Entanglement negativity as a universal non-Markovianity witness Jan - PowerPoint PPT Presentation

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

Outline Witnessing non-Markovianity with contractive functions 1 Witnessing non-Markovianity with entanglement 2 Example: eternally non-Markovian dynamics 3 2 / 18

Outline Witnessing non-Markovianity with contractive functions 1 Witnessing non-Markovianity with entanglement 2 Example: eternally non-Markovian dynamics 3 3 / 18

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

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

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 Λ t = V t , s ◦ Λ s lead to monotonic decrease of f for all 0 ≤ s ≤ t : f (Λ t [ ρ ] , Λ t [ σ ]) = f ( V t , s ◦ Λ s [ ρ ] , V t , s ◦ Λ s [ σ ]) ≤ f (Λ s [ ρ ] , Λ s [ σ ]) 4 / 18

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 Λ t = V t , s ◦ Λ s lead to monotonic decrease of f for all 0 ≤ s ≤ t : f (Λ t [ ρ ] , Λ t [ σ ]) = f ( V t , s ◦ Λ s [ ρ ] , V t , s ◦ Λ s [ σ ]) ≤ f (Λ s [ ρ ] , Λ s [ σ ]) • Witness of non-Markovianity: d d t f (Λ t [ ρ ] , Λ t [ σ ]) > 0 4 / 18

Witnessing non-Markovianity with contractive functions Consider P-divisible dynamics Λ t = V t , s ◦ Λ t such that V t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 [ ρ T ] (1) 5 / 18

Witnessing non-Markovianity with contractive functions Consider P-divisible dynamics Λ t = V t , s ◦ Λ t such that V t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 [ ρ T ] (1) Theorem: For any non-Markovian evolution Λ t = V t , s ◦ Λ s with V t , s fulfilling Eq. (1) it holds that: d d t f (Λ t [ ρ ] , Λ t [ σ ]) ≤ 0 for any contractive function f ( ρ, σ ) and any single-qubit states ρ and σ . 5 / 18

Witnessing non-Markovianity with contractive functions Consider P-divisible dynamics Λ t = V t , s ◦ Λ t such that V t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 [ ρ T ] (1) Theorem: For any non-Markovian evolution Λ t = V t , s ◦ Λ s with V t , s fulfilling Eq. (1) it holds that: d d t f (Λ t [ ρ ] , Λ t [ 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

Proof of the theorem • For any two single-qubit states ρ and σ there exists a unitary U such that ρ T = U ρ U † , σ T = U σ U † . 6 / 18

Proof of the theorem • For any two single-qubit states ρ and σ there exists a unitary U such that ρ T = U ρ U † , σ T = U σ U † . • ⇒ For any two single-qubit states ρ and σ there exists a CPTP map Φ t , s such that V t , s [ ρ ] = Φ t , s [ ρ ] , V t , s [ σ ] = Φ t , s [ σ ] , where V t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 [ ρ T ] , � U ρ U † � Φ t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 . 6 / 18

Proof of the theorem • For any two single-qubit states ρ and σ there exists a unitary U such that ρ T = U ρ U † , σ T = U σ U † . • ⇒ For any two single-qubit states ρ and σ there exists a CPTP map Φ t , s such that V t , s [ ρ ] = Φ t , s [ ρ ] , V t , s [ σ ] = Φ t , s [ σ ] , where V t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 [ ρ T ] , � U ρ U † � Φ t , s [ ρ ] = p E 1 [ ρ ] + ( 1 − p ) E 2 . • Combining these results, we obtain: f (Λ t [ ρ ] , Λ t [ σ ]) = f ( V t , s ◦ Λ s [ ρ ] , V t , s ◦ Λ s [ σ ]) = f (Φ t , s ◦ Λ s [ ρ ] , Φ t , s ◦ Λ s [ σ ]) ≤ f (Λ s [ ρ ] , Λ s [ σ ]) 6 / 18

Outline Witnessing non-Markovianity with contractive functions 1 Witnessing non-Markovianity with entanglement 2 Example: eternally non-Markovian dynamics 3 7 / 18

Quantifying entanglement a Postulates on entanglement monotones E : a Vedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78 , 2275 (1997) b Vidal and Werner, Phys. Rev. A 65 , 032314 (2002) 8 / 18

Quantifying entanglement a Postulates on entanglement monotones E : • E A | B ( ρ AB ) ≥ 0 with equality on non-entangled (separable) states ρ AB i p i ρ A i ⊗ ρ B sep = � i a Vedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78 , 2275 (1997) b Vidal and Werner, Phys. Rev. A 65 , 032314 (2002) 8 / 18

