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Entanglement negativity as a universal non-Markovianity witness Jan Koody nski, Swapan Rana, Alexander Streltsov Centre for Quantum Optical Technologies IRAU, Centre of New Technologies, University of Warsaw, Poland Symposium on

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SLIDE 1

Entanglement negativity as a universal non-Markovianity witness

Jan Kołody´ nski, Swapan Rana, Alexander Streltsov

Centre for Quantum Optical Technologies IRAU, Centre of New Technologies, University of Warsaw, Poland

Symposium on Mathematical Physics, Toru´ n June 18, 2019

1 / 18

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SLIDE 2

Outline

1

Witnessing non-Markovianity with contractive functions

2

Witnessing non-Markovianity with entanglement

3

Example: eternally non-Markovian dynamics

2 / 18

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SLIDE 3

Outline

1

Witnessing non-Markovianity with contractive functions

2

Witnessing non-Markovianity with entanglement

3

Example: eternally non-Markovian dynamics

3 / 18

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SLIDE 4

Witnessing non-Markovianity with contractive functions

  • Consider two-state function f(ρ, σ) such that

f(Λ[ρ], Λ[σ]) ≤ f(ρ, σ) for all CPTP maps and all states ρ, σ

4 / 18

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SLIDE 5

Witnessing non-Markovianity with contractive functions

  • Consider two-state function f(ρ, σ) such that

f(Λ[ρ], Λ[σ]) ≤ f(ρ, σ) for all CPTP maps and all states ρ, σ

  • Examples: trace distance, quantum relative entropy

4 / 18

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SLIDE 6

Witnessing non-Markovianity with contractive functions

  • Consider two-state function f(ρ, σ) such that

f(Λ[ρ], Λ[σ]) ≤ f(ρ, σ) for all CPTP maps and all states ρ, σ

  • Examples: trace distance, quantum relative entropy
  • Markovian dynamics Λt = Vt,s ◦ Λs lead to monotonic

decrease of f for all 0 ≤ s ≤ t: f(Λt[ρ], Λt[σ]) = f(Vt,s ◦ Λs[ρ], Vt,s ◦ Λs[σ])

≤ f(Λs[ρ], Λs[σ])

4 / 18

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SLIDE 7

Witnessing non-Markovianity with contractive functions

  • Consider two-state function f(ρ, σ) such that

f(Λ[ρ], Λ[σ]) ≤ f(ρ, σ) for all CPTP maps and all states ρ, σ

  • Examples: trace distance, quantum relative entropy
  • Markovian dynamics Λt = Vt,s ◦ Λs lead to monotonic

decrease of f for all 0 ≤ s ≤ t: f(Λt[ρ], Λt[σ]) = f(Vt,s ◦ Λs[ρ], Vt,s ◦ Λs[σ])

≤ f(Λs[ρ], Λs[σ])

  • Witness of non-Markovianity: d

dt f(Λt[ρ], Λt[σ]) > 0

4 / 18

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SLIDE 8

Witnessing non-Markovianity with contractive functions

Consider P-divisible dynamics Λt = Vt,s ◦ Λt such that Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT] (1)

5 / 18

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SLIDE 9

Witnessing non-Markovianity with contractive functions

Consider P-divisible dynamics Λt = Vt,s ◦ Λt such that Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT] (1) Theorem: For any non-Markovian evolution Λt = Vt,s ◦ Λs with Vt,s fulfilling Eq. (1) it holds that:

d dt f(Λt[ρ], Λt[σ]) ≤ 0

for any contractive function f(ρ, σ) and any single-qubit states ρ and σ.

5 / 18

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SLIDE 10

Witnessing non-Markovianity with contractive functions

Consider P-divisible dynamics Λt = Vt,s ◦ Λt such that Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT] (1) Theorem: For any non-Markovian evolution Λt = Vt,s ◦ Λs with Vt,s fulfilling Eq. (1) it holds that:

d dt f(Λt[ρ], Λt[σ]) ≤ 0

for any contractive function f(ρ, σ) and any single-qubit states ρ and σ.

→contractive functions of two single-qubit states cannot witness all

non-Markovianity

5 / 18

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SLIDE 11

Proof of the theorem

  • For any two single-qubit states ρ and σ there exists a unitary

U such that

ρT = UρU†, σT = UσU†.

