The Quantum Conspiracy:What Popularizers of QMDon’t Want You to Know

Ron Garret6 January 2011

Disclaimers●The title of this talk is intended as ironic

humor. There is no conspiracy (as far as I know :-)

● IANAPhysicist●This talk is about a way to think about

QM that hasn’t gotten much attention…

“No one understands quantum mechanics.”– Richard Feynman

What does it mean to “measure” something?

Measurements are consistent across space and time

T0: It’sGreen!

T1: Yep,it’s

Green!

?

“The most incomprehensible thing about the universe is that it is comprehensible”

— Albert Einstein

A deep mystery

●It could be that measurements are consistent across space and time because there is an underlying (meta-)physical reality “out there” which is being accurately reflected

●But it turns out we can demonstrate that this is not so…

Road map

●Step 1: Review the usual QM story●Step 2: Show how it leads to a

contradiction●Step 3: Do some math and show how

that resolves the contradiction●Step 4: Tell a new story based on the

math●Step 5: Profit!

Quantum mystery #1The two-slit experiment

Two-slit experiment results

Waves

Particles

This is not intractably weird (yet)

●Light (and electrons) might be particles that are moved around by an underlying wave

●Randomness might be due to “hidden variables”

●But we can eliminate this possibility…

Adding detectors to the slits

No detectors ==> interference

Detectors ==> no interference

Wave-particle duality

●Any modification to the experiment that allows us to determine — even in principle —which slit the particle went through destroys the interference

●Conclusion: something must be “at both slits at once” to produce interference

●This holds for any particle and any “measurement” (and any “two-slit” or split/combine experiment)

(Aside: wave-particle duality is an inherent aspect (indeed part of the definition) of waves and particles, not QM)

This is still not intractably weird

●Maybe measurement “does something” to the system to make it stop behaving like a wave and start behaving like a particle

●Maybe the wave “collapses” and “becomes” a particle (Copenhagen interpretation)

●But… how – and, more importantly, when – does “collapse” happen?

Quantum mystery #2The “Quantum Eraser”

Split Combine

Reflect

Reflect 0

1

Particle

Source

Interference

Quantum mystery #2The “Quantum Eraser”

Split Combine

Particle

Source

“Measure” 0.

5

0.5

Interference“destroyed”

Measure=rotate 90˚

Quantum mystery #2The “Quantum Eraser”

Split Combine

Particle

Source

“Measure”

0

1

“Erase”

Measure=rotate 90˚ Erase=filter at 45˚

“Erase”

Interference“restored”

Shroedinger’s Cat●When (and where) does “collapse”

happen?●At the “measurement” site?●At the detector?●In the mind? (Whose mind?)

Quantum mystery #3: Entanglement

Quantum Entanglement

UV laser& Down-Converte

r

Split

Split

LU RU

LD RD

LU/RD and LD/RU are perfectly correlated(because of conservation laws)

“Spooky action at a distance”

●Particle isn’t “really” at either detector until it is actually “measured” (whatever that means)

●“When an aspect of one photon’s quantum state is measured, the other photon changes in response, even when the two photons are separated by large distances.” (Wired, June 2010)

Now it’s intractably weird!

●Instantaneous effects are supposed to be impossible!

●Randomness precludes transmitting information using entanglement

●Or does it?

Road map

●Step 1: Review the usual QM story●Step 2: Show how it leads to a

contradiction

Taking stock●A split/combine experiment produces

interference●Any which-way measurement destroys

interference●Some which-way “proto measurements” can

be erased, restoring interference●Measurements on entangled particles are

perfectly (anti)correlated

Taking stock

What they don’t want you to know:

All of these things cannot possibly be true!

●A split/combine experiment produces interference

●Any which-way measurement destroys interference

●Some which-way “proto measurements” can be erased

●Measurements on entangled particles are perfectly (anti)correlated

The EPRG* Paradox

*Einstein-Podolsky-Rosen-Garret

The EPRG* Paradox

If we “measure” on the left, do we destroy interference on the right?

The EPRG Paradox

●If the answer is “yes” then we have FTL communications

●But if the answer is “no” then we know the position of the particle but we have interference nonetheless, which violates QM

One last possibility…

●Maybe there was no interference to begin with!

