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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
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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
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600
700
2005 2006 2007 2008 2009 2010 2011 2012 2013
quant-ph
arXiv papers with “entanglement” in the title
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50
100
150
200
250
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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
Tweets
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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.