6 / 18

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SLIDE 12

Proof of the theorem

  • For any two single-qubit states ρ and σ there exists a unitary

U such that

ρT = UρU†, σT = UσU†.

  • ⇒For any two single-qubit states ρ and σ there exists a CPTP

map Φt,s such that Vt,s[ρ] = Φt,s[ρ], Vt,s[σ] = Φt,s[σ], where Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT],

Φt,s[ρ] = pE1 [ρ] + (1 − p)E2

  • UρU†

.

6 / 18

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SLIDE 13

Proof of the theorem

  • For any two single-qubit states ρ and σ there exists a unitary

U such that

ρT = UρU†, σT = UσU†.

  • ⇒For any two single-qubit states ρ and σ there exists a CPTP

map Φt,s such that Vt,s[ρ] = Φt,s[ρ], Vt,s[σ] = Φt,s[σ], where Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT],

Φt,s[ρ] = pE1 [ρ] + (1 − p)E2

  • UρU†

.

  • Combining these results, we obtain:

f (Λt [ρ] , Λt [σ]) = f (Vt,s ◦ Λs [ρ] , Vt,s ◦ Λs [σ])

= f (Φt,s ◦ Λs [ρ] , Φt,s ◦ Λs [σ]) ≤ f (Λs [ρ] , Λs [σ])

6 / 18

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SLIDE 14

Outline

1

Witnessing non-Markovianity with contractive functions

2

Witnessing non-Markovianity with entanglement

3

Example: eternally non-Markovian dynamics

7 / 18

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Quantifying entanglementa

Postulates on entanglement monotones E:

aVedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78, 2275 (1997) bVidal and Werner, Phys. Rev. A 65, 032314 (2002) 8 / 18

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SLIDE 16

Quantifying entanglementa

Postulates on entanglement monotones E:

  • EA|B(ρAB) ≥ 0 with equality on non-entangled (separable)

states ρAB

sep = i piρA i ⊗ ρB i

aVedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78, 2275 (1997) bVidal and Werner, Phys. Rev. A 65, 032314 (2002) 8 / 18

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SLIDE 17

Quantifying entanglementa

Postulates on entanglement monotones E:

  • EA|B(ρAB) ≥ 0 with equality on non-entangled (separable)

states ρAB

sep = i piρA i ⊗ ρB i

  • Monotonicity under local operations and classical

communication: EA|B(ΛLOCC[ρAB]) ≤ EA|B(ρAB)

aVedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78, 2275 (1997) bVidal and Werner, Phys. Rev. A 65, 032314 (2002) 8 / 18

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Quantifying entanglementa

Postulates on entanglement monotones E:

  • EA|B(ρAB) ≥ 0 with equality on non-entangled (separable)

states ρAB

sep = i piρA i ⊗ ρB i

  • Monotonicity under local operations and classical

communication: EA|B(ΛLOCC[ρAB]) ≤ EA|B(ρAB)

  • Entanglement negativityb:

EA|B(ρAB) = ||ρTB||1 − 1 2 with trace norm ||M||1 = Tr

√

M†M and partial transpose TB

aVedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78, 2275 (1997) bVidal and Werner, Phys. Rev. A 65, 032314 (2002) 8 / 18

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Witnessing non-Markovianity with entanglementa

aRivas, Huelga, Plenio, Phys. Rev. Lett. 105, 050403 (2010) bDe Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99, 012303 (2019) 9 / 18

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SLIDE 20

Witnessing non-Markovianity with entanglementa

Local Markovian dynamics Λt = Vt,s ◦ Λs lead to monotonic de- crease of entanglement: EA|B(ΛA

t ⊗ 1B[ρAB]) = EA|B(VA t,s ◦ ΛA s ⊗ 1B[ρAB])

≤ EA|B(ΛA

s ⊗ 1B[ρAB])

for 0 ≤ s ≤ t

aRivas, Huelga, Plenio, Phys. Rev. Lett. 105, 050403 (2010) bDe Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99, 012303 (2019) 9 / 18

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SLIDE 21

Witnessing non-Markovianity with entanglementa

Witness of non-Markovianitya:

d dt EA|B(ΛA

t ⊗ 1B[ρAB]) > 0

for some entanglement monotone E and some t

aRivas, Huelga, Plenio, Phys. Rev. Lett. 105, 050403 (2010) bDe Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99, 012303 (2019) 9 / 18