●Maybe entanglement “counts” as a “proto-measurement” that destroys interference

●But then we can do FTL communications by creating interference with a quantum eraser!

●Conclusion: either FTL communications is possible, or something in this story is wrong

Road map

●Step 1: Review the usual QM story●Step 2: Show how it leads to a

contradiction●Step 3: Do some math and show how

that resolves the contradiction

Math (don’t panic)

● �(x,t) is the “quantum wave function”○ Complex-valued function of space and time○ Evolves according to the Schroedinger wave equation

● |�(x,t)|2 is the probability of measuring a particle at position X at time T

Things to note about the math

●Distinguishes between amplitudes (complex numbers) and probabilities (real numbers)○ Particles can interfere because complex numbers

with modulus greater than zero can add to zero.●Continuous, time-symmetric, fully

deterministic (and hence reversible) dynamics●No randomness, no “collapse”. Going from

amplitudes to probabilities has no physical justification. It’s purely a hack. (But it works!)

Two-slit math

State of the photon without “measurement”:

(�U + �L)/√2

(Note the √2. It will be important later.)

Resulting probability (|�|2):[|�U |2 + |�L|

2 + (�U*�L + �L

*�U)]/2Interference term

Two slits with detectors●Probability amplitude:

(�U |DU> + �L |DL>)/√2(|DU> is the amplitude of the detector indicating

a particle at the upper slit)

Two slits with detectors●Probability amplitude:

(�U |DU> + �L |DL>)/√2(|DU> is the amplitude of the detector indicating

a particle at the upper slit)

●Resulting probability:[|�U |2 + |�L|

2 +

(�U*�L <DU|DL> + �L

*�U <DL|DU>)]/2

Two slits with detectors●Probability amplitude:

(�U |DU> + �L |DL>)/√2(|DU> is the amplitude of the detector indicating

a particle at the upper slit)

●Resulting probability:[|�U |2 + |�L|

2 +

(�U*�L <DU|DL> + �L

*�U <DL|DU>)]/2Interference term (!)

Measurement and interference

● <DU|DL> is the amplitude of the detector switching

spontaneously from the U state to the L state● If the detector is working properly, this amplitude is 0● Then the resulting wave function is:

(|�U |2 + |�L|2)/2

● Note: no interference term!

“Measurement” is a continuum!

Entangled particles

●Wave function:(|��> + |��>)/√2

●Equivalent to:(|�>|�> + |�>|�>)/√2

(|LU>|RD> + |LD>|RU>)/√2(�LU |RD> + �LD |RU>)/√2

…which should look familiar.

Entanglement and measurement are the same

phenomenon!●Wave function of entangled particles is exactly the same as a “measured” particle

●They are in fact the same physical phenomenon (more on this in a moment)

●There is no interference in the EPRG experiment

●But… can we create interference with a quantum eraser?

Quantum eraser revisited

Split Combine

Particle

Source

“Measure”

0

1

“Erase”

Measure=rotate 90˚ Erase=filter at 45˚

“Erase”

Interference“restored”

Quantum eraser math

●Wave function after “measurement” (but before “erasure”):

(|U>|H> + |L>|V>)/√2●Wave function after “erasure”:

(|U> + |L>)(|H> + |V>)/2√2

|V>

|H>

|H> + |V> means polarized at 45˚|H> + |V>

Quantum eraser math

●Before “erasure”: no interference●After “erasure”: interference… but●Remember that √2 term? It’s there to

make the total probability come out to 1.●But the total probability isn’t 1, it’s 1/2!●Either we’ve made a mistake, or half

our photons are missing

Quantum eraser math●Half of our photons have gone missing!●They were filtered out●Filtered photons have a different wave

function:(|U> + |L>)(|H> - |V>)/2√2

|V>

|H>

|H> + |V>

|H> - |V>

So much for our Nobel prize

●Photons that pass through the filter display interference fringes

●Photons that don’t pass through the filter also display interference “anti-fringes”

●Sum together to produce “non-interference”●So quantum erasers don’t “erase” anything,

and they don’t “produce” interference, they just “filter out” interference that was already there

