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Quantifying the Unextendibility of Entanglement Kun WANG Shenzhen - - PowerPoint PPT Presentation

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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)

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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

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Resource theory of unextendibility

1

Free states: k-extendible quantum states1 A quantum state ρAB is k-extendible w.r.t. system B if

▶ State extension: exists a state σAB1···Bk such that

TrB2···Bk σAB1···Bk = ρAB

▶ Permutation invariance: the extension state σAB1···Bk is invariant w.r.t.

permutations on B systems

∞-extendible (separable) · · · 2-extendible all states

  • 1A. C. Doherty et al., Physical Review Letters 88, 187904 (2002), A. C. Doherty et al., Physical Review A 69, 022308 (2004).
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Resource theory of unextendibility (cont.)

2

Free operations: k-extendible quantum channels2 A bipartite quantum channel NAB→A′B′ is k-extendible if

▶ Channel extension: exists a channel MAB1···Bk→AB′ 1···B′ k such that for

arbitrary quantum state ρAB1···Bk: TrB′

2···B′ k MAB1···Bk→AB′ 1···B′ k(ρAB1···Bk) = NAB→A′B′(ρAB) ▶ Permutation covariance: the extension channel MAB1···Bk→AB′ 1···B′ k is

permutation covariant

A B A′ B′

N

≡

A B1 B2 Bk A′ B′

1

M

  • 2E. Kaur et al., Physical Review Letters 123, 070502 (2019).
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Resource theory of unextendibility (cont.)

Resource theory of unextendibility is important! k-extendible states form a complete hierarchy of bipartite quantum states3 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 rates4 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).
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Generalized unextendible entanglement: Motivation

Entanglement monogamy: The more a bipartite state ρAB is entangled, the less each of its individual systems can be entangled with a third party. For a tripartite state ρABB′, ρAB more entangled ⇒ ρAB′ is less entangled The free set of states dependent on ρAB: F(ρAB) :=

  • TrB[ρABB′]
  • ρAB = TrB′[ρABB′]
  • .

If ρAB is 2-extendible, then ρAB ∈ F(ρAB) Otherwise, ρAB is outside of F(ρAB) This distance witnesses unextendibility!

ρAB F(ρAB)

  • 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.

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Generalized unextendible entanglement: Definition

A functional D : S(A) × S(A) → R ∩ {+∞} is a generalized divergence if6 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 ρAB is defined as Eu(ρAB) := 1 2 inf

ρAB′∈F(ρAB) D(ρAB∥ρAB′).

D ρAB F(ρAB)

  • 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).
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Generalized unextendible entanglement: Faithfulness

Proposition 2.

When D is both strongly faithful and continuous, we have Eu(ρAB) = 0 ⇔ ρAB is two-extendible

2-extendible Eu(σAB) = 0 Eu(ρAB) > 0

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Generalized unextendible entanglement: Monotonicity

Proposition 3.

Eu does not increase under two-extendible channels. That is, for arbitrary quantum state ρAB and two-extendible quantum channel NAB→A′B′, Eu(ρAB) ⩾ Eu(NAB→A′B′(ρAB)) Intuitively, free operations cannot increase resource7 The monotonicity holds for arbitrary divergence satisfying data-processing This justifies Eu as valid entanglement measures8

  • 7E. Chitambar, G. Gour, Reviews of Modern Physics 91, 025001 (2019).
  • 8R. Horodecki et al., Reviews of Modern Physics 81, 865 (2009).
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α-unextendible entanglement: Definition

We consider three concrete and widely investigated divergences

1

The Petz-Rényi relative entropy Dα (see Ref. (8))

2

The sandwiched Rényi relative entropy Dα (see Ref. (9, 10))

3

The geometric Rényi relative entropy Dα (see Ref. (11)) The α-unextendible entanglement family is defined as Eu

α(ρAB) := 1

2 inf

ρAB′∈F(ρAB) Dα(ρAB∥ρAB′),

α ∈ [0, 2]

  • Eu

α(ρAB) := 1

2 inf

ρAB′∈F(ρAB)

  • Dα(ρAB∥ρAB′),

α ∈ [1/2, ∞)

  • Eu

α(ρAB) := 1

2 inf

ρAB′∈F(ρAB)

  • Dα(ρAB∥ρAB′),

α ∈ [0, 2] The α-unextendible entanglement satisfies faithfulness and monotonicity

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α-unextendible entanglement: Selective monotonicity

A selective two-extendible operation consists of a set

  • f completely positive maps {N y

AB→A′B′} such that

  • y N y

AB→A′B′ is trace-preserving

Each N y

AB→A′B′ is two-extendible

ρAB p1σ1

A′B′

pyσ|Y|

A′B′

N 1

AB→A′B′

N |Y|

AB→A′B′

Operating on ρAB yields an ensemble {py, σy

A′B′}, where

py := Tr[N y

AB→A′B′(ρAB)],

σy

A′B′ := N y AB→A′B′(ρAB)/py.