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SLIDE 22

Witnessing non-Markovianity with entanglementa

  • Witness not universalb: there exist non-Markovian evolutions

with d

dt EA|B(ΛA t ⊗ 1B[ρAB]) ≤ 0 for all t

  • Example: Λt =

       Et for t ≤ 1 ˜ Et−1 ◦ E1 for t > 1

with Markovian evolution Et s.t. E1 is entanglement breaking

aRivas, Huelga, Plenio, Phys. Rev. Lett. 105, 050403 (2010) bDe Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99, 012303 (2019) 9 / 18

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SLIDE 23

Witnessing non-Markovianity with entanglement

Extension to tripartite setting:

d dt EAB|C(ΛA

t ⊗ 1BC[ρABC]) > 0

potentially universal witness of non-Markovianity?

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

10 / 18

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SLIDE 24

Witnessing non-Markovianity with entanglement

Extension to tripartite setting:

d dt EAB|C(ΛA

t ⊗ 1BC[ρABC]) > 0

potentially universal witness of non-Markovianity? Theorema: For any invertible non-Markovian evolution Λt there exists a quantum state ρABC such that

d dt EAB|C(ΛA

t ⊗ 1BC[ρABC]) > 0

for some t > 0. For single-qubit evolutions the statement also holds for non-invertible dynamics.

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

10 / 18

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SLIDE 25

Proof of the theorema

  • Consider the initial state

ρABC = p1ρAB1

1

⊗ |Ψ+Ψ+|

B2C + p2ρAB1 2

⊗ |Ψ−Ψ−|B2C

with |Ψ± = (|01 ± |10)/

√

2

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

11 / 18

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SLIDE 26

Proof of the theorema

  • Consider the initial state

ρABC = p1ρAB1

1

⊗ |Ψ+Ψ+|

B2C + p2ρAB1 2

⊗ |Ψ−Ψ−|B2C

with |Ψ± = (|01 ± |10)/

√

2

  • The time-evolved state takes the form

τABC

t

= p1ΛA

t

  • ρAB1

1

  • ⊗|Ψ+Ψ+|

B2C +p2ΛA t

  • ρAB1

2

  • ⊗|Ψ−Ψ−|B2C

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

11 / 18

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SLIDE 27

Proof of the theorema

  • Consider the initial state

ρABC = p1ρAB1

1

⊗ |Ψ+Ψ+|

B2C + p2ρAB1 2

⊗ |Ψ−Ψ−|B2C

with |Ψ± = (|01 ± |10)/

√

2

  • The time-evolved state takes the form

τABC

t

= p1ΛA

t

  • ρAB1

1

  • ⊗|Ψ+Ψ+|

B2C +p2ΛA t

  • ρAB1

2

  • ⊗|Ψ−Ψ−|B2C
  • Partially transposed state:

τTC

t

= 1

2ΛA

t

  • p1ρAB1

1

+ p2ρAB1

2

  • ⊗
  • |0101|B2C + |1010|B2C

+ 1

2ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • ⊗
  • |Φ+Φ+|

B2C − |Φ−Φ−|B2C

with |Φ± = (|00 ± |11)/

√

2

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

11 / 18

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SLIDE 28

Proof of the theorema

  • Partially transposed state:

τTC

t

= 1

2ΛA

t

  • p1ρAB1

1

+ p2ρAB1

2

  • ⊗
  • |0101|B2C + |1010|B2C

+ 1

2ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • ⊗
  • |Φ+Φ+|

B2C − |Φ−Φ−|B2C

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

12 / 18

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SLIDE 29

Proof of the theorema

  • Partially transposed state:

τTC

t

= 1

2ΛA

t

  • p1ρAB1

1

+ p2ρAB1

2

  • ⊗
  • |0101|B2C + |1010|B2C

+ 1

2ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • ⊗
  • |Φ+Φ+|

B2C − |Φ−Φ−|B2C

  • Evaluating trace norm of τTC

t :

  • τTC

t

  • 1 = 1 +
  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

12 / 18

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SLIDE 30

Proof of the theorema

  • Partially transposed state:

