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![Page 1: Indefinite causal order in quantum mechanics Faculty of Physics, University of Vienna & Institute for Quantum Optics and Quantum Information, Vienna Mateus.](https://reader035.fdocuments.us/reader035/viewer/2022062401/5a4d1add7f8b9ab05997593b/html5/thumbnails/1.jpg)
Indefinite causal order in quantum mechanics
Faculty of Physics, University of Vienna &Institute for Quantum Optics and Quantum Information, Vienna
Mateus Araujo, Cyril Branciard, Fabio Costa, Adrian Feix, Christina Giarmatzi, Ognyan Oreshkov, Magdalena Zych
The 11th “Vienna Central European Seminar on Particle Physics and Quantum Field Theory”
Časlav Brukner
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Possible causal influences
𝑏 𝑎 𝑏𝑎
𝑎𝑎 𝑏 𝑐 causes causes and have a
common cause
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Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?
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Dynamical causal structure of general relativity
Superposition principle of quantum mechanics+
Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?
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Dynamical causal structure of general relativity
Superposition principle of quantum mechanics+
Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?
![Page 6: Indefinite causal order in quantum mechanics Faculty of Physics, University of Vienna & Institute for Quantum Optics and Quantum Information, Vienna Mateus.](https://reader035.fdocuments.us/reader035/viewer/2022062401/5a4d1add7f8b9ab05997593b/html5/thumbnails/6.jpg)
Dynamical causal structure of general relativity
Superposition principle of quantum mechanics+
Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?
![Page 7: Indefinite causal order in quantum mechanics Faculty of Physics, University of Vienna & Institute for Quantum Optics and Quantum Information, Vienna Mateus.](https://reader035.fdocuments.us/reader035/viewer/2022062401/5a4d1add7f8b9ab05997593b/html5/thumbnails/7.jpg)
Outline
“Causal inequalities“ Device-independent approach to causality
Framework for quantum mechanics with no assumed global causal structure:Device-dependent approach to causalityCausally non-separable processes
Quantum computation with indefinite order of gatesComputational task that cannot be accomplished on a
computer with fixed order of gates
Physical realization of causally non-separable processes
Linear optical schemes & via superposition of large masses
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“Correlation does not imply causation”
and are correlated, but …does cause ,
or cause , or both have a common cause ?
𝑏 𝑎 𝑏𝑎
𝑎𝑎 𝑏 𝑐
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The notion of “causation”
𝑎 ,𝑥
Necessity of interventions, or free variables, statistically independent of “the rest of the
experiment”
𝑏 , 𝑦
),,( yxbap
yxbpyxbapa
,),,( xapyxbap
b
),,(One-directional
signalling
Cause
Effect
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The notion of “causation”
𝑎 ,𝑥
Necessity of interventions, or free variables, statistically independent of “the rest of the
experiment”
𝑏 , 𝑦
),,( yxbap
One-directional signalling
𝑥𝑎
𝑦𝑏
yxbpyxbapa
,),,( xapyxbap
b
),,(
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Definite causal order
No-signalling One-directional
signalling
Space-like separated
Time-like separated Alice before Bob
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Causal inequalities(Device-independent approach to causality)
𝑦𝑏𝑥
𝑎
Causal correlations: either A signals to B or B signals to A, or no-signalling or a convex combination of these situations.
System enters each laboratory only once. The laboratories are shielded and interact with “the
outside world” only through the system entering and exiting it.
Assumptions:
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Causal inequalities(Device-independent approach to causality)
Causal correlations: either A signals to B or B signals to A, or no-signalling or a convex combination of these situations.
