Quantum reference frames for space and...
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Quantum reference frames for space and time
Flaminia Giacomini
Joint work with: A. Belenchia, Č. Brukner, E. Castro Ruiz, P. A. Höhn, A. Vanrietvelde
F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1712.07207, 2017 published in Nat. Commun. 10(494), 2019
A. Vanrietvelde, P. A. Höhn, F. Giacomini, E. Castro Ruiz, arXiv:1809.00556, 2018A. Vanrietvelde, P. A. Höhn, F. Giacomini, arXiv:1809.05093, 2018
F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228, 2018E. Castro Ruiz et al., arXiv190(?).XXXXX, 2019
Naples, 1-3 July 2019Obervers in quantum gravity II
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Reference frames
!2
Space and time are relational✓
AB
CWhen we describe a physical property,
we take a specific point of view
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Reference frames
!2
Space and time are relational
Our “rods” and “clocks” are physical systems
✓A
B
CWhen we describe a physical property,
we take a specific point of view
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Reference frames
!3
Physical systems are ultimately quantum
γ1
γ2
1
2( |γ1⟩ + |γ2⟩)
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Reference frames
!3
Physical systems are ultimately quantum
γ1
γ2
1
2( |γ1⟩ + |γ2⟩)
Can we “attach” a reference frame to an object whose state is in a superposition of classical states (in some basis)?
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Reference frames
!3
Physical systems are ultimately quantum
γ1
γ2
1
2( |γ1⟩ + |γ2⟩)
Can we “attach” a reference frame to an object whose state is in a superposition of classical states (in some basis)?
Quantum reference frames
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Reference frames
!3
Physical systems are ultimately quantum
Disclaimer: Does not describe spacetime fuzziness, classical reference frames
which are in a quantum relationship
γ1
γ2
1
2( |γ1⟩ + |γ2⟩)
Can we “attach” a reference frame to an object whose state is in a superposition of classical states (in some basis)?
Quantum reference frames
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Outlinequantum reference frames for space
quantum reference frames for time
Overview of the formalism
Results
F. Giacomini, E. Castro Ruiz, Č. Brukner, Nat. Commun. 10(494), 2019, arXiv:1712.07207, 2017F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228, 2018
E. Castro Ruiz et al., arXiv190(?).XXXXX, 2019
Motivation
Formalism
Phenomenological consequences- Relativity of interactions- Superposition of causal orders
- Frame dependence of entanglement and superposition- Extension of the covariance of quantum mechanics- Operational definition of rest frame
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quantum reference frames for space
1
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No absolute space
!6(see also Philipp’s talk)
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207 A. Vanrietvelde, P. A. Höhn, FG, E. Castro Ruiz (2018)
A. Vanrietvelde, P. A. Höhn, FG (2018)
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No absolute space
!6(see also Philipp’s talk)
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207 A. Vanrietvelde, P. A. Höhn, FG, E. Castro Ruiz (2018)
A. Vanrietvelde, P. A. Höhn, FG (2018)
✓A
B
C
Relational approach: only relative quantities are considered.
No need of an absolute reference frame.
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No absolute space
!6(see also Philipp’s talk)
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207 A. Vanrietvelde, P. A. Höhn, FG, E. Castro Ruiz (2018)
A. Vanrietvelde, P. A. Höhn, FG (2018)
✓A
B
C
Relational approach: only relative quantities are considered.
No need of an absolute reference frame.
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No absolute space
!6(see also Philipp’s talk)
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207 A. Vanrietvelde, P. A. Höhn, FG, E. Castro Ruiz (2018)
A. Vanrietvelde, P. A. Höhn, FG (2018)
✓A
B
C
Relational approach: only relative quantities are considered.
No need of an absolute reference frame.
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No absolute space
!6(see also Philipp’s talk)
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207 A. Vanrietvelde, P. A. Höhn, FG, E. Castro Ruiz (2018)
A. Vanrietvelde, P. A. Höhn, FG (2018)
✓A
B
C
Relational approach: only relative quantities are considered.
No need of an absolute reference frame.