Quantifying entanglement a Postulates on entanglement monotones E : • E A | B ( ρ AB ) ≥ 0 with equality on non-entangled (separable) states ρ AB i p i ρ A i ⊗ ρ B sep = � i • Monotonicity under local operations and classical communication: E A | B (Λ LOCC [ ρ AB ]) ≤ E A | B ( ρ AB ) a Vedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78 , 2275 (1997) b Vidal and Werner, Phys. Rev. A 65 , 032314 (2002) 8 / 18

Quantifying entanglement a Postulates on entanglement monotones E : • E A | B ( ρ AB ) ≥ 0 with equality on non-entangled (separable) states ρ AB i p i ρ A i ⊗ ρ B sep = � i • Monotonicity under local operations and classical communication: E A | B (Λ LOCC [ ρ AB ]) ≤ E A | B ( ρ AB ) • Entanglement negativity b : E A | B ( ρ AB ) = || ρ T B || 1 − 1 2 √ with trace norm || M || 1 = Tr M † M and partial transpose T B a Vedral, Plenio, Rippin, Knight, Phys. Rev. Lett. 78 , 2275 (1997) b Vidal and Werner, Phys. Rev. A 65 , 032314 (2002) 8 / 18

Witnessing non-Markovianity with entanglement a a Rivas, Huelga, Plenio, Phys. Rev. Lett. 105 , 050403 (2010) b De Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99 , 012303 (2019) 9 / 18

Witnessing non-Markovianity with entanglement a Local Markovian dynamics Λ t = V t , s ◦ Λ s lead to monotonic de- crease of entanglement: E A | B (Λ A t ⊗ 1 B [ ρ AB ]) = E A | B ( V A t , s ◦ Λ A s ⊗ 1 B [ ρ AB ]) ≤ E A | B (Λ A s ⊗ 1 B [ ρ AB ]) for 0 ≤ s ≤ t a Rivas, Huelga, Plenio, Phys. Rev. Lett. 105 , 050403 (2010) b De Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99 , 012303 (2019) 9 / 18

Witnessing non-Markovianity with entanglement a Witness of non-Markovianity a : d d t E A | B (Λ A t ⊗ 1 B [ ρ AB ]) > 0 for some entanglement monotone E and some t a Rivas, Huelga, Plenio, Phys. Rev. Lett. 105 , 050403 (2010) b De Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99 , 012303 (2019) 9 / 18

Witnessing non-Markovianity with entanglement a • Witness not universal b : there exist non-Markovian evolutions with d d t E A | B (Λ A t ⊗ 1 B [ ρ AB ]) ≤ 0 for all t  E t for t ≤ 1  • Example: Λ t =   ˜  E t − 1 ◦ E 1 for t > 1   with Markovian evolution E t s.t. E 1 is entanglement breaking a Rivas, Huelga, Plenio, Phys. Rev. Lett. 105 , 050403 (2010) b De Santis, Johansson, Bylicka, Bernardes, Acín, PRA 99 , 012303 (2019) 9 / 18

Witnessing non-Markovianity with entanglement Extension to tripartite setting: d d t E AB | C (Λ A t ⊗ 1 BC [ ρ ABC ]) > 0 potentially universal witness of non-Markovianity? a Kołody´ nski, Rana, Streltsov, arXiv:1903.08663 10 / 18

Witnessing non-Markovianity with entanglement Extension to tripartite setting: d d t E AB | C (Λ A t ⊗ 1 BC [ ρ ABC ]) > 0 potentially universal witness of non-Markovianity? Theorem a : For any invertible non-Markovian evolution Λ t there exists a quantum state ρ ABC such that d d t E AB | C (Λ A t ⊗ 1 BC [ ρ ABC ]) > 0 for some t > 0 . For single-qubit evolutions the statement also holds for non-invertible dynamics. a Kołody´ nski, Rana, Streltsov, arXiv:1903.08663 10 / 18

Proof of the theorem a • Consider the initial state B 2 C + p 2 ρ AB 1 ρ ABC = p 1 ρ AB 1 ⊗ | Ψ − �� Ψ − | B 2 C ⊗ | Ψ + �� Ψ + | 1 2 √ with | Ψ ± � = ( | 01 � ± | 10 � ) / 2 a Kołody´ nski, Rana, Streltsov, arXiv:1903.08663 11 / 18

Proof of the theorem a • Consider the initial state B 2 C + p 2 ρ AB 1 ρ ABC = p 1 ρ AB 1 ⊗ | Ψ − �� Ψ − | B 2 C ⊗ | Ψ + �� Ψ + | 1 2 √ with | Ψ ± � = ( | 01 � ± | 10 � ) / 2 • The time-evolved state takes the form B 2 C + p 2 Λ A � ρ AB 1 � � ρ AB 1 � ⊗| Ψ − �� Ψ − | B 2 C τ ABC = p 1 Λ A ⊗| Ψ + �� Ψ + | t t t 1 2 a Kołody´ nski, Rana, Streltsov, arXiv:1903.08663 11 / 18