τTC

t

= 1

2ΛA

t

  • p1ρAB1

1

+ p2ρAB1

2

  • ⊗
  • |0101|B2C + |1010|B2C

+ 1

2ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • ⊗
  • |Φ+Φ+|

B2C − |Φ−Φ−|B2C

  • Evaluating trace norm of τTC

t :

  • τTC

t

  • 1 = 1 +
  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1
  • Negativity of τABC

t

: EAB|C

τABC

t

  • =

||τTC

t ||1 − 1

2

= 1

2

  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

12 / 18

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SLIDE 31

Proof of the theorema

  • Negativity of τABC

t

: EAB|C

τABC

t

  • =

||τTC

t ||1 − 1

2

= 1

2

  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

bChru´

sci´ nski, Kossakowski, Rivas, Phys. Rev. A 83, 052128 (2011)

cBylicka, Johansson, Acín, Phys. Rev. Lett. 118, 120501 (2017) dChakraborty and Chru´

sci´ nski, Phys. Rev. A 99, 042105 (2019)

13 / 18

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SLIDE 32

Proof of the theorema

  • Negativity of τABC

t

: EAB|C

τABC

t

  • =

||τTC

t ||1 − 1

2

= 1

2

  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1
  • For any invertible non-Markovian dynamics Λt there exist

probabilities pi and states ρi such thatbc

d dt

  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1 > 0

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

bChru´

sci´ nski, Kossakowski, Rivas, Phys. Rev. A 83, 052128 (2011)

cBylicka, Johansson, Acín, Phys. Rev. Lett. 118, 120501 (2017) dChakraborty and Chru´

sci´ nski, Phys. Rev. A 99, 042105 (2019)

13 / 18

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SLIDE 33

Proof of the theorema

  • Negativity of τABC

t

: EAB|C

τABC

t

  • =

||τTC

t ||1 − 1

2

= 1

2

  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1
  • For any invertible non-Markovian dynamics Λt there exist

probabilities pi and states ρi such thatbc

d dt

  • ΛA

t

  • p1ρAB1

1

− p2ρAB1

2

  • 1 > 0
  • For single-qubit dynamics Λt the statement holds also for

non-invertible evolutionsd

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

bChru´

sci´ nski, Kossakowski, Rivas, Phys. Rev. A 83, 052128 (2011)

cBylicka, Johansson, Acín, Phys. Rev. Lett. 118, 120501 (2017) dChakraborty and Chru´

sci´ nski, Phys. Rev. A 99, 042105 (2019)

13 / 18

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SLIDE 34

Outline

1

Witnessing non-Markovianity with contractive functions

2

Witnessing non-Markovianity with entanglement

3

Example: eternally non-Markovian dynamics

14 / 18

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SLIDE 35

Example: eternally non-Markovian dynamics

  • Eternally non-Markovian dynamicsa:

dρ(t) dt =

3

  • i=1

γi(t) [σiρ(t)σi − ρ(t)]

(2) with γ1 = γ2 = α c

2, γ3(t) = −α c 2 tanh(ct) with α ≥ 1 and c > 0

aHall, Cresser, Li, Andersson, Phys.Rev. A 89, 042120 (2014) bKołody´

nski, Rana, Streltsov, arXiv:1903.08663

15 / 18

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SLIDE 36

Example: eternally non-Markovian dynamics

  • Eternally non-Markovian dynamicsa:

dρ(t) dt =

3

  • i=1

γi(t) [σiρ(t)σi − ρ(t)]

(2) with γ1 = γ2 = α c

2, γ3(t) = −α c 2 tanh(ct) with α ≥ 1 and c > 0

  • Evolution (2) can be written as Λt = Vt,s ◦ Λs with

Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT]

aHall, Cresser, Li, Andersson, Phys.Rev. A 89, 042120 (2014) bKołody´

nski, Rana, Streltsov, arXiv:1903.08663

15 / 18

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SLIDE 37

Example: eternally non-Markovian dynamics

  • Eternally non-Markovian dynamicsa:

dρ(t) dt =

3

  • i=1

γi(t) [σiρ(t)σi − ρ(t)]