),|,()1(),|,(),,( yxbapyxbapyxbap ABBA
ybpyxbap BA
a
BA ),,(
xapyxbap BA
b
BA ),,(
ybpyxbap BA
a
BA ),,(
yxapyxbap BA
b
BA ,),,(
No-signalling One-directional signalling: Alice to Bob
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The simplest causal inequality
𝑦𝑏𝑥
𝑎
Causal correlations satisfy causal inequalities, which are facets of the causal polytope
1 bit input, 1 bit output
1 bit input, 1 bit output
Guess Your Neighbour’s Input (GYNI) game:
21),( xbyap
C. Branciard, M. Araújo, A. Feix, F. Costa, Č. B., to appear in New J. Phys. (2015)
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𝑦𝑏𝑥
𝑎
“Non-causal correlations”(violating causal inequalities)
Interpretation: Both A signals to B and B signals to A, although the system enters only once the laboratory and the laboratories are shielded.
yxbpyxbapa
,),,(
yxapyxbapb
,),,( Two-directional signalling
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Past space-like surface
Future space-like surface Output Hilbert spaceℋ2
Input Hilbert spaceℋ1
Local quantum laboratory(Device-dependent approach to causality)
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Past space-like surface
Future space-like surface Output Hilbert spaceℋ2
Input Hilbert spaceℋ1
Local quantum laboratory(Device-dependent approach to causality)
𝑀 𝑎∈ℒ (ℋ 1⊗ℋ2)
Transformations = completely positive (CP) trace-nonincreasing maps
𝑎
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𝑎 cb
Many local quantum laboratories
Process matrix:
𝑝 (𝑎 ,𝑏 ,𝑐 ,… )=Tr ¿
𝐴1
𝐴2
𝐵1
𝐵2
𝐶1
𝐶2
)
O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)
Probabilities:
Quantum mechanics locally valid.(no globar causal order fixed)
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𝑎 cb
Conditions on W
Normalization:
𝑝 (𝑎 ,𝑏 ,𝑐 ,… )≥0
𝐴1
𝐴2
𝐵1
𝐵2
𝐶1
𝐶2
O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)
Positivity:
Quantum mechanics locally valid.(no global causal order fixed)
∑𝑎 ,𝑏 ,𝑐
𝑝 (𝑎 ,𝑏 ,𝑐 ,… )=1 and has a specific form
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Forbidden processes
Single Loops
𝑊 𝐴1 𝐴2⊗𝑊 𝐵1𝐵 2
Double Loops
𝑊 𝐴1𝐵 2⊗𝑊 𝐴2 𝐵1
𝐴1
𝐴2 𝐵2
𝐵1 𝐵1𝐴1
𝐴2 𝐵2
O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)
No “grandfather paradox”
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Admissible processes
𝑊 𝐴1 𝐴2 𝐵1⊗ Ι 𝐵2
Channels from Alice to Bob
𝐵1
𝐴1
𝐴2
Time-like separated
𝑊 𝐴1𝐵1⊗ Ι 𝐴2 𝐵2
States
𝐴1 𝐵1
Space-like separated
O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)
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There are causally non-separable processes
Causally separable processes
𝑊 𝐴1 𝐴2 𝐵1𝐵 2=𝜆𝑊 𝐴≼𝐵+(1− 𝜆 )𝑊 𝐵≼𝐴
𝑊 𝐴≼𝐵 A signals to B, or no signaling
Most general processes compatible with definite causal structure (convex mixtures of ordered processes):
𝑊 𝐵≼𝐴 B signals to A, or no signaling
Ordered processes:
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Alice
Bob
¿0 ⟩∨𝜓 ⟩
Channel from Alice to Bob
Alice
Bob
¿1⟩∨𝜓 ⟩
Channel from Bob to Alice
Quantum switch –quantum control of causal order
Path degree of freedom (quantum control)
Internal degree of freedom
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Quantum switch –quantum control of causal order
|0 ⟩∨𝜓 ⟩❑
|1 ⟩∨𝜓 ⟩❑
𝐵1 𝐵2
𝐴2𝐴1
𝐶1
Alice
Bob
Cleve
Quantum switch is a causally nonseparable process𝑊 ≠ 𝜆𝑊 𝐴≼𝐵≼𝐶+(1− 𝜆 )𝑊 𝐵≼𝐴≼𝐶
Idea of the proof: The switch allows two-way signalling (contains both and terms)
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Quantum switch –quantum control of causal order
|0 ⟩∨𝜓 ⟩❑
|1 ⟩∨𝜓 ⟩❑
𝐵1 𝐵2
𝐴2𝐴1
𝐶1