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Quantum reference frames
!7
B
Cα xB
A
B
C
xB
xA
xB ↦ qB − qC
xA ↦ − qC
Transformation to relative coordinates
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207
eiℏ α pB |x⟩B = |x − α⟩B
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Quantum reference frames
!7
B
Cα xB
A
B
C
xB
xA
xB ↦ qB − qC
xA ↦ − qC
Transformation to relative coordinates
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207
eiℏ α pB |x⟩B = |x − α⟩B
A
xA
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Quantum reference frames
!7
B
Cα xB
A
B
C
xB
xA
xB ↦ qB − qC
xA ↦ − qC
Transformation to relative coordinates
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207
eiℏ α pB |x⟩B = |x − α⟩B
eiℏ xA pB |ϕ⟩A |ψ⟩B
A
xA
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Quantum reference frames
!7
B
Cα xB
A
B
C
xB
xA
xB ↦ qB − qC
xA ↦ − qC
Transformation to relative coordinates
FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019) arXiv:1712.07207
eiℏ α pB |x⟩B = |x − α⟩B
eiℏ xA pB |ϕ⟩A |ψ⟩B
parity-swap operator
Sx = 𝒫ACeiℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
𝒫AC xA𝒫†AC = − qC
A
xA
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A: new reference frame; B: quantum system; C: old reference frame
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Product state and spatial superposition
CBA
x
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Product state and spatial superposition
CBA
x
A BC
q
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Product state and spatial superposition
CBA
x
A BC
q
Entangled state
L LC
A B
x
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Product state and spatial superposition
CBA
x
A BC
q
Entangled state
L LC
A B
x
AC B
q
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Product state and spatial superposition
CBA
x
A BC
q
Entangled state
L LC
A B
x
AC B
q
EPR state
CBA
x
Zdx|xiA|x+XiB
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frame; B: quantum system; C: old reference frame
Localised state of A
BAC
x
AC
Bq
Product state and spatial superposition
CBA
x
A BC
q
AC B
q
Entangled state
L LC
A B
x
AC B
q
EPR state
CBA
x
Zdx|xiA|x+XiB
Relative statesSx = 𝒫ACe
iℏ xA pB ρ(A)
BC = Sxρ(C)AB
S†x
8
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A: new reference frameB: quantum systemC: old reference frame
Schrödinger equation in C’s reference frame
i~d⇢(C)ABdt =
hH
(C)AB , ⇢
(C)AB(t)
i
Extended covariance
9FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019)
arXiv:1712.07207
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A: new reference frameB: quantum systemC: old reference frame
Schrödinger equation in C’s reference frame
i~d⇢(C)ABdt =
hH
(C)AB , ⇢
(C)AB(t)
i
To change to the frame of A we apply the transformation
i~d⇢(A)BCdt =
hH
(A)BC , ⇢
(A)BC(t)
iS
Extended covariance
9FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019)
arXiv:1712.07207
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A: new reference frameB: quantum systemC: old reference frame
Schrödinger equation in C’s reference frame
i~d⇢(C)ABdt =
hH
(C)AB , ⇢
(C)AB(t)
i
The evolution in the new reference frame is unitary.
H(A)BC = SH
(C)AB S
† + i~dSdt
S†
⇢(A)BC = S⇢(C)
AB S†
To change to the frame of A we apply the transformation
i~d⇢(A)BCdt =
hH
(A)BC , ⇢
(A)BC(t)
iS
Extended covariance
9FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019)
arXiv:1712.07207
![Page 31: Quantum reference frames for space and timepeople.na.infn.it/~mercatif/observers2/speakers/Flaminia.pdf · Outline quantum reference frames for space quantum reference frames for](https://reader033.fdocuments.us/reader033/viewer/2022052923/5f03d6577e708231d40b0299/html5/thumbnails/31.jpg)
A: new reference frameB: quantum systemC: old reference frame
Schrödinger equation in C’s reference frame
i~d⇢(C)ABdt =
hH
(C)AB , ⇢
(C)AB(t)
i
The evolution in the new reference frame is unitary.
