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

Reference frames

Space and time are relational✓

CWhen we describe a physical property,

we take a specific point of view

Reference frames

Space and time are relational

Our “rods” and “clocks” are physical systems

CWhen we describe a physical property,

we take a specific point of view

Reference frames

Physical systems are ultimately quantum

2( |γ1⟩ + |γ2⟩)

Reference frames

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

2( |γ1⟩ + |γ2⟩)

Quantum reference frames

Reference frames

Disclaimer: Does not describe spacetime fuzziness, classical reference frames

which are in a quantum relationship

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

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

Relational approach: only relative quantities are considered.

No need of an absolute reference frame.

No absolute space

Cα xB

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

Cα xB

xB ↦ qB − qC

xA ↦ − qC

Cα xB

xB ↦ qB − qC

xA ↦ − qC

eiℏ xA pB |ϕ⟩A |ψ⟩B

Cα xB

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

𝒫AC xA𝒫†AC = − qC

A: new reference frame; B: quantum system; C: old reference frame

Relative statesSx = 𝒫ACe

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

Localised state of A

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

Product state and spatial superposition

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

Entangled state

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

Entangled state

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

Entangled state

EPR state

Zdx|xiA|x+XiB

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

Entangled state

EPR state

Zdx|xiA|x+XiB

iℏ xA pB ρ(A)

BC = Sxρ(C)AB

A: new reference frameB: quantum systemC: old reference frame

Schrödinger equation in C’s reference frame

i~d⇢(C)ABdt =

(C)AB , ⇢

(C)AB(t)

Extended covariance

9FG, E. Castro Ruiz, C. Brukner, Nat Commun. (2019)

arXiv:1712.07207

i~d⇢(C)ABdt =

(C)AB , ⇢

(C)AB(t)

To change to the frame of A we apply the transformation

i~d⇢(A)BCdt =

(A)BC , ⇢

(A)BC(t)

Extended covariance

arXiv:1712.07207

i~d⇢(C)ABdt =

(C)AB , ⇢

(C)AB(t)

The evolution in the new reference frame is unitary.

H(A)BC = SH

(C)AB S

† + i~dSdt

⇢(A)BC = S⇢(C)

AB S†

i~d⇢(A)BCdt =

(A)BC , ⇢

(A)BC(t)

Extended covariance

arXiv:1712.07207

i~d⇢(C)ABdt =

(C)AB , ⇢

(C)AB(t)

The evolution in the new reference frame is unitary.

H(A)BC = SH

(C)AB S

† + i~dSdt

⇢(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 =

(A)BC , ⇢

(A)BC(t)

Extended covariance

arXiv:1712.07207

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

Quantum 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

Spin is unambiguous in the rest frame

Quantum rest frame

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

of velocities.

Quantum rest frame

of velocities.

Quantum rest frame

of velocities.

Spin is unambiguous in the rest frameQRF transformation to the rest

frame of a quantum particle

Quantum rest frame

of velocities.

superposition of Lorentz boosts

SL = P(v)CAUA(⇤⇡C )

Quantum rest frame

of velocities.

Operational way of finding a covariant spin operator.

Ξi = SL(IC ⊗ σi) S†L

Quantum rest frame

of velocities.

Operational way of finding a covariant spin operator.

Ξi = SL(IC ⊗ σi) S†L

Opens to practical applications.

Quantum rest frame

quantum reference frames for time

A simple clock model

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)

2( |E0⟩ + |E1⟩)

H = HA + HB −G

c4xHAHB

2( |E0⟩ + |E1⟩)

H = HA + HB −G

c4xHAHB

t⊥ =πℏ

(E1 − E0)

2( |E0⟩ + |E1⟩)

Δt =G(E1 − E0)

H = HA + HB −G

c4xHAHB

t⊥ =πℏ

(E1 − E0)

2( |E0⟩ + |E1⟩)

Δt =G(E1 − E0)

H = HA + HB −G

c4xHAHB

t⊥ =πℏ

(E1 − E0) t⊥Δt =πℏGtc4x

QM with no time parameter?E.Castro Ruiz, FG, C Brukner, PNAS (2017)

E1 x R � x

Option 1: Far-away observer

QM with no time parameter?E.Castro Ruiz, FG, C Brukner, PNAS (2017)

E1 x R � x

Option 1: Far-away observer

Option 2: Reference frames for time evolution (this talk)

Can we “stand” on different clocks and describe quantum dynamics from their point of view?

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

|Ψ⟩ph ∝ ∫ dαeiℏ Cα |ϕ⟩

C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

i⟨ti |Ψ⟩Ph = |ψ(ti)⟩(i)

Perspective of clock iCi

iℏ 1 + ∑k≠i

λikHkd |ψ(ti)⟩(i)

dti= ∑

Hk + ∑j<k

λjkHjHk |ψ(ti)⟩(i)

C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

iℏ 1 + ∑k≠i

dti= ∑

Hk + ∑j<k

C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

λik → 0 Clock hamiltonian from far-away observer

iℏ 1 + ∑k≠i

dti= ∑

Hk + ∑j<k

C |Ψ⟩ph = 0 C =N

∑k=1

Hk + ∑j<k

λjkHjHk

λjk = −G

λik → 0 Clock hamiltonian from far-away observer

iℏd |ψ(ti)⟩(i)

dti= ∑

Hk + ∑j<k

λjkHjHk |ψ(ti)⟩(i)λik small

Hk = Hk(1 − λikHk)λjk = λjk − λij − λik

Relativity of interactionsH(i) = ∑

Hk + ∑j<k

λjkHjHk

Hk = Hk(1 − λikHk)

λjk = λjk − λij − λik

λ12 λ14

Hk + ∑j<k

λjkHjHk

λ12 λ14

Perspective of clock 3

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

Hk + ∑j<k

λjkHjHk

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

Introducing the measurement

F Hellmann, M Mondragon, A Perez, C Rovelli PRD (2007) V Giovannetti, S Lloyd, L Maccone, PRD (2015)

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

Solution: “Purify” the measurement

C2System S

Ancilla M

Clocks 1 and 2

C2System S

Ancilla M

Clocks 1 and 2

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

δ( T2 − ti)KMSi

Previous Hamiltonian

System S

Ancilla M

Clocks 1 and 2

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)

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

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

The gravitational switch

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

τA = 2 τB = 2

UAUB |ψ⟩S |L⟩E

τA = 2 τB = 2

UBUA |ψ⟩S |R⟩E

τA = 2 τB = 2

( |L⟩E + |R⟩E)

2|ψ⟩S

UAUB |ψ⟩S |L⟩E + UBUA |ψ⟩S |R⟩E

( |L⟩E + |R⟩E)

2|ψ⟩S

Relative localisation of events

Far-away observer

Relative localisation of eventsC = ∑

i=A,B,C

Hi(1 + ϕi) + ∑i=A,B

δ( Ti − t*)KSi (1 + ϕi) ϕi = −

Far-away observer

i=A,B,C

Hi(1 + ϕi) + ∑i=A,B

δ( Ti − t*)KSi (1 + ϕi) ϕi = −

Far-away observer

Distance between E and the clocks

i=A,B,C

Hi(1 + ϕi) + ∑i=A,B

δ( Ti − t*)KSi (1 + ϕi) ϕi = −

From C’s point of view

M(C)LA B

t⇤ � �

t⇤ + �

t⇤ � �

t⇤ + ✏t⇤ � ✏

i=A,B,C

Hi(1 + ϕi) + ∑i=A,B

δ( Ti − t*)KSi (1 + ϕi) ϕi = −

From A’s point of view

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

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