Does time-symmetry in quantum theory imply
retrocausality?
Matthew Leifer
Chapman University
Joint work with Matt Pusey (Perimeter)
June 22, 2016
Concepts and Paradoxes in a Quantum Universe – Perimeter Institute – 1 / 22
Introduction
Collapse and
Time-Symmetry
Operational Framework
Ontological Framework
Our Assumptions
Main Results
Introduction
CPQU 06/22/2016 – 2 / 22
Collapse and Time-Symmetry
Introduction
Collapse and
Time-Symmetry
Is the collapse of the wavefunction ia fundamental time-asymmetry?
Yakir showed1 that we can restore time-symmetry by introducing a
backwards evolving state that depends on a future measurement
choice in addition to the usual forwards evolving state-vector.
Operational Framework
Ontological Framework
Our Assumptions
| hφ| Pa |ψi |2
.
p(a|ψ, φ) = P
2
′
|
hφ|
P
|ψi
|
a
a′
Main Results
Does this mean we have to accept that there is retrocausality in nature
if we want time-symmetry?
Huw Price has argued that a time-symmetric realist account of
quantum theory should be retrocausal2 , but his argument assumes
that quantum states are real.
1
Y. Aharonov, P. Bergmann and J. Lebowitz, Phys. Rev. B 134, 141016 (1964).
2
H. Price, Stud. Hist. Phil. Mod. Phys. 43:75–83 (2012).
CPQU 06/22/2016 – 3 / 22
Introduction
Operational Framework
PTM Experiments
Quantum Experiments
Operational Time
Symmetry
Ontological Framework
Our Assumptions
Operational Framework
Main Results
CPQU 06/22/2016 – 4 / 22
PTM Experiments
Introduction
Operational Framework
Consider experiments involving a preparation P , a transformation T ,
and a measurement M in a definite time order.
PTM Experiments
Quantum Experiments
Operational Time
Symmetry
a
b
Ontological Framework
Our Assumptions
P
Main Results
T
x
M
y
An operational theory consists of a set of possible experiments
(P, T, M ) and, for each experiment, a prediction
pP T M (a, b|x, y).
CPQU 06/22/2016 – 5 / 22
Example: Quantum Experiments
Introduction
A preparation P is associated with:
Operational Framework
PTM Experiments
Quantum Experiments
Operational Time
Symmetry
A Hilbert space HA .
A set of subnormalized density operators ρaA|x on HA .
A transformation T is associated with:
Ontological Framework
Our Assumptions
Main Results
An input Hilbert space Hin and an output Hilbert space Hout .
A Completely-Positive, Trace-Preserving map Eout|in from the
density operators on Hin to the density operators on Hout .
A measurement M is associated with:
A Hilbert space HB .
A set of POVMs Eb|yB on HB .
(P, T, M ) is an experiment if Hin = HA and Hout = HB
Quantum theory then predicts:
pP T M (a, b|x, y) = Tr Eb|yB EB|A ρaA|x
.
CPQU 06/22/2016 – 6 / 22
Operational Time Symmetry
Introduction
Operational Framework
PTM Experiments
Quantum Experiments
Operational Time
Symmetry
An experiment (P, T, M ) has an operational time reverse if there
exists P ′ , T ′ , M ′ such that
pP T M (a, b|x, y) = pP ′ T ′ M ′ (b, a|y, x).
Ontological Framework
a
Our Assumptions
Main Results
P
b
T
M
x
y
b
a
P0
y
T0
M0
x
t
CPQU 06/22/2016 – 7 / 22
Operational Time Symmetry
Introduction
A theory is operationally time symmetric if every experiment has an
operational time reverse.
Most operational theories are not operationally time-symmetric
because we can signal into the future but not into the past.
