Distinguishing multi-partite states by local measurements

Transcription

1 Distinguishing multi-partite states by local measurements Guillaume Aubrun / Andreas Winter Université Claude Bernard Lyon 1 / Universitat Autònoma de Barcelona Cécilia Lancien UCBL1 / UAB Madrid OA and QI - June 17 th 2013 Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

2 Outline 1 Introduction 2 Distinguishability norm associated with a single local measurement on a multi-partite system 3 Distinguishability norm associated with whole classes of locally restricted measurements on a multi-partite system 4 Conclusion and open questions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

3 Outline 1 Introduction 2 Distinguishability norm associated with a single local measurement on a multi-partite system 3 Distinguishability norm associated with whole classes of locally restricted measurements on a multi-partite system 4 Conclusion and open questions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

4 Distinguishability norms (1) Situation considered System that can be in 2 quantum states, ρ or σ, with respective prior probabilities q and 1 q. Decide in which one it is most likely, based on accessible experimental data, i.e. on the outcomes of a POVM( M = (M x ) x X performed on it. ) Probability of error : P E = 1 ( (qρ ) ) 1 Tr (1 q)σ Mx 2 x X Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

5 Distinguishability norms (1) Situation considered System that can be in 2 quantum states, ρ or σ, with respective prior probabilities q and 1 q. Decide in which one it is most likely, based on accessible experimental data, i.e. on the outcomes of a POVM( M = (M x ) x X performed on it. ) Probability of error : P E = 1 ( (qρ ) ) 1 Tr (1 q)σ Mx 2 x X Distinguishability (semi)-norm associated with the POVM M = (M x ) x X : M := ( ) Tr M x x X Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

6 Distinguishability norms (1) Situation considered System that can be in 2 quantum states, ρ or σ, with respective prior probabilities q and 1 q. Decide in which one it is most likely, based on accessible experimental data, i.e. on the outcomes of a POVM( M = (M x ) x X performed on it. ) Probability of error : P E = 1 ( (qρ ) ) 1 Tr (1 q)σ Mx 2 x X Distinguishability (semi)-norm associated with the POVM M = (M x ) x X : M := ( ) Tr M x x X Set M of POVMs : M := sup M M M Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

7 Distinguishability norms (1) Situation considered System that can be in 2 quantum states, ρ or σ, with respective prior probabilities q and 1 q. Decide in which one it is most likely, based on accessible experimental data, i.e. on the outcomes of a POVM( M = (M x ) x X performed on it. ) Probability of error : P E = 1 ( (qρ ) ) 1 Tr (1 q)σ Mx 2 x X Distinguishability (semi)-norm associated with the POVM M = (M x ) x X : M := ( ) Tr M x x X Set M of POVMs : M := sup M M M Holevo-Helstrom : ALL = 1 := Tr Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

8 Distinguishability norms (2) Problem When only POVMs from a restricted set M are allowed, how much smaller than 1 is M? What kind of restrictions? On a multi-partite system, experimenters are not able to implement any observable (measurements on their own sub-system). M is often defined by these locality constraints (e.g. LOCC SEP PPT). Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

9 Distinguishability norms (2) Problem When only POVMs from a restricted set M are allowed, how much smaller than 1 is M? What kind of restrictions? On a multi-partite system, experimenters are not able to implement any observable (measurements on their own sub-system). M is often defined by these locality constraints (e.g. LOCC SEP PPT). Motivation : Existence of Data-Hiding states (DiVincenzo/Leung/Terhal) Orthogonal states (hence perfectly distinguishable by a suitable measurement) that are barely distinguishable by PPT (and even more so LOCC) measurements. Ex in the bipartite case : Completely symmetric and antisymmetric states on C D C D, σ := 1 1 (1 + F) and α := (1 F). D 2 +D D 2 D := 1 2 σ 1 2 α is s.t. PPT = 2 D = 1. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

10 Outline 1 Introduction 2 Distinguishability norm associated with a single local measurement on a multi-partite system 3 Distinguishability norm associated with whole classes of locally restricted measurements on a multi-partite system 4 Conclusion and open questions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