“Filtering out” interference in an EPR experiment

UDUDD…

=“U” photons+“D” photons

“Select”

Road map

●Step 1: Review the usual QM story●Step 2: Show how it leads to a

contradiction●Step 3: Do some math and show how

that resolves the contradiction●Step 4: Tell a new story based on the

math

Interpretations of QM

●Copenhagen (scientifically untenable)●Relative-state (“Multiple worlds”,

“Decoherence”)○ Scientifically tenable but intuitively troublesome

●Transactional (Cramer)○ Physically real waves moving backwards in time

(predicted by Maxwell’s equations)●Quantum information theory (“Zero-worlds”)

○ Extension of classical information theory with complex numbers

Classical Information Theory

●Shannon entropy of system A:

H(A) = -�P(a) log P(a)

● P(a) is probability that A is in state a● H(A) is a measure of the “randomness” of system A

○ When system has equal probability of being in one of N states, H(A) is log(N)

○ When N is 1 (system is definitely in a single state) H(A) = 0

Classical Information theory●Joint entropy of multiple systems:

H(AB) = -�p(ab) log p(ab)●Conditional entropy:

H(A|B) = -�p(a|b) log p(a|b)● Information entropy:

I(A:B) = I(B:A) = H(A) – H(A|B)= H(A) + H(B) – H(AB)

= H(AB) – H(A|B) – H(B|A)●I(A:B) is the amount of information about A

contained in B (0 <= I(A:B) <= 1)

Entropies of classical systems

Quantum information theory

●Von Neuman entropy:S(A) = -TrA(�A log �A)

● �A is the quantum density matrix

● TrA is a “trace” operator

●Details beyond the scope of this presentation●Main point: complex numbers =>

Information entropy is no longer restricted to the range [0,1]

Entropy diagram of an entangled pair of particles

-1

2 -1

● Information entropy > 1● Particles are better than perfectly

correlated● Total entropy is zero ==> No

randomness

Measurement●To describe a measurement we need at

least three mutually entangled particles○The one being measured○At least two more to describe the

measurement apparatus

Entropy diagram of three mutually entangled particles

Entropy diagram of three mutually entangled particles

Entropy diagram of 1023 particles

Reversibility●Quantum measurements are reversible, but

only by “undoing” all of the associated entanglements

●This can only be done by returning the entangled particles to close physical proximity

●To reverse a macroscopic measurement we would have to “undo” 1023 entanglements

●Possible in principle, not in practice

Philosophical implications●The classical universe is not “real”

○ There is no (one) classical universe○ There is only the quantum universe (which can be

viewed as an infinite collection of classical universes)

●This is not (quite) as strange as it seems○ Even classical reality is not as we perceive it○ “We” are not made of atoms, we are made of

(classical) bits (“Correlations without correlata” -- David Mermin)

Some pithy quotes●... the particle-like behavior of quantum

systems is an illusion created by the incomplete observation of a quantum (entangled) system with a macroscopic number of degrees of freedom.

● ... randomness is not an essential cornerstone of quantum measurement but rather an illusion created by it.

-- Nicholas Cerf and Chris Adami

Take-home message

●“The most incomprehensible thing about the universe is that it is comprehensible”

-- Albert Einstein●QIT explains why the universe is

comprehensible!●“Spooky action at a distance” is no more

(and no less) mysterious than “spooky action across time.” Both are produced by the same physical mechanism.

Two monks were arguing about a flag. One said, "The flag is moving." The other said, "The wind is moving." The sixth patriarch happened to be passing by. He told them, "Not the wind, not the flag. Mind is moving.”

-- Mumon, “The Gateless Gate”

Backup slides

Polarization and Picket Fences

Light is a wave●It can be polarized●It travels at the speed of

electromagnetic waves (because it is an electromagnetic wave)

●It can produce interference

The photoelectric effect

●Shining light on matter produces electrons

Two weird features of the photoelectric effect

●The number of electrons produced is proportional to the intensity of the light

●The energy of the electrons produced is (inversely) proportional to the wavelength of the light

●This is not what one would expect if light is a wave

Diffraction

DefiniteVelocity

IndefinitePosition

DefinitePosition

IndefiniteVelocity

Time/frequency duality