H(A)BC = SH
(C)AB S
† + i~dSdt
S†
⇢(A)BC = S⇢(C)
AB S†
We define an extended symmetry transformation as:
SH ({mi, xi, pi}i=A,B) S† + i~dS
dtS† = H ({mi, xi, pi}i=B,C)
To change to the frame of A we apply the transformation
i~d⇢(A)BCdt =
hH
(A)BC , ⇢
(A)BC(t)
iS
Extended covariance
9FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019)
arXiv:1712.07207
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
Spin is unambiguous in the rest frame
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
QRFs allow us to transform to the rest frame of a particle in a superposition
of velocities.
Spin is unambiguous in the rest frame
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
QRFs allow us to transform to the rest frame of a particle in a superposition
of velocities.
Spin is unambiguous in the rest frame
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
v1
v2
C
C
A
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
QRFs allow us to transform to the rest frame of a particle in a superposition
of velocities.
Spin is unambiguous in the rest frameQRF transformation to the rest
frame of a quantum particle
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
v1
v2
C
C
A
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
QRFs allow us to transform to the rest frame of a particle in a superposition
of velocities.
Spin is unambiguous in the rest frameQRF transformation to the rest
frame of a quantum particle
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
v1
v2
C
C
A
superposition of Lorentz boosts
SL = P(v)CAUA(⇤⇡C )
Quantum rest frame
10
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F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
QRFs allow us to transform to the rest frame of a particle in a superposition
of velocities.
Spin is unambiguous in the rest frameQRF transformation to the rest
frame of a quantum particle
Operational way of finding a covariant spin operator.
Ξi = SL(IC ⊗ σi) S†L
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
v1
v2
C
C
A
superposition of Lorentz boosts
SL = P(v)CAUA(⇤⇡C )
Quantum rest frame
10
![Page 40: Quantum reference frames for space and timepeople.na.infn.it/~mercatif/observers2/speakers/Flaminia.pdf · Outline quantum reference frames for space quantum reference frames for](https://reader033.fdocuments.us/reader033/viewer/2022052923/5f03d6577e708231d40b0299/html5/thumbnails/40.jpg)
F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
QRFs allow us to transform to the rest frame of a particle in a superposition
of velocities.
Spin is unambiguous in the rest frameQRF transformation to the rest
frame of a quantum particle
Operational way of finding a covariant spin operator.
Ξi = SL(IC ⊗ σi) S†L
Lack of an operational definition of spin (Stern Gerlach experiment) in special relativistic quantum mechanics.
(Pauli-Lubanski, Wigner-Pryce, Foldy-Wouthuysen, Chakrabarti, Czachor, Fradkin-Good, Fleming,…)
Opens to practical applications.
v1
v2
C
C
A
superposition of Lorentz boosts
SL = P(v)CAUA(⇤⇡C )
Quantum rest frame
10
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quantum reference frames for time
2
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A simple clock model
!12
E0
E1
=
HC = E0 |E0⟩⟨E0 | + E1 |E1⟩⟨E1 |1
2( |E0⟩ + |E1⟩)
t⊥ =πℏ
(E1 − E0)
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Gravitating clocks lead to a non-classical spacetime
!13 E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1
1
2( |E0⟩ + |E1⟩)
H = HA + HB −G
c4xHAHB
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Gravitating clocks lead to a non-classical spacetime
!13 E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1
1
2( |E0⟩ + |E1⟩)
H = HA + HB −G
c4xHAHB
t⊥ =πℏ
(E1 − E0)
![Page 45: Quantum reference frames for space and timepeople.na.infn.it/~mercatif/observers2/speakers/Flaminia.pdf · Outline quantum reference frames for space quantum reference frames for](https://reader033.fdocuments.us/reader033/viewer/2022052923/5f03d6577e708231d40b0299/html5/thumbnails/45.jpg)
Gravitating clocks lead to a non-classical spacetime
!13 E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1
1
2( |E0⟩ + |E1⟩)
Δt =G(E1 − E0)
c4xt
H = HA + HB −G
c4xHAHB
t⊥ =πℏ
(E1 − E0)
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Gravitating clocks lead to a non-classical spacetime
!13 E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1
1
2( |E0⟩ + |E1⟩)
Δt =G(E1 − E0)
c4xt
H = HA + HB −G
c4xHAHB
t⊥ =πℏ
(E1 − E0) t⊥Δt =πℏGtc4x
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!14
QM with no time parameter?E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1 x R � x
Option 1: Far-away observer
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!14
QM with no time parameter?E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1 x R � x
Option 1: Far-away observer
Option 2: Reference frames for time evolution (this talk)
S
C1
C2
C3C4
Can we “stand” on different clocks and describe quantum dynamics from their point of view?
