Download Quantum mechanics as a representation of classical conditional

Quantum mechanics as a representation
of classical conditional probabilities
Gábor Hofer-Szabó
Research Centre for the Humanities, Budapest
– p. 1
Project
Questions:
1. How to represent conditional probabilities in a
noncommutative setting?
2. How to reconstruct the quantum state by measuring
simply conditional probabilities?
3. How to represent observables and states such that the
Born rule yields us the conditional probabilities?
– p. 2
Project
Outline:
I. The formalism of QM
II. Empirical foundations
III. Representing measurements by operators
IV. How to assign operators to measurements?
– p. 3
I. The formalism of QM
– p. 4
The formalism of QM
General scheme:
Operator assignment:
System
Measurement: a
Outcomes: Ai
State: s
−→ H: Hilbert space
−→ Oa : self-adjoint operator
−→ Pai : projections
−→ Ws : density operator
Born rule:
ps (Ai |a) =: pia = Tr(Ws Pai )
– p. 5
The formalism of QM
Yes-no questions:
Operator assignment:
System −→ C2
Measurement: a −→ Oa = aσ
a ∈ R3 , |a| = 1
σ = (σx , σy , σz )
Outcomes: A± −→ Pa± = 21 (1 ± aσ)
State: s −→ Ws = 12 (1 + sσ) s ∈ R3 , |s| 6 1
|s| = 1: pure, |s| < 1: mixed state
Born rule:
±
=
Tr
(W
P
ps (A± |a) =: p±
s a )
a
– p. 6
The formalism of QM
Outcomes: A± −→ Pa± = 21 (1 ± aσ)
z
State: s −→ Ws = 12 (1 + sσ)
a
1
=
Born rule: p±
a
2 (1 ± sa)
s
y
x
– p. 7
II. Empirical foundations
– p. 8
Empirical foundations
1. Measurement and state preparation are different things.
2. States refer to ensembles not to individual systems.
3. In a series of papers Band and Park provided means to
“empirically” specify the state of a system:
(a) If dim H = 2, then any three noncommuting self-adjoint
operators suffice to specify Ws .
(b) If dim H = n, then a set of n2 − 1 self-adjoint operators
is needed, but to specify the set is difficult.
– p. 9
Empirical foundations
4. In quantum tomography one is looking for a sufficient set
of self-adjoint operators (called the quorum) by which
the density operator of the system can be reconstructed.
5. However, in quantum tomography it is taken for granted
that the
measurement −→ self-adjoint operator;
measurement outcome −→ (orthogonal) projection
assignment is already given.
6. Can this assignment can be empirically grounded?
– p. 10
III. Representing measurements by operators
– p. 11
Representing measurements by operators
One measurement:
!
Ws =
p+
a
p−
a
Pa+
=
1
!
0
Pa−
=
0
!
1
Tr(Ws Pa± ) = p±
a
The off-diagonals do not matter
Ws is not necessarily a density matrix
– p. 12
Representing measurements by operators
Two commensurable measurements:


p++
ab







Ws = 






p+−
ab
p−+
ab
p−−
ab
p+
a
p−
a
p+
b
p−
b














– p. 13
Representing measurements by operators
Two commensurable measurements:
Note that
+−
+
p++
+
p
=
p
a
ab
ab
−+
+
p++
+
p
=
p
ab
ab
b
does not hold a priori. If it does, then the upper block is
enough:

 ++
pab


Ws = 

p+−
ab
p−+
ab
p−−
ab




– p. 14
Representing measurements by operators
Two incommensurable measurements:
 +

