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Quantum Stein’s lemma for correlated states and asymptotic entanglement transformations Fernando G.S.L. Brandão and Martin B. Plenio arXiv 0810.0026 (chapter 4) arXiv 0902.XXXX QIP 2009, Santa Fe Multipartite entangled states Non-entangled states p j ... 1 j j Can be created by LOCC (Local Operations and Classical Communication) k j Multipartite entangled states LOCC asymptotic entanglement conversion r is an achievable rate if A B lim min || ( nr ) n ||1 0 n LOCC LOCC optimal conversion rate R( ) inf r : achievable n LOCC n n ( r o ( n )) Asymptotically non-entangled states is asymptotically non-entangled if there is a state such that R( ) 0 Is every entangled state asymptotically entangled? • For distillable states: R( EPR ) 0 Hence, they must be asymptotically entangled • For bound entangled states, (Horodecki, Horodecki, Horodecki 98) Are they asymptotically entangled? R( EPR ) 0 Asymptotically non-entangled states is asymptotically non-entangled if there is a state such that R( ) 0 Is every entangled state asymptotically entangled? • For distillable states: R( EPR ) 0 Hence, they must be asymptotically entangled • For bound entangled states, (Horodecki, Horodecki, Horodecki 98) Are they asymptotically entangled? R( EPR ) 0 Asymptotically non-entangled states • We can use entanglement measures to analyse the problem: • Let r be an achievable rate: lim min || ( nr ) n ||1 0 n LOCC 1 1 1 nr nr E ( ) E (( )) E ( n ) (|| ( nr ) n ||1 ) n n n LOCC monotonicity Asymptotic continuity 1 rE ( ) E ( ), E ( ) : lim E ( n ) n n R R( ) E ( ) / E ( ) • If E ( ) 0 , then is asymptotically entangled Asymptotically non-entangled states • We can use entanglement measures to analyse the problem: • Let r be an achievable rate: lim min || ( nr ) n ||1 0 n LOCC 1 1 1 nr nr E ( ) E (( )) E ( n ) (|| ( nr ) n ||1 ) n n n LOCC monotonicity Asymptotic continuity 1 rE ( ) E ( ), E ( ) : lim E ( n ) n n R R( ) E ( ) / E ( ) • If E ( ) 0 , then is asymptotically entangled Asymptotically non-entangled states • Every bipartite entangled state is asymptotically entangled (Yang, Horodecki, Horodecki, Synak-Radtke 05) • Entanglement cost: EF ( n ) R(EPR ) EC ( ) lim n n Bennett, DiVincenzo, Smolin, Wootters 96, Hayden, Horodecki, Terhal 00 • EC ( ) 0 for every bipartite entangled states Asymptotically non-entangled states • This talk: Every multipartite entangled state is asymptotically entangled • The multipartite case is not implied by the bipartite: there are entangled states which are separable across any bipartition ex: State derived from the Shift Unextendible-Product-Basis (Bennett, DiVincenzo, Mor, Shor, Smolin, Terhal 98) Asymptotically non-entangled states • Regularized relative entropy of entanglement: (Vedral and Plenio 98, Vollbrecht and Werner 00) ER ( ) min S ( || ) min tr( (log log )) S n E ( ) R ER ( ) lim n n S R( ) ER ( ) / ER ( ) ER ( ) 0 for every entangled state E We show that by linking R to a certain quantum • Rest of the talk: • hypothesis testing problem Same result has been found by Marco Piani, with different techniques Quantum Hypothesis Testing Given n copies of a quantum state, with the promise that it is described either by or , determine the identity of the state. Measure two outcome POVM . An , I An Error probabilities Alternative Null - Type I error: hypothesis hypothesis n ( An ) : tr( ( I An )) n - Type II error: n ( An ) : tr( n An ) Quantum Hypothesis Testing Given n copies of a quantum state, with the promise that it is described either by or , determine the identity of the state. Measure two outcome POVM Error probabilities state is - Type I The error: An , I An The state is n ( An ) : tr( ( I An )) n - Type II error: n ( An ) : tr( n An ) Quantum Hypothesis Testing Given n copies of a quantum state, with the promise that it is described either by or , determine the identity of the state Measure two outcome POVM Error probabilities An , I An - Type I error: n ( An ) : tr( ( I An )) - Type II error: n ( An ) : tr( n n