Theorem 4.

The α-sandwiched unextendible entanglement does not increase under selective two-extendible channels for α ∈ [1, ∞),

  • Eu

α(ρAB) ⩾

  • y

py Eu

α(σy A′B′). ✔ Similar result holds also for Eu

α(ρAB) and

Eu

α(ρAB) for certain range of α

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α-unextendible entanglement: More properties

The α-unextendible entanglement family satisfies many desirable properties for a reasonable entanglement measure9:

▶ 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).
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Min-unextendible entanglement

Setting D ≡ Dmin (the min-relative entropy10), we get the min-unextendible entanglement Eu

min(ρAB) := 1

2 inf

ρAB′∈F(ρAB) Dmin(ρAB∥ρAB′).

Efficiently computable via semidefinite program (SDP) 2−2Eu

min(ρAB) = max Tr[ΠρABρAB′]

s.t. TrB′ ρABB′ = ρAB, ρABB′ ⩾ 0. Satisfies the additivity property Eu

min (ρA1B1 ⊗ ρA2B2) = Eu min (ρA1B1) + Eu min (ρA2B2) . ✔ 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).
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Probabilistic entanglement distillation overhead

Probabilistic distillation: A (n, m, p) protocol distilling m copies of Bell state from n copies of ρAB with probability p using two-extendible operations11

ρ⊗n

AB

Φ⊗m

2

with prob. p two-extendible operation

DAnBn→A′B′

Probabilistic distillation overhead: Dov(ρAB, m) := inf

n∈N,p∈(0,1]

n p

  • ∃(n, m, p) protocol
  • .

Theorem 5.

Eu(ρAB) lower bounds the distillation overhead: Dov(ρAB, m) ⩾ m/Eu(ρAB).

✔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).

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Perfect entanglement distillation

Perfect distillation: A (n, m) protocol distilling deterministically and perfectly m copies of Bell state from n copies of ρAB using two-extendible operations12

ρ⊗n

AB

Φ⊗m

2

two-extendible operation

DAnBn→A′B′

Perfect distillation rate: Dp(ρAB) := sup

  • lim inf

n→∞

mn n

  • ∃(n, mn) protocol
  • Theorem 6.

Eu

min upper bounds the distillation rate: Dp(ρAB) ≤ Eu min(ρAB). ✔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).

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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

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Thank you for your attention!

See arxiv:1911.07433 for more details.

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Bibliography I

  • A. C. Doherty, P. A. Parrilo, F. M. Spedalieri, Physical Review Letters 88,

187904 (2002).

  • A. C. Doherty, P. A. Parrilo, F. M. Spedalieri, Physical Review A 69, 022308

(2004).

  • E. Kaur, S. Das, M. M. Wilde, A. Winter, Physical Review Letters 123,

070502 (2019).

  • V. Coffman, J. Kundu, W. K. Wootters, Physical Review A 61, 052306

(2000).

  • Y. Polyanskiy, S. Verdú, presented at the 48th Annual Allerton Conference
  • n Communication, Control, and Computing (Allerton), pp. 1327–1333.
  • N. Sharma, N. A. Warsi, Physical Review Letters 110, 080501 (2013).
  • E. Chitambar, G. Gour, Reviews of Modern Physics 91, 025001 (2019).
  • D. Petz, Reports on Mathematical Physics 23, 57–65 (1986).
  • M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel, Journal of

Mathematical Physics 54, 122203 (2013).

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Bibliography II

  • M. M. Wilde, A. Winter, D. Yang, Communications in Mathematical Physics

331, 593–622 (2014).

  • K. Matsumoto, presented at the Nagoya Winter Workshop: Reality and

Measurement in Algebraic Quantum Theory, arXiv:1311.4722, pp. 229–273.

  • R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Reviews of

Modern Physics 81, 865 (2009).

  • N. Datta, IEEE Transactions on Information Theory 55, 2816–2826 (2009).
  • C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, Physical

Review A 54, 3824 (1996). J.-W. Pan, C. Simon, Č. Brukner, A. Zeilinger, Nature 410, 1067–1070 (2001).

  • E. T. Campbell, S. C. Benjamin, Physical Review Letters 101, 130502

(2008).

  • F. Rozpędek, T. Schiet, D. Elkouss, A. C. Doherty, S. Wehner, et al.,

Physical Review A 97, 062333 (2018).

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Bibliography III

  • E. M. Rains, IEEE Transactions on Information Theory 47, 2921–2933

(2001).

  • R. Duan, Y. Feng, Z. Ji, M. Ying, 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).