Group theoretical study of LOCC-detection of maximally entangled - PowerPoint PPT Presentation

Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing MH arXiv:0810.3380 Accepted for publication for NJP. Graduate School of Information Sciences Tohoku University Masahito Hayashi

Contents • Motivation • Formulation • Testing of binomial distributions • Global tests • A-B locality • Two different information sources • A-B locality and sample locality

Contents • Motivation • Formulation • Testing of binomial distributions • Global tests • A-B locality • Two different information sources • A-B locality and sample locality

Problem • You find a Bell pair source in a shop. ? • Shall you buy this? Real Bell pair source ? Precision? Bell pair generator How to check ? JPY 1,000,000 Tomography?, witness? Bell’s inequality? Coincidence count? By K. Matsumoto NO .. !!! NO .. !!! PLEASE USE OUR METHOD PLEASE USE OUR METHOD

Problems with existing methods Careless treatment of the error. Often, the conclusion is meaningless Here, we introduce notion of hypothesis test which is used in statistics. These methods are not optimal. There are more precise methods, which is easy to implement We search for optimal test. Conventional visibility and witness have bias. (The quality depends not only on the fidelity but also on the angle.) We propose symmetric testing.

Who needs precise and efficient test? Huge number of data Even poor test gives is available. a precise decision! However, this is not always true. 1. Bell pair sharing in the long distance (repeater, entanglement distillation) 2. At the development stage, some new Bell pair sources may be very unstable. Huge number Efficient test Poor test gives of data is needed! a bad decision! is not available.

Contents • Motivation • Formulation • Testing of binomial distributions • Global tests • A-B locality • Two different information sources • A-B locality and sample locality

Quantum Hypothesis Testing I • Two Hypotheses ρ ∈ S H : null hypothesis 0 0 ρ ∈ S H : alternative hypothesis 1 1 { } = − • Two-valued measurement M T I , T T : Accept H 0 − I T : Accept H (Reject H ) 1 0 • Two error probabilities First error probability ( ) α ρ − ρ ρ ∈ � S T , 1 Tr T ( ) 0 Second error probability ( ) β ρ ρ ρ ∈ � S T , Tr T ( ) 1

Quantum Hypothesis Testing II α T • Test is called level- ( ) α ρ ≤ α ∀ ∈ S ρ T , if { } ( ) 0 ≤ ≤ ∀ ∈ ρ α ρ ≤ α � T T S S T 0 T I , , T , α S , 0 ( ) ( ) 0 β ρ β ρ � S S , , min T α 0 ∈ T T α S , 0 α T • Test is called a UMP level- test ( ) ( ) β ρ = β ρ ∀ ∈ ρ , , S S S if T , α 0 1 α T • Test is called a UMP level- test C C , ( ) ( ) 1 2 β ρ = β ρ ∀ ∈ ρ C , , S S S T , 2 if α , C 0 1 1 ⎧ ⎫ ⎪ ⎪ T satisfies ( ) ( ) β ρ β ρ � C ⎨ ⎬ S S , , min T 2 α , C 0 ⎪ ⎪ ∈ C C and ⎩ ⎭ T 1 T α S , 0 1 2 C C : conditions , 1 2

Our setting • Is the state close to the maximally entangled ∑ − d 1 φ H � 0 state on ? i i A B , = A B , A B ρ = σ ⊗ i 0 n • We assume that i.i.d. condition i.e., n • Our hypotheses are { } σ ∈ σ − φ σ φ ≤ ε � 0 0 S H : 1 ≤ ε 0 A B , A B , σ ∈ c S H : ≤ ε 1 α • The set of level- tests { } ⊗ ≤ ≤ ∀ σ ∈ − σ ≤ α � n n T T S S T 0 T I , ,1 Tr T α ≤ ε ≤ ε ,

Group invariance on H A B , • U(1)-action ( ) θ φ φ + − φ φ � i 0 0 0 0 U e I θ A B , A B , A B , A B , • SU(d)-action ⊗ ∀ ∈ � U g ( ) g g , g SU d ( ) • SU(d)xU(1)-action θ ∀ θ ∈ × � U g ( , ) U g U ( ) , ( , ) g SU d ( ) U (1) θ • U(d 2 -1)-action ( ) φ φ + − φ φ � 0 0 0 0 V g ( ) g I , A B , A B , A B , A B , ∀ ∈ − 2 g U d ( 1)

Locality restriction ⎧ ⎫ ( ) T G : -inv. ( ) ⊗ β ε σ β σ n � C ⎨ ⎬ , min T α , , n G ∈ ⎩ T satisfies C ⎭ T T α S ≤ ε , = × − 2 G U (1), SU d ( ), SU d ( ) U (1), U d ( 1) where ∅ : No condition S A B ( , ) : A B - separable � L A ( B ) : 2-way LOCC → → L A ( B ) :1-way LOCC A B … … … … S A ( , , A B , , , B ) :separable on A , , A B , , , B 1 n 1 n 1 n 1 n … … L A ( , , A B , , , B ) : 1 n 1 n … … 2-way LOCC on A , , A B , , , B 1 n 1 n … → … L A ( , , A B , , B ) : 1 n 1 n … → … 1-way LOCC A , , A B , , B 1 n 1 n

