Taming Quantum Physics

1
Taming Quantum Physics

• What has to be accepted, and what can be
  understood ?

2
A Healthy Look on Physics
Learn to distinguish

• The description of a system
• State properties that the system has, with
  certainty, at a given time
• from its evolution
• How it will evolve in time, because of
  interactions with other systems.

• The physical reality
• A particle, a device, an evolution under some
  interaction etc.
• from the mathematical objects that represent it
• Vectors, operators,

3
Postulates of Quantum Physics

• Description
• The system is described by a Hilbert space
• i.e. the space of states is a vector space.

• Evolution
• In the absence of measurement, the evolution is
  reversible.

These are the only postulates. We accept them
because they work very well ?
4
Notions

• Part 1

5
Logical Flow

• Description
• Hilbert space

• Evolution
• Reversible.

Schrödinger equation

• Gleasons theorem
• Pure states are represented by subspaces of
  dimension 1, i.e. by
• either the projector
• or a normalized vector, defined up to a global
  phase.
• The probability of finding state f in a
  measurement, having state y, is p(fy )?fy?2
  (historically called Borns rule)
• Proof very technical, so accept this as a
  postulate too ?

• All the rest
• ideal measurements
• physical quantities
• compatible quantities
• mixed states
• composite systems

6
Ideal Measurements
Rule
The possible outcomes of an ideal measurements
are associated to a basis of the Hilbert space.
Reasons

• Denote fk? the set of possible outcomes
• In an ideal measurement, the outcomes should be
  distinguishable, i.e. p(fjfk ) djk ? the
  fk? must be orthogonal
• An ideal measurement should always give an
  answer ? the fk? must span the whole Hilbert
  space.

Remarks

• An ideal measurement gives the same outcome if
  immediately repeated.
• Non-ideal measurements CAN be described in
  quantum physics as noisy versions of ideal ones
  fortunately, because real measurements are
  never ideal.

7
Physical Quantities
Rule
A physical quantity is associated to a Hermitian
operator.
Reasons

• Denote fk? the set of possible outcomes of a
  measurement
• if fk? corresponds to the value ak of a
  physical quantity A, then this quantity is
  represented by ? k ak fk ?? fk .
• For convenience, one often chooses the
  eigenvalues ak to be real.

Consequences

• The existence of non-compatible quantities is
  represented by the fact that two operators in
  general do not commute ? dont have the same
  eigenvectors (see next).
• Borns rule implies the rule for average values
  of a physical quantity in a repeated measurement
  on identically prepared systems ?A??fAf?

8
Compatible Quantities
Rule
Two physical quantities are compatible if and
only if the corresponding operators commute.
Reasons

• Let ak, resp. bj, be the set of possible
  values of quantity A, resp. B. The quantities A
  and B are called compatible if there exists a
  family of states such that both the value of A
  and of B are certain, i.e. ak,bj?.
• Mathematically, this means that A and B must
  have a common set of eigenvectors, and this
  happens if and only if A,B0.

Remarks

• Compatibility can be checked
• in a serial experiment (measure A, then B, the
  again A)
• in a parallel experiment (send N particles to
  A, N to B, then make statistics) this is the
  case envisaged in the uncertainty relations.

9
Mixed states
Rule
A mixed state is represented by a convex
combination of projectors (density matrix)
Reasons

• Consider the mixture state y1? with probability
  p1, state y2? with p2
• On the one hand, on such state, ?A? p1 ?A?1
  p2 ?A?2
• On the other hand, ?A?kTr(APk).
• Combining these, one has , ?A?kTr(Ar) with
  rp1P1 p2P2.

Remarks

• Example of mixed states
• N particles in thermal equilibrium mixture of
  states with definite energy, with probabilities
  given by Boltzmann rule.
• Output of a non-ideal (noisy) measurement.
• The state of a sub-system in an entangled state
  (see next).

10
Composite systems
Rule
The Hilbert space that describes a composite
system is the tensor product of the Hilbert
spaces of the sub-systems
Reasons

• We must describe classical states like system
  1 is in state y, system 2 in state f.
• The whole space of states must be a Hilbert
  space.

Consequences

• Among the possible states, some (actually, most
  of them) are not of the classical form these
  are called entangled.
• Entanglement describes a situation in which
  global properties are sharply defined, but
  individual properties are not.
• Entangled states give rise to correlations that
  cannot be explained by either classical mechanism
  (communication or pre-established strategy)

11
Evolution (1)
Rule
Evolution in time is represented by the action of
a unitary operator on the initial state
y(t)?U(t)y(0)?.
Reasons

• We have postulated that evolution is reversible,
  i.e. no information is created or destroyed by
  it. On a Hilbert space, this means that scalar
  products must be preserved.
• The full proof is hard, but the idea is that
  ?y1y2? is somehow the distinguishability
  between the two states. Then
• Suppose that a reversible evolution y??y? is
  such that ?y1y2? ??y1y2? and leads to better
  distinguishability
• Then one could do the following procedure
  Evolve Distinguish Evolve backwards. The
  evolution is reversible, so you go back to the
  initial state.
• But this means that there is a procedure to
  distinguish the two states better than what their
  scalar product allows, leading to a contradiction.