An ) Several different instances depending on the constraints imposed in the error probabilities: Chernoff distance, Hoeffding bound, Stein’s Lemma, etc... Quantum Stein’s Lemma Asymmetric hypothesis testing rn ( ) : min n ( An ) : n ( An ) 0 An I Quantum Stein’s Lemma (Hiai and Petz 91; Ogawa and Nagaoka 00) log rn ( ) 0, lim S ( || ) n n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n we have where 4. If 5. If we have an unknown n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If n I n / dim( H ) n n1 , then trn1 ( ) n n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n we have where 4. If 5. If we have an unknown n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If n I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n we have where 4. If 5. If we have an unknown n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If n I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n we have where 4. If 5. If we have an unknown n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If n I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n we have where 4. If 5. If we have an unknown n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If n I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n we have where 4. If 5. If we have an unknown n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If n I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis - Null hypothesis: For every n we have n n we have an unknown n n D( H n ) , where satisfies S n (*)n n P * P - Alternative hypothesis: For every state 1. Each n SYM ( n ) is closed and convex 2. Each contains the maximally mixed state 3. If 4. If 5. If I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma Consider the following two hypothesis n n n S ( H1 n H ... H 2 k ) n - Null hypothesis: For every n we have - Alternative hypothesis: For every state n n n D( H n ) , 1. Each n where 4. If 5. If n n satisfies is closed and convex 2. Each contains the maximally mixed state 3. If we have an unknown I n / dim( H ) n n1 , then trj ( ) n j 1,..., n 1 n and m , then n m n , then S n ( ) n A generalization of Quantum Stein’s Lemma theorem: Given - (Direct Part) n n 0 satisfying properties 1-5 and there is a D(H ) An , I An n s.t. lim tr( An n ) 1 n n n n , tr( Ann ) 2 n ( E ( ) ) n S ( || ) E ( ) lim min n n n A generalization of Quantum Stein’s Lemma theorem: Given n n - (Strong Converse) satisfying properties 1-5 and D (H ) 0, An , I An n , if n n n s.t. tr( Ann ) 2 lim tr( An ) 0 n n n ( E ( ) ) A generalization of Quantum Stein’s Lemma theorem: Given n n - (Strong Converse) satisfying properties 1-5 and D (H ) 0, An , I An n , if n n n s.t. tr( Ann ) 2 n ( E ( ) ) lim tr( An ) 0 n n Proof: Exponential de Finetti theorem (Renner 05) + duality convex optimization + quantum Stein’s lemma; see arXiv 0810.0026 Corollary: strict positiveness of ER∞ For an entangled state we construct a sequence of POVMs s.t. lim tr( An ) 1 n n n Sn n , tr( Ann ) 2 n , 0 Corollary: strict positiveness of ER∞ How we construct the An’s : we measure each copy with a local informationally complete POVM M to obtain an empirical estimate n of the state. If || n ||1 : min || ||1 / 2 S we accept, otherwise we reject M Corollary: strict positiveness of ER∞ • By Chernoff-Hoeffding’s bound, it’s clear that for some tr ( Ann ) 2 n n (d ) , n for 0 n of the form , with supported on separable states Corollary: strict positiveness of ER∞ • In general, by the exponential de Finneti theorem, (Renner 05) tr1,..., n ( S n (n )) (d )" (1 ) n " almost power states • trA " (1 ) n n • We show that " 2 ( n) , for || ||1 (1 ) n ( n) ( d ) tr ( A " " ) 2 n || ||1 which implies the result Corollary: strict positiveness of ER∞ • Let’s show that ( d )tr( A " n (1 ) n ") 2 ( n) || ||1 • We measure an info-complete POVM on all copies of tr1,..., n ( S n (n )) M expect the first • The estimated state is close from the post-selected state with probability 1 2 ( n ) • As we only used LOCC, the post-selected state must be separable and hence 2 far apart from Thank you! •x