Contents • Motivation • Formulation • Testing of binomial distributions • Global tests • A-B locality • Two different information sources • A-B locality and sample locality

Binomial distribution k • The data obeys the distribution ⎛ ⎞ n H H − − � n n k k ⎜ ⎟ P ( ) k (1 p ) p 0 1 ε p ⎝ ⎠ p k 0 � • The test is described by a map from T … {0,1, , } n [0,1] to { } � � β ε ∀ ∈ ε − ≤ α � n n n ( q ) min P ( ) T p [0, ],1 P ( ) T α q p � T ∑ � � n � n n P ( ) T P ( ) ( ) k T k = p p k 0 � n T ε α • The test (will be fined in the next slide) , � β = β ε ∀ > ε n n n ( T , P ) ( q ), p . is UMP test, i.e., ε α α , q

� n Definition of T ε α , ⎧ < n 1 if k l ε α H H , ⎪ � 0 1 γ = � n ⎨ n n T ( ) k if k l n k ε α ε α ε α 0 l ε α , , , ⎪ , > n 0 if k l , ⎩ ε α , γ γ n n n l ε α where and are defined as H is supported with pro ε α ε α , , , 0 − n n l 1 l ε α ε α , , ∑ ∑ < − α ≤ n n P ( ) k 1 P ( ), k ε ε = = k 0 k 0 − n l 1 ε α , ∑ γ = − α − n n n n P ( l ) 1 P ( ). k ε α ε ε α ε , , = k 0

Asymptotic theory (small deviation) • When the true parameter is close to 0, the distribution goes to Poisson distribution. k t − = � n t lim P ( ) k P k ( ) e t n / t →∞ k ! n β δ = β δ n lim ( / n t '/ ) n ( t ') α α →∞ n ⎧ ⎫ ∀ ∈ δ ⎪ ⎪ t [0, ], � β δ � ⎨ ⎬ ( t ') min P T ( ) � α − ≤ α t ' � ⎪ ⎪ 1 P T ( ) ⎩ ⎭ T t

Contents • Motivation • Formulation • Testing of binomial distributions • Global tests • A-B locality • Two different information sources • A-B locality and sample locality

Global Test ( samples) n Theorem β ε σ = β ε n ( ) ( p ) α α , , n G ( ) ( ) σ ⊗ = β φ φ ε n 0 0 n T , , α A B , A B , = × − 2 G U (1), SU d ( ) U (1), U d ( 1) where − n l 1 ε α , ( ) ∑ ε + γ � n n n n T T , P ( ) T P ( ) T α ε α n k , l ε α , = k 0 − ⊗ ⊗ − ⊗ ⊗ ⊗ � � � n P ( ) T ( I T ) ( I T ) T T �� �� � ���� ���� � k k − n k � + ⊗ ⊗ ⊗ − ⊗ ⊗ − � � + T T ( I T ) ( I T ). �� �� � ���� ���� � k − n k

Contents • Motivation • Formulation • Testing of binomial distributions • Global tests • A-B locality • Two different information sources • A-B locality and sample locality

A-B locality (One sample) { } = M p u u • For a rank-one POVM i i i ∑ ⊗ ⊗ � the test T M ( ) p u u u u i i i i i i is constructed as follows: { } = M p u u 1. Alice performs POVM . i i i 2. Bob performs two-valued POVM { } − u u , I u u i i i i u u 3. If is observed, accept. i i Otherwise, reject

A-B locality (One sample) 1 M • Covariant POVM cov ϕ ϕ ϕ ν ϕ � 1 M cov ( d ) d ( d ) ν where is invariant measure. 1 M • The POVM can be realized by randomly cov choosing measuring basis = φ φ 1 0 0 T M ( ) cov A B , A B , ( ) + + − φ φ 0 0 1 ( d 1) I A B , A B , σ = − + 1 Tr ( T M ) 1 dp ( d 1), cov − φ σ φ � 0 0 p 1 A B . A B . − ε + � 1, A B 1 1 T T T M ( ( ), d ( d 1)) • The test ε α α , cov α is level- .

A-B locality (One sample) − β ε σ = σ C 1, A B ( ) Tr T α ε α ,1, G , ⎧ ε ε ⎛ ⎞ ⎛ ⎞ d dp d α ≤ α ⎜ ⎟ ⎜ ⎟ ⎪ (1- ) 1- 1- if + + ⎪ + ⎝ ⎠ ⎝ ⎠ d 1 d 1 d 1 ⎨ = α ε ⎪ p d − > α 1 if ⎪ ⎩ ε + d 1 = × − where 2 G SU d ( ), SU d ( ) U (1), U d ( 1) = → � C S A B ( , ), ( L A B ), ( L A B )