Quantum reference frames for space and...

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


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


γ1

γ2

1

2( |γ1⟩ + |γ2⟩)


Quantum reference frames

Reference frames

!3


Disclaimer: Does not describe spacetime fuzziness, classical reference frames

which are in a quantum relationship

γ1

γ2

1

2( |γ1⟩ + |γ2⟩)



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.


!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


!7

B

Cα xB

A

B

C

xB

xA

xB ↦ qB − qC

xA ↦ − qC




A

xA


!7

B

Cα xB

A

B

C

xB

xA

xB ↦ qB − qC

xA ↦ − qC




eiℏ xA pB |ϕ⟩A |ψ⟩B

A

xA


!7

B

Cα xB

A

B

C

xB

xA

xB ↦ qB − qC

xA ↦ − qC




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


Localised state of A

BAC

x


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq

Product state and spatial superposition

CBA

x


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq


CBA

x

A BC

q


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq


CBA

x

A BC

q

Entangled state

L LC

A B

x


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq


CBA

x

A BC

q

Entangled state

L LC

A B

x

AC B

q


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq


CBA

x

A BC

q

Entangled state

L LC

A B

x

AC B

q

EPR state

CBA

x

Zdx|xiA|x+XiB


iℏ xA pB ρ(A)

BC = Sxρ(C)AB

S†x

8



BAC

x

AC

Bq


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


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



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


arXiv:1712.07207



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†


i~d⇢(A)BCdt =

hH

(A)BC , ⇢

(A)BC(t)

iS

Extended covariance


arXiv:1712.07207



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)


i~d⇢(A)BCdt =

hH

(A)BC , ⇢

(A)BC(t)

iS

Extended covariance


arXiv:1712.07207

F. Giacomini, E. Castro Ruiz, Č. Brukner, arXiv:1811.08228 (2018)

Quantum rest frame

10


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


Spin is unambiguous in the rest frame



Quantum rest frame

10


QRFs allow us to transform to the rest frame of a particle in a superposition

of velocities.




Quantum rest frame

10



of velocities.




v1

v2

C

C

A

Quantum rest frame

10



of velocities.

Spin is unambiguous in the rest frameQRF transformation to the rest

frame of a quantum particle



v1

v2

C

C

A

Quantum rest frame

10



of velocities.





v1

v2

C

C

A

superposition of Lorentz boosts

SL = P(v)CAUA(⇤⇡C )

Quantum rest frame

10



of velocities.



Operational way of finding a covariant spin operator.

Ξi = SL(IC ⊗ σi) S†L



v1

v2

C

C

A



Quantum rest frame

10



of velocities.



Operational way of finding a covariant spin operator.

Ξi = SL(IC ⊗ σi) S†L



Opens to practical applications.

v1

v2

C

C

A



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



E0

E1

1

2( |E0⟩ + |E1⟩)

H = HA + HB −G

c4xHAHB

t⊥ =πℏ

(E1 − E0)



E0

E1

1

2( |E0⟩ + |E1⟩)

Δt =G(E1 − E0)

c4xt

H = HA + HB −G

c4xHAHB

t⊥ =πℏ

(E1 − E0)



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



C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

c4xjk

!15



|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩

C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

c4xjk

!15




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




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



!15




iℏ 1 + ∑k≠i


dti= ∑

k≠i

Hk + ∑j<k


C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

c4xjk

λik → 0 Clock hamiltonian from far-away observer



!15




iℏ 1 + ∑k≠i


dti= ∑

k≠i

Hk + ∑j<k


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



!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


k≠i

Hk + ∑j<k

λjkHjHk



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


k≠i

Hk + ∑j<k

λjkHjHk



C1

C3

C2

C4

C5

λ12 λ14


λ5k = 0 ∀k ≠ 3 No interactions between clock 5 and the other clocks


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


[C, O] = 0 Non-evolving quantities?Restriction of observables?

S

C1

C2


!17



Solution: “Purify” the measurement

S

C1

C2


!17




S

C1

C2System S

Ancilla M

Clocks 1 and 2


!17




S

M

C1

C2System S

Ancilla M

Clocks 1 and 2


!17




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


!17




S

M

C1

C2

C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i

δ( T2 − ti)KMSi


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




S

M

C1

C2

C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i

δ( T2 − ti)KMSi


System S

Ancilla M

Clocks 1 and 2

Time of measurement controlled by clock 2

Observable on S and M


!17




S

M

C1

C2

C = H1 + H2 + HS + λH1H2 + (1 + λH1)∑i

δ( T2 − ti)KMSi


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


!18

The gravitational switch

A B

M Zych, F Costa, I Pikovski, C Brukner (2017)

!18


A B


E

!18


A B

UA

UB

τA = 2 τB = 2


E

!18


A B

UA

UB

τA = 2 τB = 2

UAUB |ψ⟩S |L⟩E


E

!19


A B


!19


A B


E

!19


A B


UA

UB

τA = 2 τB = 2

E

!19


A B


UBUA |ψ⟩S |R⟩E

UA

UB

τA = 2 τB = 2

E

!20


A B

UA UB


( |L⟩E + |R⟩E)

2|ψ⟩S

!20


A B

UA UB


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


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


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


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

Quantum reference frames for space and...

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