11 Local measurements on a multi-partite quantum system Finite-dimensional multi-partite quantum system : H = C d 1 C d K with d := dim H = d 1 d K < + Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

12 Local measurements on a multi-partite quantum system Finite-dimensional multi-partite quantum system : H = C d 1 C d K with d := dim H = d 1 d K < + M = M (1) M (K ) a local POVM on H. Comparison of M with : 1 most natural? 2 most relevant! (norm equivalence with dimension-independent constants of domination) Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

13 Local measurements on a multi-partite quantum system Finite-dimensional multi-partite quantum system : H = C d 1 C d K with d := dim H = d 1 d K < + M = M (1) M (K ) a local POVM on H. Comparison of M with : 1 most natural? 2 most relevant! (norm equivalence with dimension-independent constants of domination) Definition K -partite generalization of the 2-norm : 2(K ) := Tr TrI 2 I [K ] Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

14 Local measurements on a multi-partite quantum system Finite-dimensional multi-partite quantum system : H = C d 1 C d K with d := dim H = d 1 d K < + M = M (1) M (K ) a local POVM on H. Comparison of M with : 1 most natural? 2 most relevant! (norm equivalence with dimension-independent constants of domination) Definition K -partite generalization of the 2-norm : 2(K ) := Tr TrI 2 I [K ] Remark : On a single system, 2(1) = Tr 2 + Tr 2 reduces to 2 = Tr 2 on traceless Hermitians. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

15 Tensor product of local 4-design POVMs (1) M := ( Dp x P x )x X is a t-design POVM on CD if (p x ) x X is a probability distribution and (P x ) x X are rank-1 projectors on C D s.t. p x P t x = ψ ψ t 1 dψ = x X ψ C D, ψ ψ =1 D (D + t 1) U σ. σ S t Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

16 Tensor product of local 4-design POVMs (1) M := ( Dp x P x )x X is a t-design POVM on CD if (p x ) x X is a probability distribution and (P x ) x X are rank-1 projectors on C D s.t. p x P t x = ψ ψ t 1 dψ = x X ψ C D, ψ ψ =1 D (D + t 1) U σ. σ S t Example : U := {D ψ ψ dψ, ψ C D, ψ ψ = 1} -design POVM on C D. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

17 Tensor product of local 4-design POVMs (1) M := ( Dp x P x )x X is a t-design POVM on CD if (p x ) x X is a probability distribution and (P x ) x X are rank-1 projectors on C D s.t. p x P t x = ψ ψ t 1 dψ = x X ψ C D, ψ ψ =1 D (D + t 1) U σ. σ S t Example : U := {D ψ ψ dψ, ψ C D, ψ ψ = 1} -design POVM on C D. Lemma : Moments method M = M (1) M (K ) with M (j) := ( d j p xj P xj )x j X j 4-design POVM on C d j. an Hermitian on H, and S the random variable taking value Tr ( P x1 P xk ) with probability px1 p xk, so that M = d E S. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

18 Tensor product of local 4-design POVMs (1) M := ( Dp x P x )x X is a t-design POVM on CD if (p x ) x X is a probability distribution and (P x ) x X are rank-1 projectors on C D s.t. p x P t x = ψ ψ t 1 dψ = x X ψ C D, ψ ψ =1 D (D + t 1) U σ. σ S t Example : U := {D ψ ψ dψ, ψ C D, ψ ψ = 1} -design POVM on C D. Lemma : Moments method M = M (1) M (K ) with M (j) := ( d j p xj P xj )x j X j 4-design POVM on C d j. ( an Hermitian on H ), and S the random variable taking value Tr P x1 P xk with probability px1 p xk, so that M = d E S. By Jensen : E S E(S 2 ), and by Hölder : E S (E(S2 )) 3 E(S 4 ). Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