Operational Framework
PTM Experiments
Quantum Experiments
Operational Time
Symmetry
Ontological Framework
Our Assumptions
pP T M (a|x, y) = pP T M (a|x, y ′ )
Main Results
pP T M (b|x, y) 6= pP T M (b|x′ , y)
We can, however, artificially restrict attention to the no-signaling sector
of a theory i.e. only consider experiments for which
pABJ (a|x, y) = pABJ (a|x, y ′ )
pABJ (b|x, y) = pABJ (b|x′ , y)
CPQU 06/22/2016 – 8 / 22
Operational Time Symmetry: Quantum Case
Introduction
In quantum theory, no-signaling into the future corresponds to
Operational Framework
X
PTM Experiments
Quantum Experiments
ρaA|x′ = ρA ,
a
a
Operational Time
Symmetry
i.e. x is the choice of an ensemble decomposition of a fixed density
operator.
Ontological Framework
Our Assumptions
Main Results
ρaA|x =
X
The no-signaling sector of quantum theory is operationally time
symmetric3 .
Ea|xA =
− 12
− 21
ρA ρaA|x ρA
ρB = EB|A (ρA )
1
2
1
2
ρbB|y = ρB Eb|yB ρB
1
1
1
1
−2
−2
2 †
EA|B (·) = ρA EB|A ρB (·) ρB
ρA2 .
3
M. Leifer and R. Spekkens, Phys. Rev. A 88:052130 (2013).
CPQU 06/22/2016 – 9 / 22
Introduction
Operational Framework
Ontological Framework
Ontic Extensions
Ontological Time
Symmetry
Our Assumptions
Main Results
Ontological Framework
CPQU 06/22/2016 – 10 / 22
Ontic Extensions
Introduction
Operational Framework
We now assume that the system has some ontological properties between P
and M , denoted by λ, known as the system’s ontic state.
Ontological Framework
Ontic Extensions
b
a
Ontological Time
Symmetry
P
Our Assumptions
T
M
Main Results
x
λ
y
An ontic extension of an experiment is a joint distribution pP T M (a, b, λ|x, y)
such that
X
pP T M (a, b, λ|x, y) = pP T M (a, b|x, y).
λ
A ontic extension of a theory is an assignment of such a distribution to every
experiment.
CPQU 06/22/2016 – 11 / 22
Ontological Time Symmetry
Introduction
Operational Framework
An experiment (P, T, M ) has an ontological time reverse if there
exists P ′ , T ′ , and M ′ such that
Ontological Framework
pP T M (a, b, λ|x, y) = pP ′ T ′ M ′ (b, a, λ|y, x).
Ontic Extensions
Ontological Time
Symmetry
a
Our Assumptions
Main Results
P
b
T
x
λ
y
y
a
b
P0
M
T0
λ
M0
x
t
An ontic extension of a theory is ontologically time symmetric if every
experiment has an ontological time reverse.
CPQU 06/22/2016 – 12 / 22
Introduction
Operational Framework
Ontological Framework
Our Assumptions
The Time Symmetry
Assumption
No Retrocausality
λ-mediation
Our Assumptions
Weak λ-mediation
Main Results
CPQU 06/22/2016 – 13 / 22
The Time Symmetry Assumption
Introduction
Operational Framework
Ontological Framework
If a theory is operationally time symmetric then it should have an
extension that is ontologically time symmetric.
Our Assumptions
The Time Symmetry
Assumption
pP T M (a, b|x, y) = pP ′ T ′ M ′ (b, a|y, x)
No Retrocausality
λ-mediation
Weak λ-mediation
Main Results
⇒
pP T M (a, b, λ|x, y) = pP ′ T ′ M ′ (b, a, λ|y, x)
CPQU 06/22/2016 – 14 / 22
No Retrocausality
Introduction
Operational Framework
x and y are free choices and the model has the following causal
structure:
Ontological Framework
Our Assumptions
The Time Symmetry
Assumption
a
b
No Retrocausality
λ-mediation
λ
Weak λ-mediation
Main Results
x
y
p(a, b, λ|x, y) = p(b|λ, a, x, y)p(λ|a, x)p(a|x)
CPQU 06/22/2016 – 15 / 22
λ-mediation
Introduction
All correlations between P and M are mediated by λ.
