Quantifying the Unextendibility of Entanglement Kun WANG Shenzhen Institute for Quantum Science and Engineering (SIQSE) Southern University of Science and Technology Joint work with Xin WANG (Baidu) and Mark M. WILDE (LSU) (arxiv:1911.07433) ISIT 2020, Los Angeles, USA 1/19
Resource theory of unextendibility Free states: k -extendible quantum states 1 1 A quantum state ρ AB is k -extendible w.r.t. system B if ▶ State extension: exists a state σ AB 1 ··· B k such that Tr B 2 ··· B k σ AB 1 ··· B k = ρ AB ▶ Permutation invariance: the extension state σ AB 1 ··· B k is invariant w.r.t. permutations on B systems ∞ -extendible 2 -extendible all states · · · (separable) 1A. C. Doherty et al. , Physical Review Letters 88 , 187904 (2002), A. C. Doherty et al. , Physical Review A 69 , 022308 (2004). 2/19
Resource theory of unextendibility (cont.) Free operations: k -extendible quantum channels 2 2 A bipartite quantum channel N AB → A ′ B ′ is k -extendible if ▶ Channel extension: exists a channel M AB 1 ··· B k → AB ′ k such that for 1 ··· B ′ arbitrary quantum state ρ AB 1 ··· B k : k M AB 1 ··· B k → AB ′ k ( ρ AB 1 ··· B k ) = N AB → A ′ B ′ ( ρ AB ) Tr B ′ 2 ··· B ′ 1 ··· B ′ ▶ Permutation covariance: the extension channel M AB 1 ··· B k → AB ′ k is 1 ··· B ′ permutation covariant A ′ A ′ A A N B ′ B ′ B 1 B 1 M ≡ B 2 B k 2E. Kaur et al. , Physical Review Letters 123 , 070502 (2019). 3/19
Resource theory of unextendibility (cont.) Resource theory of unextendibility is important! k -extendible states form a complete hierarchy of bipartite quantum states 3 Resource theory of unextendibility relaxes resource theory of entanglement and thus offers a good approximation This approximation leads to tighter upper bounds on quantum communication rates 4 The question: Is it possible for unextendibility measures to bound other quantum information tasks? Our results: Introduce a family of unextendibility measures and find novel applications in entanglement/secret-key distillation! 3A. C. Doherty et al. , Physical Review Letters 88 , 187904 (2002), A. C. Doherty et al. , Physical Review A 69 , 022308 (2004). 4E. Kaur et al. , Physical Review Letters 123 , 070502 (2019). 4/19
Generalized unextendible entanglement: Motivation Entanglement monogamy: The more a bipartite state ρ AB is entangled, the less each of its individual systems can be entangled with a third party. For a tripartite state ρ ABB ′ , ρ AB more entangled ⇒ ρ AB ′ is less entangled The free set of states dependent on ρ AB : � � � � F ( ρ AB ) := Tr B [ ρ ABB ′ ] � ρ AB = Tr B ′ [ ρ ABB ′ ] . If ρ AB is 2 -extendible, then ρ AB ∈ F ( ρ AB ) ρ AB F ( ρ AB ) Otherwise, ρ AB is outside of F ( ρ AB ) This distance witnesses unextendibility! V. Coffman et al. , Physical Review A 61 , 052306 (2000). Figure credit: H. S. Dhar et al. , in Lectures on General Quantum Correlations and their Applications (Springer, 2017), pp. 23-64. 5/19
inf Generalized unextendible entanglement: Definition A functional D : S ( A ) × S ( A ) → R ∩ { + ∞} is a generalized divergence if 6 D ( ρ ∥ σ ) ⩾ D ( N ( ρ ) ∥N ( σ )) known as the data-processing inequality Definition 1 (Generalized unextendible entanglement). The generalized unextendible entanglement of a bipartite state ρ AB is defined as E u ( ρ AB ) := 1 ρ AB ′ ∈F ( ρ AB ) D ( ρ AB ∥ ρ AB ′ ) . 2 ρ AB F ( ρ AB ) D 6Y. Polyanskiy, S. Verdú, presented at the 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1327–1333, N. Sharma, N. A. Warsi, Physical Review Letters 110 , 080501 (2013). 6/19
Generalized unextendible entanglement: Faithfulness Proposition 2. When D is both strongly faithful and continuous, we have E u ( ρ AB ) = 0 ⇔ ρ AB is two-extendible 2 -extendible E u ( σ AB ) = 0 E u ( ρ AB ) > 0 7/19
Generalized unextendible entanglement: Monotonicity Proposition 3. E u does not increase under two-extendible channels. That is, for arbitrary quantum state ρ AB and two-extendible quantum channel N AB → A ′ B ′ , E u ( ρ AB ) ⩾ E u ( N AB → A ′ B ′ ( ρ AB )) Intuitively, free operations cannot increase resource 7 The monotonicity holds for arbitrary divergence satisfying data-processing This justifies E u as valid entanglement measures 8 7E. Chitambar, G. Gour, Reviews of Modern Physics 91 , 025001 (2019). 8R. Horodecki et al. , Reviews of Modern Physics 81 , 865 (2009). 8/19
