hole. 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. Holographic entanglement entropy

minimal To compute entropy of region A in bulk bulk the boundary field theory, find surface minimal area of the bulk surface with the same boundary: S A 1 ( ))= min∂m =∂ A area(m + 4GN Ryu and Takayanagi, 2006

Recover, for example, in 1+1 dimensional conformal field theory: boundary c SAL( (L ))= l og( / a) + 3 Strong subadditivity from holography » … minimal S(A) + S(B) ≥ S(A B) + S(A B) bulk Headrick and Takayanagi, 2007 surface

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 Building spacetime from quantum entanglement

−β Ei /2 E e e−β i /2 E E ∑ ∑ |i〉⊗ | i 〉 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).

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. C. A huge but finite amount of stuff, which is also outside the black hole.

singularity singularity S in out out

time time M M

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). Horowitz-Maldacena Proposal (2003)

out 2 out 1 singularity

S in 2 out

in 1 =

time M M 1 2 M

N 1 out Considering dividing the infalling matter 〈 〈 〈 0 | 0 | 0 | into a relatively small subsystem M 1 (matter that collapses quickly) and a larger

subsystem M2 (which collapses slowly).

If M 2 is initially in a fixed (vacuum) state, U then a generic final state boundary condition, will project onto a very nearly M M 1 2 in maximally entangled state of M 1 and the outgoing radiation; hence the black hole S- | 0 〉 matrix will be very nearly unitary.

1/2 |HM |  L1 norm deviation from unitarity: 1 ≈exp −S /2 + O ( m 3/2 ) H  ()BH |in |  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. 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. 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 1958.

Another eyewitness account: Jeremy Bernstein, The life it brings , 1987, p. 139 Frontiers of Physics short distance long distance complexity

Higgs boson Large scale structure “More is different”

Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory

Freeman Dyson on discussion with Bohr in San Diego, 1959.

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. Niels Bohr @bohr Theoretical Physicist

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