Time reference frames and gravitating quantum clocks

Esteban Castro Ruiz, Flaminia Giacomini, Alessio Belenchia & Caslav Brukner

Causality in the Quantum World Anacapri, September 2019

Outline

1.Events (introduction)

2. Gravitating quantum clocks (motivation)

3. Quantum clocks as temporal reference frames

4. Outlook

1. Events (introduction)

1.1 Relative localisation and causal order of events

A B

C

D

1.1 Localisation and causal order of events

A B

C

D

t

x

A B

C

D

t

x

1.3 Clocks tick according to the spacetime metric

A B

C

D

t

x

1.3 Clocks track time evolution and events

2. Gravitating quantum clocks (motivation)

2.1 A simple clock model

E0

E1

t? =~⇡

E1 � E0

=

| 0i =|E0i+ |E1ip

2

2.2 Gravitating clocks lead to a non-fixed metric background

E0

E1

��(x) =G(E1 � E0)

c

4x

t?�t =⇡~Gt

c

4x

ECR, F Giacomini, C Brukner, PNAS (2017)

2.3 Entanglement of Quantum clocks through gravity

H = HA +HB � G

c4xHAHB

| i = 1p2|0i

1p2

⇣|0i+ e�

it

~ �E |1i⌘�

+1p2|1i

1p2

⇣|0i+ e�

it

~ �E(1�G�E

c

4x

)|1i⌘�

ECR, F Giacomini, C Brukner, PNAS (2017)

2.4 The general problem: gravitating quantum systems make the metric “fuzzy“

i~d| idt

= H| i

x

What is ?d

dt

(x)

3. Quantum clocks as temporal reference frames

3.1 “Jumping“ onto a (quantum) clock’s reference frame?

S

C1

C2

C3C4

C| i = 0

| i =Z

d↵ e�i↵C|'i

�IJ = � G

c

4xIJ

S

C1

C2

C3C4

3.3 Timeless approach for multiple clocks

| i =Z

dt |ti3 ⌦ U3| 3(0)i3

S

C1

C2

C3C4

3.3 Timeless approach for multiple clocks

| i =Z

dt |ti2 ⌦ U2| 2(0)i2

S

C1

C2

C3C4

3.3 Timeless approach for multiple clocks

What if the clocks get entangled?

3.2 Analogy: quantum reference frames for space

x

� �

| i =1p2|0i|�i |1� �i|1i+

1p2

(x)

F Giacomini, ECR, C Brukner, Nat. Commun. (2019)

x

�

| i = |�i

F Giacomini, ECR, C Brukner, Nat. Commun. (2019)

3.2 Analogy: quantum reference frames for space

3.3 Operational definition of events

(HA + HB + fA(TA) + fB(TB))| i = 0

A Ba b

S

A B

S

aS

b

S

3.4 The simplest case: non-interacting clocks

(HA + HB + �(TA � t⇤A)K(A) + �(TB � t⇤B)K

(B)| i = 0

3.4 The simplest case: non-interacting clocks

| i =Z

dtA |tAiAe�itAHB Te�iR tA0 ds (fA(s)+fB(s+TB))| A(0)iA

3.4 The simplest case: non-interacting clocks

baS

BA

S

A B

��

3.5 Entanglement of quantum clocks through gravity (once more)

✓HA + HB + HC � G

c4xHAHB + f(TA)(1�

G

c4xHB)

◆| i = 0

3.5 Entanglement of quantum clocks through gravity (once more)

UC = e�i⌧C(HA+HB+�HAHB) Te�iR ⌧C0 ds(1+�HB) f(s(1+�HB)+TA)

3.5 Entanglement of quantum clocks through gravity (once more)

�

1/�

⌧A = ⌧B = 0

⌧A = ⌧B = 0

A

B

S

a

3.5 Entanglement of quantum clocks through gravity (once more)

UA = Te�i

R ⌧A0 ds

⇣HB+HC1+�HB

+f(s)⌘

t⇤A

S

M.Zych, F. Costa, I. Pikovski, C. Brukner https://arxiv.org/abs/1708.00248v2

3.6 Gravitational quantum switch X

I

HI(1 + �I) +X

I

fI(TI)(1 + �I)

!| i = 0

S

M.Zych, F. Costa, I. Pikovski, C. Brukner https://arxiv.org/abs/1708.00248v2

3.6 Gravitational quantum switch X

I

HI(1 + �I) +X

I

fI(TI)(1 + �I)

!| i = 0

3.6 Gravitational quantum switch

A B

t⇤

t⇤

M(C)R

M(C)LA B

t⇤ � �

t⇤ + �

t⇤ + �

t⇤ � �

| i =Z

dt |tiC ⌦�UaSA U bS

B |�i|LiM + U bSB UaS

A | i|RiM�

G Chiribella, GM D’Ariano, P Perinotti, B Valiron (2009)

| i =Z

dt |tiA ⌦�UaSA (t� t⇤)U bS

B |�i|LiM + U bSB UaS(t� t⇤)A| i|RiM

�

3.6 Gravitational quantum switch

A B

t⇤

t⇤

A B

M(A)R

M(A)L

t⇤ + ✏t⇤ � ✏

4. Outlook

• Beyond superpositions of semiclassical states?

• Inclusion of spatial quantum reference

frames?

• Connection with indefinite causal structures via causal reference frames?

O Oreshkov, arXiv:1801.07594 (2018)

P Allard Guérin, C Brukner NJP (2018)

F Giacomini, ECR, C Brukner, Nat. Commun. (2019)

Thank you