inf inf inf α -unextendible entanglement: Definition We consider three concrete and widely investigated divergences The Petz-Rényi relative entropy D α (see Ref. ( 8 )) 1 The sandwiched Rényi relative entropy � D α (see Ref. ( 9 , 10 )) 2 The geometric Rényi relative entropy � D α (see Ref. ( 11 )) 3 The α -unextendible entanglement family is defined as α ( ρ AB ) := 1 E u ρ AB ′ ∈F ( ρ AB ) D α ( ρ AB ∥ ρ AB ′ ) , α ∈ [0 , 2] 2 α ( ρ AB ) := 1 E u � � D α ( ρ AB ∥ ρ AB ′ ) , α ∈ [1/2 , ∞ ) 2 ρ AB ′ ∈F ( ρ AB ) α ( ρ AB ) := 1 � � E u D α ( ρ AB ∥ ρ AB ′ ) , α ∈ [0 , 2] 2 ρ AB ′ ∈F ( ρ AB ) The α -unextendible entanglement satisfies faithfulness and monotonicity 9/19
α -unextendible entanglement: Selective monotonicity A selective two-extendible operation consists of a set N 1 p 1 σ 1 AB → A ′ B ′ A ′ B ′ of completely positive maps {N y AB → A ′ B ′ } such that � y N y ρ AB AB → A ′ B ′ is trace-preserving Each N y AB → A ′ B ′ is two-extendible N |Y| p y σ |Y| AB → A ′ B ′ A ′ B ′ Operating on ρ AB yields an ensemble { p y , σ y A ′ B ′ } , where p y := Tr [ N y σ y A ′ B ′ := N y AB → A ′ B ′ ( ρ AB )] , AB → A ′ B ′ ( ρ AB )/ p y . Theorem 4. The α -sandwiched unextendible entanglement does not increase under selective two-extendible channels for α ∈ [1 , ∞ ) , � � p y � α ( σ y E u E u α ( ρ AB ) ⩾ A ′ B ′ ) . y ✔ Similar result holds also for E u α ( ρ AB ) and � E u α ( ρ AB ) for certain range of α 10/19
α -unextendible entanglement: More properties The α -unextendible entanglement family satisfies many desirable properties for a reasonable entanglement measure 9 : ▶ Normalization , ▶ Convexity , and ▶ Subadditivity It has a simple expression for pure states in terms of Rényi entropy 9R. Horodecki et al. , Reviews of Modern Physics 81 , 865 (2009). 11/19
inf Min-unextendible entanglement Setting D ≡ D min (the min-relative entropy 10 ), we get the min-unextendible entanglement min ( ρ AB ) := 1 E u ρ AB ′ ∈F ( ρ AB ) D min ( ρ AB ∥ ρ AB ′ ) . 2 Efficiently computable via semidefinite program (SDP) min ( ρ AB ) = max Tr [Π ρ AB ρ AB ′ ] 2 − 2 E u s.t. Tr B ′ ρ ABB ′ = ρ AB , ρ ABB ′ ⩾ 0 . Satisfies the additivity property E u min ( ρ A 1 B 1 ⊗ ρ A 2 B 2 ) = E u min ( ρ A 1 B 1 ) + E u min ( ρ A 2 B 2 ) . ✔ Likewise, we can use the max-relative entropy and fidelity to obtain other SDP computable measures 10N. Datta, IEEE Transactions on Information Theory 55 , 2816–2826 (2009). 12/19
inf Probabilistic entanglement distillation overhead Probabilistic distillation: A ( n, m, p ) protocol distilling m copies of Bell state from n copies of ρ AB with probability p using two-extendible operations 11 ρ ⊗ n Φ ⊗ m D A n B n → A ′ B ′ with prob. p AB 2 two-extendible operation Probabilistic distillation overhead: � n � � � � � ∃ ( n, m, p ) protocol D ov ( ρ AB , m ) := . p n ∈ N ,p ∈ (0 , 1] Theorem 5. E u ( ρ AB ) lower bounds the distillation overhead: D ov ( ρ AB , m ) ⩾ m / E u ( ρ AB ) . ✔ This technique can also be used to study probabilistic secret key distillation. 11C. H. Bennett et al. , Physical Review A 54 , 3824 (1996), J.-W. Pan et al. , Nature 410 , 1067–1070 (2001), E. T. Campbell, S. C. Benjamin, Physical Review Letters 101 , 130502 (2008), F. Rozpędek et al. , Physical Review A 97 , 062333 (2018). 13/19
lim inf Perfect entanglement distillation Perfect distillation: A ( n, m ) protocol distilling deterministically and perfectly m copies of Bell state from n copies of ρ AB using two-extendible operations 12 ρ ⊗ n Φ ⊗ m D A n B n → A ′ B ′ AB 2 two-extendible operation Perfect distillation rate: � � � m n � � ∃ ( n, m n ) protocol D p ( ρ AB ) := sup n n →∞ Theorem 6. E u min upper bounds the distillation rate: D p ( ρ AB ) ≤ E u min ( ρ AB ) . ✔ This technique can also be used to study perfect secret key distillation. 12E. M. Rains, IEEE Transactions on Information Theory 47 , 2921–2933 (2001), R. Duan et al. , Physical Review A 71 , 022305 (2005), W. Matthews, A. Winter, Physical Review A 78 , 012317 (2008), X. Wang, R. Duan, Physical Review A 94 , 050301 (2016), X. Wang, R. Duan, Physical Review A 95 , 062322 (2017). 14/19
Conclusions What we have done? A systematic way of quantifying the unextendibility of bipartite states Introduced a family of measures called unextendible entanglement These measures bear nice properties such as (selective) monotonicity, normalization, additivity, reduction on pure states, et al. These measures find novel applications in distillation tasks Research directions: Generalize these entanglement measures to the k -extendibility regime More applications of these divergence-based entanglement measures 15/19
Thank you for your attention! See arxiv:1911.07433 for more details. 16/19
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