19 Tensor product of local 4-design POVMs (1) M := ( Dp x P x )x X is a t-design POVM on CD if (p x ) x X is a probability distribution and (P x ) x X are rank-1 projectors on C D s.t. p x P t x = ψ ψ t 1 dψ = x X ψ C D, ψ ψ =1 D (D + t 1) U σ. σ S t Example : U := {D ψ ψ dψ, ψ C D, ψ ψ = 1} -design POVM on C D. Lemma : Moments method M = M (1) M (K ) with M (j) := ( d j p xj P xj )x j X j 4-design POVM on C d j. ( an Hermitian on H ), and S the random variable taking value Tr P x1 P xk with probability px1 p xk, so that M = d E S. By Jensen : E S E(S 2 ), and by Hölder : E S (E(S2 )) 3 E(S 4 ). (E(S 2 )) 3 So : d E(S 4 M d E(S2 ). ) Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

20 Tensor product of local 4-design POVMs (2) Theorem (L/Winter) M = M (1) M (K ) with M (j) 4-design POVM on C d j. E(S 2 K 1 ) = j=1 d j (d j + 1) ( Tr Tr I ) 2 I [K ] [ E(S 4 K 1 ) j=1 d j (d j + 3) 18K ( Tr Tr I ) 2 I [K ] ] 2 Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

21 Tensor product of local 4-design POVMs (2) Theorem (L/Winter) M = M (1) M (K ) with M (j) 4-design POVM on C d j. E(S 2 K 1 ) = j=1 d j (d j + 1) ( Tr Tr I ) 2 I [K ] [ E(S 4 K 1 ) j=1 d j (d j + 3) 18K ( Tr Tr I ) 2 I [K ] 1 So : 18 K /2 2(K ) M 2(K ). ] 2 Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

22 Tensor product of local 4-design POVMs (2) Theorem (L/Winter) M = M (1) M (K ) with M (j) 4-design POVM on C d j. E(S 2 K 1 ) = j=1 d j (d j + 1) ( Tr Tr I ) 2 I [K ] [ E(S 4 K 1 ) j=1 d j (d j + 3) 18K ( Tr Tr I ) 2 I [K ] 1 So : 18 K /2 2(K ) M 2(K ). Conclusion : If M is a sufficiently symmetric local POVM on H (tensor product of local 4-design POVMs), then M is essentially equivalent to 2(K ) (dimension-independent constants of domination). ] 2 Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

23 Related work and applications Remark : Other 4 th vs 2 nd order moment inequalities may be obtained from hypercontractive inequalities (Montanaro). Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

24 Related work and applications Remark : Other 4 th vs 2 nd order moment inequalities may be obtained from hypercontractive inequalities (Montanaro). Previously known results : Let be a traceless Hermitian on H. K = 1 : M 2 for M a 4-design POVM. Applications in quantum algorithms (Ambainis/Emerson) and quantum dimensionality reduction (Harrow/Montanaro/Short). 1 K = 2 : M for M a tensor product of two 4-design POVMs (Matthews/Wehner/Winter). Applications in entanglement theory (Brandão/Christandl/Yard). Results for any K and non-necessarily traceless useful too? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

25 Implementable POVM with good discriminating power (1) Problem A 4-design POVM on C d must have at least Ω(d 4 ) outcomes. No explicit constructions of 4-design POVMs are known. Minimal number of outcomes for a randomly chosen POVM M on C d so that with high probability M 2(1) (dimension-independently)? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

26 Implementable POVM with good discriminating power (1) Problem A 4-design POVM on C d must have at least Ω(d 4 ) outcomes. No explicit constructions of 4-design POVMs are known. Minimal number of outcomes for a randomly chosen POVM M on C d so that with high probability M 2(1) (dimension-independently)? Strategy : Draw independently P 1,...,P n uniformly distributed rank-1 projectors on C d, set S := n P k and P := ( Pk := S 1/2 P k S 1/2 ) 1 k n. k=1 P is a random POVM on C d with n outcomes s.t. P = n k=1 Tr( Pk ). Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

27 Implementable POVM with good discriminating power (1) Problem A 4-design POVM on C d must have at least Ω(d 4 ) outcomes. No explicit constructions of 4-design POVMs are known. Minimal number of outcomes for a randomly chosen POVM M on C d so that with high probability M 2(1) (dimension-independently)? Strategy : Draw independently P 1,...,P n uniformly distributed rank-1 projectors on C d, set S := n P k and P := ( Pk := S 1/2 P k S 1/2 ) 1 k n. k=1 P is a random POVM on C d with n outcomes s.t. P = n k=1 Theorem (Aubrun/L) ( α > 0, C α > 0 : n C α d 2 P Tr( Pk ) (1) P (1) ) 1 α Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

