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Deriving the Qubit from Entropy Principles
Adam Brandenburger and Pierfrancesco La Mura
March 17, 2015
The Mathematics of Quantum Mechanics
Image: Shutterstock
And Beyond
From:
To:
“I don’t believe in Hilbert space anymore.”
Quoted by Birkhoff, G., in Dilworth, R. (ed.), Lattice Theory, Proceedings of the Second Symposium in Pure
Mathematics of the American Mathematical Society, AMS, 1961
Quantum Quest
Can we find physically intelligible principles from which the
mathematical structure of quantum mechanics emerges?
Image: Nature, 09/12/13, p.154
Axioms for QM
“The special theory of relativity, we know, can be deduced in its
entirety from two axioms: the equivalence of inertial reference
frames, and the constancy of the speed of light. Aharanov
{unpublished lecture notes} has proposed such a logical structure
for quantum theory.”
Popescu, S., and D. Rohrlich, “Quantum Nonlocality as an Axiom,” Foundations of Physics, 24, 1994, 379-385
Heisenberg Uncertainty Principle
∆Q · ∆P ≥ ~/2
We begin with the uncertainty principle and turn it into an axiom
Figure from Busch, P., P. Lahti, and R. Werner, “Proof of Heisenberg’s Error-Disturbance Relation,” Physical
Review Letters, 111, 2013, 160405
Empirical Model
z-direction
Outcomes
0
1
Frequencies
fz
1 − fz
y-direction
x-direction
Outcomes
0
1
Frequencies
fx
1 − fx
Outcomes
0
1
Frequencies
fy
1 − fy
Phase Space
Outcome in
x-direction
Outcome in
y-direction
Outcome in
z-direction
q000
0
0
0
q001
0
0
1
q010
0
1
0
q011
0
1
1
q100
1
0
0
q101
1
0
1
q110
1
1
0
q111
1
1
1
q000 + q001 + q010 + q011 = fx
q000 + q001 + q100 + q101 = fy
q000 + q010 + q100 + q110 = fz
Negative Probabilities
A phase-space model is a local hidden-variable model (satisfying
free choice of measurements)
By Bell’s Theorem, some classical feature will need to be relaxed in
order to represent quantum systems
Abramsky and Brandenburger [2011] prove that the introduction of
negative probabilities on phase space makes it possible to
reproduce precisely the set of empirical models that obey the
no-signaling condition
Of course, we insist that all observable events have non-negative
probabilities
Really??
Or . . .
“Negative energies and probabilities should not be considered as
nonsense. They are well-defined concepts mathematically, like a
negative of money.”
– P. Dirac, Proc. Roy. Soc. London A, 1942
Image: http://johncarlosbaez.wordpress.com/2013/07/19/negative-probabilities/
R´enyi Entropy
R´enyi entropy is a family of functions measuring information (or
uncertainty), axiomatized via additivity over statistically
independent systems:
Hα (q) =

n
P

1

qiα ) if α 6= 1
log
(
−
 α−1
2
n
i=1
P


qi log2 qi
−
if α = 1
i=1
Requiring a well-behaved entropy function with signed probability
distributions yields the following family, for k = 1, 2, 3, . . .:
n
X
1
H2k (q) = −
qi2k )
log2 (
2k − 1
i=1
Uncertainty Principle
We introduce an Uncertainty Principle expressed in entropic terms
and formulated in phase space
Because R´enyi entropy is a family of measures of information, we
state our axiom as holding independently of the value of k (the
values of the lower bounds β2k will be determined by the next
axiom)
An empirical model defined by frequencies fx , fy , fz is
allowable iff for every k = 1, 2, 3, . . ., there is a
phase-space probability distribution q that represents it
and that satisfies H2k (q) ≥ β2k
Unbiasedness Principle
A set of measurements on a system is called mutually unbiased, or
complementary, if complete certainty of the measured value of the
outcome of one of them implies maximal uncertainty about the
outcomes of the others
z-direction
Outcomes
0
1
Frequencies
1/2
1/2
y-direction
Outcomes
Frequencies
0
1
1/2
1/2
x-direction
Outcomes
0
1
Frequencies
1
0
Lemma
Under the Unbiasedness Principle, the lower bounds in the
Uncertainty Principle satisfy β2k = 2 for all k
Interim Summary
We want to find all empirical probabilities fx , fy , fz that satisfy the
Uncertainty and Unbiasedness Principles — i.e., that satisfy the
requirement that, for each k, there is a phase-space probability
distribution q that represents fx , fy , fz and that has R´enyi entropy
H2k (q) ≥ 2
Frequency of
outcome 0
Frequency of
outcome 1
Measure in x-direction
½(1 + rx)
½(1 − rx)
Measure in y-direction
½(1 + ry)
½(1 − ry)
Measure in z-direction
½(1 + rz)
½(1 − rz)
Re-parameterized Empirical Model
Main Mathematical Result
Let
R2k = {(rx , ry , rz ) ∈ [−1, 1]3 : ∃ q with H2k (q) ≥ 2
which represents (rx , ry , rz )}
We want to find ∩k R2k
Note that H2k (q) is not monotone decreasing (or increasing) when
q is allowed to be a signed probability distribution!
Theorem
The sets R2k are increasing (set-wise) in k
Duality
Norm minimization problem:
min ||q||2k
q∈R8
subject to Aq = b
where

1
 1
A=
 1
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
0
1
0
1
0
0
1
1

0
0 
,
0 
1


(1 + rx )/2
 (1 + ry )/2 

b=
 (1 + rz )/2 
1
Dual problem:
max b T x
x∈R4
subject to ||AT x||
2k
2k−1
≤1
The Problem in Dual Space
bŸ
zk
Ÿ
Ÿ
yk
zk+1
Ÿ
Ÿ
wk
Ÿ
yk+1
0
Ck+1
Ck
Ck = {x ∈ R4 : ||AT x||
2k
2k−1
≤ 1},
Ck+1 = {x ∈ R4 : ||AT x|| 2k+2 ≤ 1}
The proof works by showing that
||z k+1 ||2
1 1
≤ ( ) k(k+1)
k
2
||z ||2
2k+1
The Bloch Sphere
+1
+1
−1
−1
The set R2 = ∩k R2k is the unit ball in R3 :
R2 = {(rx , ry , rz ) ∈ [−1, 1]3 : rx2 + ry2 + rz2 ≤ 1}
Comments
I
Our approach yields the single qubit — we believe it can be
extended to n-qubit systems
I
We do not claim that our axioms are self-evident — indeed
they are mysterious at the macroscopic level
I
But our axioms are physically intelligible — and evidently true
of microscopic systems