Emergence of the classical world from quantum physics: Schrödinger cats, entanglement, and decoherenceLuiz Davidovich Instituto de Física Universidade Federal do Rio de Janeiro

Outline of the talk

! Decoherence and the classical limit of the quantum world

! Entanglement and decoherence: new experimental results

! Multiparticle systems and decoherence

2

Schrödinger on the classical limit

! 1926: “At first sight it appears very strange to try to describe a process, which we previously regarded as belonging to particle mechanics, by a system of such proper vibrations.'' Demonstrates that “a group of proper vibrations” of high quantum number $n$ and of relatively small quantum number differences may represent a particle executing the motion expected from usual mechanics, i. e. oscillating with a constant frequency. 3

Schrödinger on the classical limit

4

! 1935: “An uncertainty originally restricted to the atomic domain has become transformed into a macroscopic uncertainty, which can be resolved through direct observation... This inhibits us from accepting in a naive way a `blurred model' as an image of reality...There is a difference between a shaky or not sharply focused photograph and a photograph of clouds and fogbanks.”

Quantum measurement

Quantum measurement

Quantum measurement

Quantum measurement

Quantum measurement

Linear evolution:

Quantum measurement

Linear evolution:

Quantum measurement

Linear evolution:

Quantum measurement

Linear evolution:

Quantum physics and localization

! “Let Ψ1 and Ψ2 be two solutions of the same Schrödinger equation. Then Ψ = Ψ1+Ψ2 also represents a solution of the Schrödinger equation, with equal claim to describe a possible real state. When the system is a macrosystem, and when Ψ1 and Ψ2 are `narrow’ with respect to the macro-coordinates, then in by far the greater number of cases, this is no longer true for Ψ. Narrowness in regard to macro-coordinates is a requirement which is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.”

Letter from Einstein to Born, January 1, 1954

Why interference cannot be seen?

Why interference cannot be seen?

! Decoherence: entanglement with the environment - same process by which quantum computers become classical computers!

Why interference cannot be seen?

! Decoherence: entanglement with the environment - same process by which quantum computers become classical computers!

! Dynamics of decoherence: related to elusive boundary between quantum and classical world

8

Decoherence dynamics

8

Decoherence dynamics

8

Decoherence dynamics

Exponential decay:

Dynamics of entanglement

! Multiparticle system, initially entangled, with individual couplings of particles to independent environments: each particle undergoes decay, dephasing, diffusion.

Dynamics of entanglement

! Multiparticle system, initially entangled, with individual couplings of particles to independent environments: each particle undergoes decay, dephasing, diffusion.

! How is local dynamics related to nonlocal loss of entanglement?

Dynamics of entanglement

! Multiparticle system, initially entangled, with individual couplings of particles to independent environments: each particle undergoes decay, dephasing, diffusion.

! How is local dynamics related to nonlocal loss of entanglement?

! How does loss of entanglement scale with number of particles?

Dynamics of entanglement

! Multiparticle system, initially entangled, with individual couplings of particles to independent environments: each particle undergoes decay, dephasing, diffusion.

! How is local dynamics related to nonlocal loss of entanglement?

! How does loss of entanglement scale with number of particles?

! Need measure of entanglement!

Entangled and separable states

! Separable states: – Pure states:

– Mixed states (R. F. Werner, PRA, 1989):

! Entangled state: non-separable

Entangled and separable states

! Separable states: – Pure states:

– Mixed states (R. F. Werner, PRA, 1989):

! Entangled state: non-separableBell states - Maximally entangled states: complete ignorance on each qubit

Entangled and separable states

! Separable states: – Pure states:

– Mixed states (R. F. Werner, PRA, 1989):

! Entangled state: non-separableBell states - Maximally entangled states: complete ignorance on each qubit

ρA,B =121 00 1

⎛⎝⎜

⎞⎠⎟

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Schrödinger on Entanglement

“This is the reason that knowledge of the individual systems can decline to the scantiest, even zero, while that of t h e c o m b i n e d s y s t e m r e m a i n s continually maximal. Best possible knowledge of a whole does not include best possible knowledge of its parts – and that is what keeps coming back to haunt us.”