28 Implementable POVM with good discriminating power (1) Problem A 4-design POVM on C d must have at least Ω(d 4 ) outcomes. No explicit constructions of 4-design POVMs are known. Minimal number of outcomes for a randomly chosen POVM M on C d so that with high probability M 2(1) (dimension-independently)? Strategy : Draw independently P 1,...,P n uniformly distributed rank-1 projectors on C d, set S := n P k and P := ( Pk := S 1/2 P k S 1/2 ) 1 k n. k=1 P is a random POVM on C d with n outcomes s.t. P = n k=1 Theorem (Aubrun/L) ( α > 0, C α > 0 : n C α d 2 P Tr( Pk ) (1) P (1) ) 1 α Remark : Optimal result since P informationally complete n d 2. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

29 Implementable POVM with good discriminating power (2) Idea of the proof : 1 Large deviation probability of d n 2 Large deviation probability of n k=1 n k=1 Tr(P k ) from 2(1)? Tr( Pk ) from d n n k=1 Tr(P k )? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

30 Implementable POVM with good discriminating power (2) Idea of the proof : 1 Large deviation probability of d n 2 Large deviation probability of n k=1 n k=1 Tr(P k ) from 2(1)? Tr( Pk ) from d n n k=1 Use twice : (i) Moments estimate : p 1, E X p ( cp) p X ψ1 c (ii) Bernstein-type tail bound for sums of i.i.d centered ψ 1 -r.v. (iii) Net argument : individual to global error term Tr(P k )? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

31 Implementable POVM with good discriminating power (2) Idea of the proof : 1 Large deviation probability of d n 2 Large deviation probability of n k=1 n k=1 Tr(P k ) from 2(1)? Tr( Pk ) from d n n k=1 Use twice : (i) Moments estimate : p 1, E X p ( cp) p X ψ1 c (ii) Bernstein-type tail bound for sums of i.i.d centered ψ 1 -r.v. (iii) Net argument : individual to global error term 1 {X k := d Tr ( P k ) U } 1 k n for a given S 2(1) 2 {X k := ψ dp k 1 ψ } 1 k n for a given unit vector ψ Tr(P k )? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

32 Implementable POVM with good discriminating power (2) Idea of the proof : 1 Large deviation probability of d n 2 Large deviation probability of n k=1 n k=1 Tr(P k ) from 2(1)? Tr( Pk ) from d n n k=1 Use twice : (i) Moments estimate : p 1, E X p ( cp) p X ψ1 c (ii) Bernstein-type tail bound for sums of i.i.d centered ψ 1 -r.v. (iii) Net argument : individual to global error term 1 {X k := d Tr ( P k ) U } 1 k n for a given S 2(1) Tr(P k )? 2 {X k := ψ dp k 1 ψ } 1 k n for a given unit vector ψ ( 1 P (1) P 15 ) 8 2(1) 1 C d 2 e cn C d e c n/d Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

33 Outline 1 Introduction 2 Distinguishability norm associated with a single local measurement on a multi-partite system 3 Distinguishability norm associated with whole classes of locally restricted measurements on a multi-partite system 4 Conclusion and open questions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

34 Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

35 Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements A tensor product of 4-design POVMs is a particular LOCC strategy. 1 Consequently : 18 K / K /2 2(K ) LOCC. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

36 Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements A tensor product of 4-design POVMs is a particular LOCC strategy. 1 Consequently : 18 K / K /2 2(K ) LOCC. On a m-partite Hilbert space, the ball of radius 2 1 m/2 for 2 (centered at 1) is fully separable (Barnum/Gurvits). 2 Consequently : 2 K /2 2 SEP and 2 PPT. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

37 Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements A tensor product of 4-design POVMs is a particular LOCC strategy. 1 Consequently : 18 K / K /2 2(K ) LOCC. On a m-partite Hilbert space, the ball of radius 2 1 m/2 for 2 (centered at 1) is fully separable (Barnum/Gurvits). 2 Consequently : 2 K /2 2 SEP and 2 PPT. Comparison of LOCC, SEP and PPT with ALL 1 1 d 18 K /2 1 LOCC d 2 K /2 1 SEP PPT 1 d Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