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!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
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!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
C |Ψ⟩ph = 0 C =N
∑k=1
Hk + ∑j<k
λjkHjHk
λjk = −G
c4xjk
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!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩
C |Ψ⟩ph = 0 C =N
∑k=1
Hk + ∑j<k
λjkHjHk
λjk = −G
c4xjk
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!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩
C |Ψ⟩ph = 0 C =N
∑k=1
Hk + ∑j<k
λjkHjHk
λjk = −G
c4xjk
i⟨ti |Ψ⟩Ph = |ψ(ti)⟩(i)
Perspective of clock iCi
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!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩
iℏ 1 + ∑k≠i
λikHkd |ψ(ti)⟩(i)
dti= ∑
k≠i
Hk + ∑j<k
λjkHjHk |ψ(ti)⟩(i)
C |Ψ⟩ph = 0 C =N
∑k=1
Hk + ∑j<k
λjkHjHk
λjk = −G
c4xjk
i⟨ti |Ψ⟩Ph = |ψ(ti)⟩(i)
Perspective of clock iCi
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!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩
iℏ 1 + ∑k≠i
λikHkd |ψ(ti)⟩(i)
dti= ∑
k≠i
Hk + ∑j<k
λjkHjHk |ψ(ti)⟩(i)
C |Ψ⟩ph = 0 C =N
∑k=1
Hk + ∑j<k
λjkHjHk
λjk = −G
c4xjk
λik → 0 Clock hamiltonian from far-away observer
i⟨ti |Ψ⟩Ph = |ψ(ti)⟩(i)
Perspective of clock iCi
![Page 55: Quantum reference frames for space and timepeople.na.infn.it/~mercatif/observers2/speakers/Flaminia.pdf · Outline quantum reference frames for space quantum reference frames for](https://reader033.fdocuments.us/reader033/viewer/2022052923/5f03d6577e708231d40b0299/html5/thumbnails/55.jpg)
!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩
iℏ 1 + ∑k≠i
λikHkd |ψ(ti)⟩(i)
dti= ∑
k≠i
Hk + ∑j<k
λjkHjHk |ψ(ti)⟩(i)
C |Ψ⟩ph = 0 C =N
∑k=1
Hk + ∑j<k
λjkHjHk
λjk = −G
c4xjk
λik → 0 Clock hamiltonian from far-away observer
iℏd |ψ(ti)⟩(i)
dti= ∑
k≠i
Hk + ∑j<k
λjkHjHk |ψ(ti)⟩(i)λik small
Hk = Hk(1 − λikHk)λjk = λjk − λij − λik
i⟨ti |Ψ⟩Ph = |ψ(ti)⟩(i)
Perspective of clock iCi
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!16
Relativity of interactionsH(i) = ∑
k≠i
Hk + ∑j<k
λjkHjHk
Hk = Hk(1 − λikHk)
λjk = λjk − λij − λik
C1
C3
C2
C4
C5
λ12 λ14
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!16
Relativity of interactionsH(i) = ∑
k≠i
Hk + ∑j<k
λjkHjHk
Hk = Hk(1 − λikHk)
λjk = λjk − λij − λik
C1
C3
C2
C4
C5
λ12 λ14
Perspective of clock 3
λ5k = 0 ∀k ≠ 3 No interactions between clock 5 and the other clocks
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!16
Relativity of interactionsH(i) = ∑
k≠i
Hk + ∑j<k
λjkHjHk
Hk = Hk(1 − λikHk)
λjk = λjk − λij − λik
C1
C3
C2
C4
C5
λ12 λ14
Perspective of clock 3
λ5k = 0 ∀k ≠ 3 No interactions between clock 5 and the other clocks
Perspective of clock 1
λ54 ≠ 0λ52 ≠ 0
Interactions between clock 5 and clocks 2 and 4
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!17
Introducing the measurement
S
C1
C2
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
S
C1
C2
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
C1
C2
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
C1
C2System S
Ancilla M
Clocks 1 and 2
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
M
C1
C2System S
Ancilla M
Clocks 1 and 2
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
M
C1
C2
C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i
δ( T2 − ti)KMSi
Previous Hamiltonian
System S
Ancilla M
Clocks 1 and 2
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
M
C1
C2
C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i
δ( T2 − ti)KMSi
Previous Hamiltonian
System S
Ancilla M
Clocks 1 and 2
Time of measurement controlled by clock 2F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007)
V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
M
C1
C2
C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i
δ( T2 − ti)KMSi
Previous Hamiltonian
System S
Ancilla M
Clocks 1 and 2
Time of measurement controlled by clock 2
Observable on S and M
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!17
Introducing the measurement
[C, O] = 0 Non-evolving quantities?Restriction of observables?
