Quantum entanglement in the 21st century

John PreskillThe 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

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.

This

Page

Blank

This

Page

Blank

This

Page

Blank

This

Page

Blank

This

Page

Blank

….….

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

Classically Easy

Quantumly Hard

Quantumly Easy

Problems

Classically Easy

Quantumly Hard

Quantumly Easy

Problems

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?

Algorithms

Spacetime

Error Correction

Quantum entanglement in the 21st century

Matter

0

100

200

300

400

500

600

700

2005 2006 2007 2008 2009 2010 2011 2012 2013

quant-ph

arXiv papers with “entanglement” in the title

0

50

100

150

200

250

300

350

400

450

2005 2006 2007 2008 2009 2010 2011 2012 2013

cond-mat hep-th gr-qc

arXiv papers with “entanglement” in the title

Classical correlations are polygamous

Betty

Adam Charlie

Quantum correlations are monogamous

unentangledfully

entangled

Betty

Adam Charlie

Quantum correlations are monogamous

fully

entangledunentangled

Betty

Adam Charlie

Monogamy is frustrating!

unentangledfully

entangled

cryptography

quantum matter

black holes

Betty

Adam Charlie

event horizon

singularity

outgoing radiation

collapsing body

Information Puzzle: Is a black hole a quantum cloner?

“time slice”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.

time(outsidehorizon)

event horizon

singularity

time(outsidehorizon)

outgoing radiation

collapsing body

“Black hole complementarity”

“time slice”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.

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).

in out≠ ⊗H H H

“No-cloning” lower bound on the information retention time

Let’s demand that verifiable cloningdoes 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

Alice

Bob

( )(Alice)

proper Planckexp / (1)S S S

r t r O rτ ≈ −∆ ≤ ×

and therefore

( )logS S S

t O r r∆ ≥

(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 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).

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).

time

V B

maximalentanglement

Alice’squbits

Bob’s decoder

blackhole

black hole

rad

iatio

n

E R B'

refe

ren

ce

syste

m

N

( ) max 1Haar

2( ) 2

2

k

B NB B N B c

k c

NdV V

Rρ ρ ρ′ ′ −

+− ⊗ ≤ = =∫

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.

event horizon

singularity

time(outsidehorizon)

outgoing radiation

Complementarity Challenged

Betty Adam

Robert

(1) For an old black hole, recently

emitted radiation (B) is highly

entangled with radiation

emitted earlier (R) by the time it

reaches Robert.

(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!

B A

R

Alice

black hole

Bob

What’s inside a black hole?

A. An unlimited amount of stuff.

singularity

time

collapsing matter

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?

-- What about AdS/CFT duality?

B. Nothing at all.

singularity

time

collapsing matter

“It is time to constrain and construct the dynamics of firewalls.” – Raphael Bousso

But …

-- “Curtains for the equivalence principle?” (Braunstein, 2009)

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).

But …

-- R could be far, far away from the black hole.

A black hole wormhole-connected to the Hawking radiation it has emitted (Maldacena and Susskind).

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

bulk

boundary

minimal

bulk

surface

To compute entropy of region A in

the boundary field theory, find

minimal area of the bulk surface with

the same boundary:

Ryu and Takayanagi, 2006

Recover, for example, in 1+1

dimensional conformal field theory:

1min area(m

4) )(

m A

NG

S A ∂ =∂= +�

l( ( )) og( /3

)c

S A LL a= +�

Strong subadditivity from holography

boundary

Headrick and Takayanagi, 2007

minimal

bulk

surface

bulk bulk

boundary

S(A) + S(B) ≥ S(A»B) + S(A…B)

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

/2i

E

i

eβ−∑ /2

| |iE

i i

i

e E Eβ− ⟩⊗ ⟩∑

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).

Alice Bob

singularity

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.

Love in a wormhole throat

time

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).

But …

-- R could be far, far away from the black hole.

A black hole wormhole-connected to the Hawking radiation it has emitted (Maldacena and Susskind).

S

=

singularity

M

outin

M

out

singularity

time time

Horowitz-Maldacena Proposal (2003)

Quantum information escapes from a black hole via postselectedteleportation. 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).

=

M

out

singularity

time

Horowitz-Maldacena Proposal (2003)

Quantum information escapes from a black hole via postselectedteleportation. 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).

S

1M

1out

1in

2M

2in

2out

1M

0 |⟨

U

| 0⟩

0 |⟨ 0 |⟨

2M in

out1N

Generic final state

Considering dividing the infalling matter into a relatively small subsystem M1

(matter that collapses quickly) and a larger subsystem M2 (which collapses slowly).

If M2 is initially in a fixed (vacuum) state, then a generic final state boundary condition, will project onto a very nearly maximally entangled state of M1 and the outgoing radiation; hence the black hole S-matrix will be very nearly unitary.

L1 norm deviation from unitarity: ( )1

1/2

3/2

in

| |exp / 2 ( )

| |

M

BHS O m

≈ − +

H

H

Such a small violation of unitarity may be an artifact of the semiclassicalframework used in the analysis, as nonperturbative quantum gravity corrections of that order are expected.

Entanglement Renormalization and Holography

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.

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).

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

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

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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@einstein Stop telling God what to do!

Niels Bohr @bohr

If quantum mechanics hasn't profoundly

shocked you, you haven't understood it yet.