(2) with γ1 = γ2 = α c

2, γ3(t) = −α c 2 tanh(ct) with α ≥ 1 and c > 0

  • Evolution (2) can be written as Λt = Vt,s ◦ Λs with

Vt,s[ρ] = pE1[ρ] + (1 − p)E2[ρT] Non-Markovianity of (2) cannot be witnessed by contractive func- tions f(ρ, σ) of single-qubit states and bipartite negativityb:

d dt EA|B(ΛA

t ⊗ 1B[ρAB]) ≤ 0

aHall, Cresser, Li, Andersson, Phys.Rev. A 89, 042120 (2014) bKołody´

nski, Rana, Streltsov, arXiv:1903.08663

15 / 18

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SLIDE 38

Example: eternally non-Markovian dynamics

Lemmaa: Negativity is monotonic under local maps of the form PA ⊗ 1B[ρ] = pEA

1 ⊗ 1B[ρ] + (1 − p)EA 2 ⊗ 1B

ρTA .

(3)

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

16 / 18

slide-39
SLIDE 39

Example: eternally non-Markovian dynamics

Lemmaa: Negativity is monotonic under local maps of the form PA ⊗ 1B[ρ] = pEA

1 ⊗ 1B[ρ] + (1 − p)EA 2 ⊗ 1B

ρTA .

(3) Proof: EA|B PA ⊗ 1B [ρ]

  • = 1

2

  • PA ⊗ 1B

ρTB

  • 1 − 1
  • ≤ 1

2

  • p
  • EA

1 ⊗ 1B

ρTB

  • 1 + (1 − p)
  • EA

2 ⊗ 1B

ρTAB

  • 1 − 1
  • = p

2

  • EA

1 ⊗ 1B

ρTB

  • 1 − 1
  • = pEA|B

EA

1 ⊗ 1B [ρ]

  • ≤ EA|B(ρ)

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

16 / 18

slide-40
SLIDE 40

Example: eternally non-Markovian dynamics

Lemmaa: Negativity is monotonic under local maps of the form PA ⊗ 1B[ρ] = pEA

1 ⊗ 1B[ρ] + (1 − p)EA 2 ⊗ 1B

ρTA .

(3) Proof: EA|B PA ⊗ 1B [ρ]

  • = 1

2

  • PA ⊗ 1B

ρTB

  • 1 − 1
  • ≤ 1

2

  • p
  • EA

1 ⊗ 1B

ρTB

  • 1 + (1 − p)
  • EA

2 ⊗ 1B

ρTAB

  • 1 − 1
  • = p

2

  • EA

1 ⊗ 1B

ρTB

  • 1 − 1
  • = pEA|B

EA

1 ⊗ 1B [ρ]

  • ≤ EA|B(ρ)

⇒ Any local dynamics ΛA

t = VA t,s ◦ ΛA s with VA t,s of the form (3) fulfills

d dt EA|B(ΛA

t ⊗ 1B[ρAB]) ≤ 0

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

16 / 18

slide-41
SLIDE 41

Example: eternally non-Markovian dynamicsa

Non-Markovianity of eternally non-Markovian dynamics can be witnessed in the tripartite setting by EAB|C(ΛA

t ⊗ 1BC[ρABC])

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

17 / 18

slide-42
SLIDE 42

Summary

Entanglement negativity in tripartite settinga: universal non-Markovianity witness for

  • invertible dynamics in any dimension
  • all (also non-invertible) qubit dynamics

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

18 / 18

slide-43
SLIDE 43

Summary

Entanglement negativity in tripartite settinga: universal non-Markovianity witness for

  • invertible dynamics in any dimension
  • all (also non-invertible) qubit dynamics

Eternally non-Markovian qubit dynamics: non-Markovianity cannot be witnessed by

  • negativity in bipartite setting
  • any contractive function of qubit states f(ρ, σ)

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

18 / 18

slide-44
SLIDE 44

Summary

Entanglement negativity in tripartite settinga: universal non-Markovianity witness for

  • invertible dynamics in any dimension
  • all (also non-invertible) qubit dynamics

Eternally non-Markovian qubit dynamics: non-Markovianity cannot be witnessed by

  • negativity in bipartite setting
  • any contractive function of qubit states f(ρ, σ)

Open question: can entanglement monotones universally witness non-Markovianity

  • f all evolutions, incuding non-invertible dynamics beyond qubits?

aKołody´

nski, Rana, Streltsov, arXiv:1903.08663

18 / 18