Solution: “Purify” the measurement
S
M
C1
C2
C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i
δ( T2 − ti)KMSi
Previous Hamiltonian
System S
Ancilla M
Clocks 1 and 2
Time dilation factor due to clock 1
Time of measurement controlled by clock 2
Observable on S and M
F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)
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!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
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!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!18
The gravitational switch
A B
UA
UB
τA = 2 τB = 2
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!18
The gravitational switch
A B
UA
UB
τA = 2 τB = 2
UAUB |ψ⟩S |L⟩E
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
UA
UB
τA = 2 τB = 2
E
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!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
UBUA |ψ⟩S |R⟩E
UA
UB
τA = 2 τB = 2
E
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!20
The gravitational switch
A B
UA UB
M Zych, F Costa, I Pikovski, C Brukner (2017)
( |L⟩E + |R⟩E)
2|ψ⟩S
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!20
The gravitational switch
A B
UA UB
M Zych, F Costa, I Pikovski, C Brukner (2017)
UAUB |ψ⟩S |L⟩E + UBUA |ψ⟩S |R⟩E
2
( |L⟩E + |R⟩E)
2|ψ⟩S
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!21
Relative localisation of events
A B
C
S
E E
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!21
Relative localisation of events
A B
C
S
Far-away observer
E E
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!21
Relative localisation of eventsC = ∑
i=A,B,C
Hi(1 + ϕi) + ∑i=A,B
δ( Ti − t*)KSi (1 + ϕi) ϕi = −
GME
c2 xi
A B
C
S
Far-away observer
E E
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!21
Relative localisation of eventsC = ∑
i=A,B,C
Hi(1 + ϕi) + ∑i=A,B
δ( Ti − t*)KSi (1 + ϕi) ϕi = −
GME
c2 xi
A B
C
S
Far-away observer
Distance between E and the clocks
E E
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!22
Relative localisation of eventsC = ∑
i=A,B,C
Hi(1 + ϕi) + ∑i=A,B
δ( Ti − t*)KSi (1 + ϕi) ϕi = −
GME
c2 xi
From C’s point of view
A B
t⇤
t⇤
M(C)R
M(C)LA B
t⇤ � �
t⇤ + �
t⇤ + �
t⇤ � �
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A B
t⇤
t⇤
A B
M(A)R
M(A)L
t⇤ + ✏t⇤ � ✏
!23
Relative localisation of eventsC = ∑
i=A,B,C
Hi(1 + ϕi) + ∑i=A,B
δ( Ti − t*)KSi (1 + ϕi) ϕi = −
GME
c2 xi
From A’s point of view
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!24
SummaryOperational and relational formalism for quantum reference frames
for space and time.
For space:Frame-dependence of entanglement and superposition
Generalisation of covarianceGeneralisation of the weak equivalence principle (not covered)
Operational definition of the rest frame of a quantum system (relativistic spin)
For time:Hamiltonian for interacting clocks (with gravitational time dilation)
Relativity of interactionsSuperposition of causal orders
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Thank you
F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1712.07207, 2017 published in Nat. Commun. 10(494), 2019
A. Vanrietvelde, P. A. Höhn, F. Giacomini, E. Castro Ruiz, arXiv:1809.00556, 2018A. Vanrietvelde, P. A. Höhn, F. Giacomini, arXiv:1809.05093, 2018
F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228, 2018E. Castro Ruiz et al., arXiv190(?).XXXXX, 2019