Alice
Bob
Cleve
Quantum switch is a causally nonseparable process𝑊 ≠ 𝜆𝑊 𝐴≼𝐵≼𝐶+(1− 𝜆 )𝑊 𝐵≼𝐴≼𝐶
However, the switch cannot violate a „causal inequality“
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„Superposition of unitarities“
|0 ⟩∨𝜓 ⟩❑
|1 ⟩∨𝜓 ⟩❑
𝐵1 𝐵2
𝐴2𝐴1
𝐶1
Alice
Bob
Cleve
|𝜓 𝑓𝑖𝑛𝑎𝑙 ⟩=1√2
(¿0 ⟩¿¿1𝑈 2𝑈 1|𝜓 ⟩2+¿1⟩1𝑈 1𝑈 2|𝜓 ⟩2)¿
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Computational advantages
1C. Branciard, M. Araújo, A. Feix, F. Costa, and C.B., arXiv:1508.01704 (2015)2T. Colnaghi, G. M. D’Ariano, S. Facchini, and P. Perinotti, Phys. Lett. A 376, 2940-2943 (2012)
3G. Chiribella, Phys. Rev. A 86, 040301(R) (2012)4M. Araujo, F. Costa, and C.B., Phys. Rev. Lett. 113, 250402 (2015)
5A. Feix, M. Araujo, and C.B. Phys. Rev. A 92, 052326 (2015)
“Guess Your Neighbour's Input” game1
Controlled permutations of unitary gates2
Discrimination between pairs of commuting or anti-commuting unitaries with a single use3
Reduction of the query complexity from to 4
Reduction of communication complexity5
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Experimental Demonstration
L. M. Procopio et. al, Nature Communication 6, 7913 (2015).
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Order of events determined by their position in space-time
A
B
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General relativity: space-time is dynamical
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Dynamics of the clock described in:
Proper times of clocks a & b
General relativity: space-time is dynamical
M. Zych, F. Costa, I. Pikovski, Č. B., Nature Communication 2:505 (2011)I. Pikovski, M. Zych, F. Costa, Č. B., Nature Physics 11, 668 (2015)
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M. Zych, F. Costa, I. Pikovski, Č. B., Nature Communication 2:505 (2011)I. Pikovski, M. Zych, F. Costa, Č. B., Nature Physics 11, 668 (2015)
Dynamics of the clock described in:
Proper times of clocks a & b
General relativity: space-time is dynamical
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Quantum control of temporal order
Gravitational quantum switch
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Summary and Outlook
Global causal order need not be a necessary element of quantum theory.
There exist physical processes that are causally nonseparable.
Causally nonseparable processes is a new resource for quantum information processing
The framework also includes processes with no (known) physical interpretation
Does a theory of quantum gravity provide a physical interpretation to such processes?
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Thank you!
quantumfoundations.org
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BA
Rr rB
Potential
r
Force-free time dilation
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Relative degrees of freedom
Clock in a superposition Fixed backgroud
Local clockSuperposition of backgrouds
=
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running clock in a superposition
QM+GR: interference cannot be observed since the which-path information is stored in the clock time
Clock in a superposition
GR: time shown by the clock depends on the path taken
QM: either path or interference
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Visibility modulation
𝒱=|cos( ∆𝐸𝑔∆ 𝑥𝑡2ℏ𝑐2 )|=|cos (∆𝜏𝑡⊥ 𝜋2 )|
Difference in proper time
Orthogonalization time
For one needs
𝑡⊥=ℏ𝜋Δ𝐸
M. Zych, F. Costa, I. Pikovski, Č. B., Nature Communication 2:505 (2011)I. Pikovski, M. Zych, F. Costa, Č. B., Nature Physics 11, 668 (2015)
∆ 𝑥