12
Evolution (2)
Rule
Alternatively, evolution can be found by solving
the Schrödinger equation i?dy/dtH(t)y.
Reasons

• Unitarity reads y(tt)U(t)y(t).
• U(t) represents a translation in time physics
  and group theory require therefore U(t)
  e-iH(t)t/? where H(t) is the generator of
  translations in time (Hamiltonian, i.e. the
  operator associated to energy) and ? is a
  constant that turns out to be Plancks constant.
• From this form, one easily derives the
  differential equation above.

Consequences

• In the case H(t)H, the eigenstates of H (i.e.
  the states of well-defined energy) are the
  stationary states.

13
Test your understanding!

• Part 2

14
FAQ 1
The probabilities given by Borns rule describe
the weird randomness of quantum physics. True or
False?

• True

• False

• In classical physics, if you have a pure state,
  all different states can be distinguished. So,
  only in quantum physics it can happen that, given
  a pure state, a different pure state is a
  possible outcome of a measurement.

• Borns rule shows that there are only two pure
  states for a spin (UP and DOWN in the direction I
  have chosen) and all the other states are
  statistical mixtures of those.

There is no statistical mixture in a pure quantum
state for instance, if I measure the spin in the
direction defined by the state, the outcome is
CERTAIN and not probabilistic. Borns rule
represents the purely quantum fact that not all
the properties can be sharply defined.
15
FAQ 2
In a d-level system (i.e., dimension of Hilbert
space d) there are only d states. For instance,
a spin ½ can only be UP or DOWN. True or False?

• True

• False

• On each spin, I can only measure along one
  direction, and the result will be either UP or
  DOWN. Therefore there is no more information than
  this.

• A spin can be prepared in an arbitrary
  direction, therefore there are infinitely many
  different states (although only pairwise
  distinguishable).

The objection here stems from a correct
observation however, by preparing MANY spins in
the same state (the same output of a
Stern-Gerlach magnet) and by measuring average
values of all Pauli matrices, one CAN reconstruct
the state. Therefore, the whole information IS
there only, it cannot be learned in a single
shot.
16
FAQ 3
If an operator is not hermitian, it cannot
represent anything that can be measured. True or
False?

• True

• False

• Only hermitian operators represent physical
  quantities and after all, one measures only real
  numbers.

• A measurement is defined by a basis if I like,
  I can associate complex numbers, or colors, or
  names to each outcome.

If you think that only real numbers can be
measured think again ? Nothing prevents indeed
from describing outcomes using colors I got
black with probability ½, red with probability ¼,
and green with probability ¼. Some DERIVED
statistical quantities (e.g. average values)
cannot be defined, but all the information IS
there. Of course, most of the usual physical
quantities (position, momentum, energy, magnetic
moment) are described by real scalars or
vectors, and in this sense, it is natural to
represent them with hermitian operators.
17
FAQ 4
The uncertainty relations represent the lack of
precision of measurement devices. True or False?

• True

• False

• The uncertainty relations derive from the
  intrinsic quantum randomness. Only close-to-ideal
  measurements can reveal the uncertainty
  relations.

• Heisenberg himself explained them as follows
  quantum objects are so small, that when a device
  interacts with them, their state is modified.

The worst possible measurement is done if one
forgets to turn on the device in this case, the
needle will always point to the same result ?
DADB0 for any state, so no uncertainty in the
statistics! More generally, rough measurements
may fail to satisfy the uncertainty relations. As
for Heisenberg well, he got the maths correctly,
but when his result became popular and he felt
forced to explain it, he invented a wrong
explanation. It happens, and it shows how quantum
physics can betray even the experts ?
18
FAQ 5
Mixed states are a typically quantum notion.
True or False?

• True

• False

• Statistical ensembles of states are also
  possible in classical physics.

• There is no statistical element in classical
  physics.

There can be a statistical element in classical
physics go back to your thermodynamics! Of
course, quantum (pure) states are different from
classical (pure) states therefore, in general, a
mixture of quantum states will be different from
a mixture of classical states. But the
randomness involved in the mixture is the usual
classical randomness, due to our ignorance, not
the intrinsic quantum randomness.
19
FAQ 6
Correlations, that are neither due to
pre-established agreement nor to communication,
exist. True or False?

• True

• False

• This is the current understanding, based on the
  quantum formalism. It does not mean that such
  weird correlations actually exist.

• Correlations between space-like separated events
  that violate Bells inequality have been
  observed.

Both criteria (space-like separation ? no
communication violation of Bells inequality ?
no pre-established agreement) are independent of
the quantum formalism. Quantum physics lead to
predict that such counter-intuitive phenomena
should exist now, we have seen them. Therefore,
any possible future theory will have to predict
this fact (just like general relativity must
still predict that apples fall).