Naturwissenschaften 23, 807 (1935)

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Measures of entanglement for pure states

reduced density matrix of A or B

Separable state (two qubits):

Maximally entangled state:

Von Neumann entropy

Linear entropy

S(ρr ) = 0

A mathematical interlude: partial transposition of a matrix

��00 �01

�10 �11

⇥�

��00 �10

�01 �11

⇥Transposition: a positive map T : �� �T

Matrix in computational basis:00 , 01 , 10 , 11{ }

Does not change eigenvalues!

A mathematical interlude: partial transposition of a matrix

��00 �01

�10 �11

⇥�

��00 �10

�01 �11

⇥Transposition: a positive map T : �� �T

|�±⌅ =1⌃2

(| ⇥⇥⌅± | ⇤⇤⌅)Partial transposition:1A � TB : �⇥ �TB

12

�

⇧⇧⇤

1 0 0 ±10 0 0 00 0 0 0±1 0 0 1

⇥

⌃⌃⌅⇥12

�

⇧⇧⇤

1 0 0 00 0 ±1 00 ±1 0 00 0 0 1

⇥

⌃⌃⌅

Φ± =1200 ± 11( )

Matrix in computational basis:00 , 01 , 10 , 11{ }

Does not change eigenvalues!

Example:

A mathematical interlude: partial transposition of a matrix

��00 �01

�10 �11

⇥�

��00 �10

�01 �11

⇥Transposition: a positive map T : �� �T

|�±⌅ =1⌃2

(| ⇥⇥⌅± | ⇤⇤⌅)Partial transposition:1A � TB : �⇥ �TB

12

�

⇧⇧⇤

1 0 0 ±10 0 0 00 0 0 0±1 0 0 1

⇥

⌃⌃⌅⇥12

�

⇧⇧⇤

1 0 0 00 0 ±1 00 ±1 0 00 0 0 1

⇥

⌃⌃⌅

Negative eigenvalue!Φ± =1200 ± 11( )

Matrix in computational basis:00 , 01 , 10 , 11{ }

Does not change eigenvalues!

Example:

Mixed states: Separability criterium

! If ρ is separable, then the partially transposed matrix is positive (Asher Peres, PRL, 1996):

! For 2X2 and 2X3 systems, ρ is separable iff it remains a density operator under the operation of partial transposition (Horodecki family 1996) - that is, it has a partial positive transpose (PPT)

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Negativity as a measure of entanglement

Negative eigenvalues of partially transposed matrix

N (⇥AB) � 2�

i

|�i�|

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Negativity as a measure of entanglement

=1 for a Bell state

Negative eigenvalues of partially transposed matrix

N (⇥AB) � 2�

i

|�i�|

15

Negativity as a measure of entanglement

=1 for a Bell state

Negative eigenvalues of partially transposed matrix

Dimensions higher than 6: =0 does not imply separability!

N (⇥AB) � 2�

i

|�i�|

Mixed states: Concurrence

! (Wootters, 1998) - for two qubits:

eingenvalues (positive), in decreasing order, of

(conjugation in the computational basis)

maximally entangled separable

Mixed states: Concurrence

! (Wootters, 1998) - for two qubits:

eingenvalues (positive), in decreasing order, of

(conjugation in the computational basis)

maximally entangled separable

Pure states:C = 2 1− Trρr

2( )

Mixed states: Concurrence

! (Wootters, 1998) - for two qubits:

eingenvalues (positive), in decreasing order, of

(conjugation in the computational basis)

maximally entangled separable

Pure states:C = 2 1− Trρr

2( )

C2 !Tangle

Relation between concurrence and negativity

Boundary of separability: independent of the entanglement measure

Two qubits! F. Verstraete et al.,

J. Phys. A: Math. Gen. 34, 10327 (2001)

Neg

ativ

ity

Concurrence

A paradigmatic example: Atomic decay

•Qubit states:

•“Amplitude channel”:

Usual master equation for decay of two-level atom, upon tracing on environment (Markovian approximation)

Weisskopf and Wigner (1930)!

M. P. Almeida et al., Science 316, 579 (2007)

A paradigmatic example: Atomic decay

•Qubit states:

•“Amplitude channel”:

Usual master equation for decay of two-level atom, upon tracing on environment (Markovian approximation)

Weisskopf and Wigner (1930)!

Our strategy: follow evolution as a function of p, not t

M. P. Almeida et al., Science 316, 579 (2007)

A paradigmatic example: Atomic decay

•Qubit states:

•“Amplitude channel”:

Usual master equation for decay of two-level atom, upon tracing on environment (Markovian approximation)

Weisskopf and Wigner (1930)!