pa


Ws = 

p−
a
p+
b
p−
b




– p. 15
Representing measurements by operators
Two incommensurable measurements:
Operator assignment:
System −→ C2
Measurements: a −→ Oa = aσ
b
−→ Ob = bσ
Outcomes: A± −→ Pa± = 12 (1 ± aσ)
B±
−→ Pb± = 21 (1 ± bσ)
State: s −→ Ws = ?
– p. 16
Representing measurements by operators
Two incommensurable measurements:
Born rule:
±
=
Tr
(W
P
ps (A± |a) =: p±
s
a )
a
±
ps (B ± |b) =: p±
=
Tr
(W
P
s
b
b )
– p. 17
Representing measurements by operators
An example:
Operator assignment:
System:
Color: a
Size: b
Shape: c
Black/White: A±
Large/Small: B ±
Round/Cubic: C ±
State: s
−→ C2
−→ Oa = aσ
−→ Ob = bσ
−→ Oc = cσ
−→ Pa± = 21 (1 ± aσ)
−→ Pb± = 12 (1 ± bσ)
−→ Pc± = 12 (1 ± cσ)
−→ Ws = ?
– p. 18
Representing measurements by operators
Do measurements a, b, c and
±
± fix W ?
,
p
probabilities p±
,
p
s
a
c
b
z
Only if the {a, b, c} 7→ {a, b, c}
assignment is fixed.
s
a
y
c
x
b
But even in this case Ws is not
necessarily a density matrix!
– p. 19
Representing measurements by operators
A counter-example: Let
z
+
+
=
p
p+
=
p
a
c =: p
b
c
and
a
y
x
b
s
a = x
b = (0, cos ϕ, − sin ϕ)
c = z
Then for any p ≈ 1 and ϕ ≈ π/2:
|s| > 1
Ws is not a density matrix.
– p. 20
IV. How to assign operators to measurements?
– p. 21
How to assign operators to measurements?
Symmetry considerations:
Spin: Let the rotation group be represented on C.
Spin measurement in direction a −→ Oa = aσ
Outcomes: A± −→ Pa± = 21 (1 ± aσ)
Then Ws gets fixed.
– p. 22
How to assign operators to measurements?
Projection postulate:
Measuring a and selecting A±
the system previously in state s −→ Wa± :=
Pa± W Pa±
Tr(Pa± W Pa± )
will go over to state sa±
– p. 23
How to assign operators to measurements?
Selecting outcome A+ , the
state
z
a
1
W = (1 + sσ)
2
s
y
x
– p. 24
How to assign operators to measurements?
Selecting outcome A+ , the
state
z
1
W = (1 + sσ)
2
s=a
y
x
turns into
Pa+
1
= (1 + aσ)
2
– p. 25
How to assign operators to measurements?
z
The probability of obtaining
outcome B + is
+
pa+ (B |b) =
s=a
Tr(Pa+ Pb+ )
1
= (1 + ab)
2
y
x
b
– p. 26
How to assign operators to measurements?
z
Similarly, for the reversed case
1
pa+ (B |b) = pb+ (A |a) = (1 + ba)
2
+
a
+
y
x
b
– p. 27
How to assign operators to measurements?
Question:
Does Ws get fixed if we assume that the conditional
probabilities emerge from underlying elements of reality?
– p. 28
How to assign operators to measurements?
Elements of reality:
An event type α± is called an element of reality with
respect to measurement a if the following hold:
ps (A± |a ∧ α± ) = δ±±
ps (a ∧ α± ) = ps (a) ps (α± )
– p. 29
How to assign operators to measurements?
Elements of reality:
If α± , β ± and γ ± are elements of reality with respect to a
measurements a, b and c, respectively, then:
ps (A± |a) = ps (α± )
ps (B ± |b) = ps (β ± )
ps (C ± |c) = ps (γ ± )
– p. 30
How to assign operators to measurements?
Answer:
z
There is a statistical ensemble
with the following probability
for atomic elements of reality:
c
a
y
x
b
s
1
ps (α ∧ β ∧ γ ) = (1 ∓ sin ϕ)
8
±
±
±
which leads to the conditional probabilities represented
by vectors a, b and c, for which
Ws is not a density operator.
– p. 31
Conclusions
Given three yes-no measurements on a statistical ensemble
1. and representing the three pairs of measurement
outcomes by three orthogonal pairs of projections in M2 ;
2. the state of the system by a matrix Ws in M2 ;
3. requiring that the conditional probabilities of the
measurement outcomes be given by the Born rule;
4. requiring that the projection postulate also holds;
5. and also requiring that the conditional probabilities
emerge from the underlying elements of reality
even if all these hold, Ws is not necessarily a density
operator. That is the formalism of QM does not apply.
– p. 32
References
G. M. D’Ariano, L. Maccone and M.G. A. Paris, “Quorum of observables for universal
quantum estimation,” J. Phys. A, 34, 93-103, 2001.
M. Gömöri and G. Hofer-Szabó, “On the meaning of EPR’s Criterion of Reality,” (in
preparation).
G. Hofer-Szabó, “ How human and nature shake hands: the role of no-conspiracy in physical
theories,” (submitted).
J. M. Jauch, Foundations of Quantum Mechanics, (New York: Addison-Wesley, 1968).
H. Margenau, “Measurements and Quantum States, Part I and Part II,” Phil. Sci., 30, 1-16
and 138-157, 1963.
J. L. Park and W. Band, “The empirical determination of quantum states,” Found. Phys., 1,
133-144 (1970).
J. L. Park and W. Band, “A general theory of empirical state determination in quantum
mechanics, Part I and Part II,” Found. Phys., 1, 211-226 and 339-357 (1971).
J. L. Park and W. Band, “Preparation and Measurement in Quantum Physics,” Found. Phys.,
22, 657-668 (1992).
L. Wessels, “The preparation problem in quantum mechanics,” in J. Earman and J. D. Norton
(eds.), The Cosmos of Science, Essays of Exploration, (University of Pittsburgh Press –
Universitätsverlag Konstanz, 243-273, 1997)
– p. 33