Post on 08-Jun-2020
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
Reference frames
!2
Space and time are relational✓
AB
CWhen we describe a physical property,
we take a specific point of view
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
Reference frames
!3
Physical systems are ultimately quantum
γ1
γ2
1
2( |γ1⟩ + |γ2⟩)
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)?
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
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
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
quantum reference frames for space
1
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)
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.
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.
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)
Quantum rest frame
10
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
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
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
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
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
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
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
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
quantum reference frames for time
2
A simple clock model
!12
E0
E1
=
HC = E0 |E0⟩⟨E0 | + E1 |E1⟩⟨E1 |1
2( |E0⟩ + |E1⟩)
t⊥ =πℏ
(E1 − E0)
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
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)
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)
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
!14
QM with no time parameter?E.Castro Ruiz, FG, C Brukner, PNAS (2017)
E0
E1 x R � x
Option 1: Far-away observer
!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?
!15
Timeless quantum mechanics
D. Page, W. Wootters, PRD (1983) M. Reisenberger, C. Rovelli, PRD (2002)
!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
!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
!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
!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
!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
!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
!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
!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
!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
!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)
!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)
!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)
!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)
!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)
!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)
!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)
!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)
!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)
!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!18
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!18
The gravitational switch
A B
UA
UB
τA = 2 τB = 2
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!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
!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
E
!19
The gravitational switch
A B
M Zych, F Costa, I Pikovski, C Brukner (2017)
UA
UB
τA = 2 τB = 2
E
!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
!20
The gravitational switch
A B
UA UB
M Zych, F Costa, I Pikovski, C Brukner (2017)
( |L⟩E + |R⟩E)
2|ψ⟩S
!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
!21
Relative localisation of events
A B
C
S
E E
!21
Relative localisation of events
A B
C
S
Far-away observer
E E
!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
!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
!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⇤ � �
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
!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
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