Our strategy: follow evolution as a function of p, not t

Apply evolution to two qubits, take trace with respect to environment degrees of freedom, find evolution of two-qubit reduced density matrix, calculate entanglement

M. P. Almeida et al., Science 316, 579 (2007)

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

p = sin2(2✓)

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

p = sin2(2✓)

Investigating the dynamics of entanglement

“Sudden death” of entanglement

N (p = 0) = 2|↵�|

Λ

“Sudden death” of entanglement

“Entanglement Sudden Death”(Yu and Eberly)

N (p = 0) = 2|↵�|

Λ

Role of environment

! Usually one traces out environment, and one looks at irreversible evolution of system

! As entanglement decays and eventually disappears, what is its imprint onto the environment?

22

Measuring the environment?

Role of environment

! Usually one traces out environment, and one looks at irreversible evolution of system

! Our setup allows in principle access to environment

! Is it useful to watch the environment?

24

25

“Quantum Darwinism describes the proliferation, in the environment, of multiple records of selected states of a quantum system” pointer states. !Detailed study of the environment uncovers essential trait of the classical world!

Simple model

S1

S2

E1

E2

Amplitude channels

O. Jiménez Farías et al., PRL 109, 150403 (2012)G. H. Aguilar et al., PRL 113, 240501 (2014)G. H. Aguilar et al., PRA 89, 022339 (2014)

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Tangles

p0

0.1

0.2

0.3

0.2 0.4 0.6 0.8 10.1 0.3 0.5 0.7 0.9

C2S1S2

C2E1E2

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Fidelity with respect to state

|Di = 1/p6(|0000i+ |1111i+ |0011i+ |1100i+ |0110i+ |1001i)

Dicke state with second and fourth qubits flipped

Fidelity

p 10.2 0.80.40

0.3

0.3

0.6

0.1 0.90.6 0.7

0.4

0.5

0.70.66

p=0.27 p=0.73

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Fidelity with respect to state

|Di = 1/p6(|0000i+ |1111i+ |0011i+ |1100i+ |0110i+ |1001i)

Dicke state with second and fourth qubits flipped

Genuine multipartite!

W. Wieczorek et al., Phys. Rev. Lett. 101, 010503 (2008)

Fidelity

p 10.2 0.80.40

0.3

0.3

0.6

0.1 0.90.6 0.7

0.4

0.5

0.70.66

p=0.27 p=0.73

Decay of entanglement for N qubits, other environments?

! Independent individual environments

Results for amplitude damping

! Bipartitions ! Critical transition probability for which negativity

vanishes (same for all partitions):

! Smaller than 1 if finite-time disappearance of entanglement

! Critical value approaches 1 when ! Does entanglement become more robust when

number of qubits increases? 30

k : N � k

pADc (k) = |�/⇥|2/N

N �⇥

State fully separable at this point!

|�/⇥| < 1�

→

31

Does entanglement become more robust with increasing N?

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Is ESD relevant for many particles?

32

Is ESD relevant for many particles?

Λ p( ) ∼ exp −pN( )Λ 0( )

Generalization: Graph states! At each vertex, place a qubit in

the state ! Apply control-Z gate between two

connected vertices:

! Universal states for measurement-based quantum computation (Raussendorf and Briegel)

0 + 1( ) / 2

0 + 1( ) / 2 ⊗ 0 + 1( ) / 2

→ 00 + 01 + 10 − 11( ) / 2

R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001)

Generalization: Graph states

•Any convex bi- or multi-partite entanglement quantifier (no increase under LOCC) •For a large class of quantum channels, and any partition, total entanglement is determined by entanglement of boundary qubits (connected by gray lines) •Lower bounds for entanglement depend only on number of boundary qubits

35

1935

Collaborators: entanglement dynamics

37

Leandro Aolita Fernando de Melo Rafel Chaves Malena Hor-Meyll Alejo Salles

Osvaldo Jiménez-Farías Gabriel Aguillar

Daniel Cavalcanti

Marcelo P. de Almeida

Antonio Acín

Andrea Valdés-Hernandéz

Paulo Souto Ribeiro Stephen Walborn Joe Eberly Xiao-Feng Qian

Collaborators: quantum metrology

Gabriel Bié Marcio Taddei Camille Latune Bruno Escher

Nicim Zagury Ruynet Matos Filho

THANKS!