Operational Framework
Ontological Framework
a
Our Assumptions
The Time Symmetry
Assumption
b
No Retrocausality
λ
λ-mediation
Weak λ-mediation
x
Main Results
y
p(a, b, λ|x, y) = p(b|λ, y)p(λ|a, x)p(a|x)
4
Taken together, the last two assumptions are equivalent to saying that
the model is an ontological model4 .
N. Harrigan and R. Spekkens, Found. Phys. 40:125 (2010).
CPQU 06/22/2016 – 16 / 22
Weak λ-mediation
Introduction
Operational Framework
a
b
Ontological Framework
Our Assumptions
λ
The Time Symmetry
Assumption
No Retrocausality
λ-mediation
x
y
Weak λ-mediation
Main Results
p(a, b, λ|x, y) = p(b|λ, x, y)p(λ|a, x)p(a|x)
CPQU 06/22/2016 – 17 / 22
Introduction
Operational Framework
Ontological Framework
Our Assumptions
Main Results
Main Theorem
Quantum Violation
Main Results
Proof
Colbeck-Renner
CPQU 06/22/2016 – 18 / 22
Main Theorem
Introduction
Operational Framework
Ontological Framework
Theorem: An ontic extension of an operationally time symmetric
experiment that satisfies Time Symmetry and No Retrocausality must
satisfy
Our Assumptions
Main Results
Main Theorem
Quantum Violation
Proof
p(λ|x, y) = p(λ)
p(a|λ, x, y) = p(a|λ, x)
p(b|λ, x, y) = p(b|λ, y).
Colbeck-Renner
If it also satisfies Weak λ-mediation then,
p(a, b|x, y) =
X
p(a|λ, x)p(b|λ, y)p(λ).
λ
CPQU 06/22/2016 – 19 / 22
Quantum Violation
Introduction
Operational Framework
Ontological Framework
Does quantum theory violate this for timelike experiments with no
signalling into the future?
Qubit Example:
Our Assumptions
Main Results
|0i
Main Theorem
Quantum Violation
Proof
Colbeck-Renner
|−i
|+i
|1i
Prepare and measure in the optimal bases for CHSH violation, with
identity dynamics in between.
CPQU 06/22/2016 – 20 / 22
Proof of p(λ|x, y) = p(λ)
Introduction
By No Retrocasality:
Operational Framework
p(a, b, λ|x, y) = p(b|a, λ, x, y)p(λ|a, x)p(a|x).
Ontological Framework
Our Assumptions
Main Results
Main Theorem
Using Bayes’ rule on p(λ|a, x) gives:
Quantum Violation
p(a, b, λ|x, y) = p(b|a, λ, x, y)p(a|λ, x)p(λ|x).
Proof
Colbeck-Renner
Sum over a and b to get: p(λ|x, y) = p(λ|x).
By Time Symmetry:
p(a, b, λ|x, y) = p(a|b, λ, x, y)p(λ|b, y)p(b|y).
Applying the same argument gives: p(λ|x, y) = p(λ|y).
Both these conditions together imply: p(λ|x, y) = p(λ).
CPQU 06/22/2016 – 21 / 22
Colbeck-Renner
Introduction
Operational Framework
Ontological Framework
Our Assumptions
If we drop Weak λ-mediation then we still have the first part of the
theorem.
We can use the Colbeck-Renner theorem5 to prove that there is an
experiment for which we must have:
Main Results
Main Theorem
p(b|a, λ, x, y) = p(b|a, x, y)
Quantum Violation
Proof
Colbeck-Renner
a
b
λ
x
5
y
This would be a fairly pointless ontic extension.
R. Colbeck and R. Renner, Nature Communications 2, 411 (2011)
CPQU 06/22/2016 – 22 / 22