In the beginning there was ���

Many-body Quantum Mechanics, ���which exploded.

Honours QM

Joe Hope 61252780, room 103

joseph.hope @ anu.edu.au

Entanglement and mixtures

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12

↑,g + i↓,g( ) unitary evolution⎯ → ⎯ ⎯ ⎯ ⎯ 12

↑,g + i↓,e( )

1. What is the density matrix?

2. Find the spin system’s reduced density matrix.

Consider the following ‘measurement’ of a spin system with a two-level atom

Review: Wave evolution is linear

Measurement device System

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A ⊗ 0 → A ⊗ Device registers AB ⊗ 0 → B ⊗ Device registers B

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A + i B( ) ⊗ 0 → A ⊗ Device registers A

+i B ⊗ Device registers B

Wavefunction collapse is random

We need density matrices just to write down what we’d expect from wavefunction collapse:

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12

1 −ii 1⎛ ⎝ ⎜

⎞ ⎠ ⎟ ⊗

1 00 0⎛ ⎝ ⎜

⎞ ⎠ ⎟

unitary evolution⎯ → ⎯ ⎯ ⎯ ⎯

12 0 0 −i

20 0 0 00 0 0 0i2 0 0 1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

12

A + i B( )⊗ 0 → 12

A ⊗ Device registers A+i B ⊗ Device registers B

⎛

⎝⎜⎞

⎠⎟

What should have happened?

We should have ended with a 50% mixture of

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121 −ii 1⎛ ⎝ ⎜

⎞ ⎠ ⎟ ⊗

1 00 0⎛ ⎝ ⎜

⎞ ⎠ ⎟

measurement⎯ → ⎯ ⎯ ⎯

12 0 0 00 0 0 00 0 0 00 0 0 1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

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A ⊗ Device registers A B ⊗ Device registers Band

Decoherence

1. What does that mean with regards to observations?

2. Does this explain wavefunction collapse?

If we ignore a system that gets entangled with one that we are interested in, it becomes a mixture rather than a superposition.

Summary

•  Without classical objects, you can’t make something work like the measurement postulate without simply causing entanglement instead

•  Any entanglement with a large system causes decoherence identical to measurement postulate

•  No subsystem in an entanglement chain can ever detect the presence of the others branches

Review: Hidden variables

J. Bell (1964): Quantum mechanics is incompatible with any local hidden variable theory.

2. “Prior agreement”: agreed result for any measurement? X Each photon must know about both measurements.

Is reality really random or are there hidden variables?

Random polarisation, but left always same as right

Running out of good options

We are back to the random wavefunction collapse. Random (nonlocal) wavefunction collapse

- appears to violate relativity - requires true action at a distance - doesn’t obey the Schrödinger equation

…and we still haven’t defined “measurement” (which is when this collapse is supposed to occur)

Chains

Device 1 System

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A + i B( ) ⊗ 01 ⊗ 02 →

A ⊗ Device 1 registers A ⊗ Device 2 registers A+i B ⊗ Device 1 registers B ⊗ Device 2 registers B

Device 2

Entanglement in measurement

Photon transmitted + Photon absorbed

A measurement device is made of waves

detector

Detector changes state Detector unexcited

Computer goes “ping” Computer doesn’t +

Schrödinger’s cat

Suppose the computer is a killer robot in a box with a cat.

A+D

1.  What happens if you open the box? 2.  How could you determine if it were A+D or A-D?

Reduction of the wavefunction

Measurements change AND into OR through “wave function collapse”.

Isn’t the cat a “measuring device”?

A+D

Continual collapse

Maybe all wavefunctions collapse all the time... –  Macroscopic entanglement ≈ 1023 particles –  Any collapse and the whole entanglement collapses

Spontaneously… Not too fast Not too slow

Continual collapse

A measurement device is made of lots of waves

Photon transmitted + Photon reflected

detector

Detector changes state Detector unexcited

Computer goes “ping” Computer doesn’t + OR

Consciousness?

Some people claim that “consciousness” collapses wavefunctions:

no ping

ping

+ So what is “consciousness”? If it really makes the world behave differently - we could make a “consciousness detector”.

Something special further down the chain

Maybe the only consciousness that collapses wavefunctions is God John:

no ping

ping

+

Everett’s idea

•  Can’t interact with them •  Not exactly “other” universes

•  No collapse at all

•  No “quantum” and “classical” worlds

•  No need to define measurement or consciousness

•  Compatible with relativity

"The first thing to realise about parallel universes ... is that they are not parallel. It is also important to realise that they are not, strictly speaking, universes either, but it is easiest if you try and realise that a little later, after you've realised that everything you've realised up to that moment is not true." - Douglas Adams