38 Data-Hiding and optimality of the lower bounds Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

39 Data-Hiding and optimality of the lower bounds M := U C d 1 U C d K tensor product of the local uniform POVMs. Tightness of d 1 M? K /2 /2 δk 0 : M d 1 with 2 π < δ < 1 Dependence on K and d of the constant relating the norms is real. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

40 Data-Hiding and optimality of the lower bounds M := U C d 1 U C d K tensor product of the local uniform POVMs. Tightness of d 1 M? K /2 /2 δk 0 : M d 1 with 2 π < δ < 1 Dependence on K and d of the constant relating the norms is real. Tightness of 1 d 1 PPT? If I [K ] : d i = d i = d, then 0 : PPT 2 d+1 1 i I i / I Dependence on d of the constant relating the norms is real. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

41 Data-Hiding and optimality of the lower bounds M := U C d 1 U C d K tensor product of the local uniform POVMs. Tightness of d 1 M? K /2 /2 δk 0 : M d 1 with 2 π < δ < 1 Dependence on K and d of the constant relating the norms is real. Tightness of 1 d 1 PPT? If I [K ] : d i = d i = d, then 0 : PPT 2 d+1 1 i I i / I Dependence on d of the constant relating the norms is real. Remark : In the special case H = (C D ) K (d = D K ) with K fixed and D +, it may be shown by volumic considerations that typically : { SEP 1 1 1/K 1 d, so that : PPT SEP 1 PPT d 1 (Aubrun/L). 1 Data-Hiding Hermitians are exceptionnal. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

42 Outline 1 Introduction 2 Distinguishability norm associated with a single local measurement on a multi-partite system 3 Distinguishability norm associated with whole classes of locally restricted measurements on a multi-partite system 4 Conclusion and open questions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

43 Conclusions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

44 Conclusions On a composite quantum system, equivalence between the distinguishability norm associated with a sufficiently symmetric local POVM and a multi-partite relative of the 2-norm, with constants of domination that depend on the number of local parties but not on their dimension. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

45 Conclusions On a composite quantum system, equivalence between the distinguishability norm associated with a sufficiently symmetric local POVM and a multi-partite relative of the 2-norm, with constants of domination that depend on the number of local parties but not on their dimension. On a d-dimensional quantum system, random POVMs made of Ω(d 2 ) independent uniformly distributed rank-1 projectors (appropriately renormalized) achieve this same discriminating power with high probability. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

46 Conclusions On a composite quantum system, equivalence between the distinguishability norm associated with a sufficiently symmetric local POVM and a multi-partite relative of the 2-norm, with constants of domination that depend on the number of local parties but not on their dimension. On a d-dimensional quantum system, random POVMs made of Ω(d 2 ) independent uniformly distributed rank-1 projectors (appropriately renormalized) achieve this same discriminating power with high probability. Comparison between M and ALL for various sets M of locally restricted POVMs : Existence of Data-Hiding states Optimality of the constants of domination relating the norms in both their number of party and their dimensional dependence. Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

47 Open questions Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

48 Open questions Performance of LOCC (or at least SEP) measurements? Do there exist constants C > 0 and α < 1 s.t. for any number of parties K and any global dimension d there exists an Hermitian 0 s.t. LOCC C αk d 1? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

49 Open questions Performance of LOCC (or at least SEP) measurements? Do there exist constants C > 0 and α < 1 s.t. for any number of parties K and any global dimension d there exists an Hermitian 0 s.t. LOCC C αk d 1? Starting with a POVM that already behaves like the uniform POVM (e.g. a 4-design POVM), how many rank-1 projectors must be drawn from its probability distribution to get a random POVM that, again, behaves like the uniform POVM (with high probability)? Probably Ω(d 2 ) is not enough, but perhaps Ω(d 2 (logd) β ) or Ω(d 2+ε )...? Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19

50 References D.P. DiVincenzo, D. Leung, B.M. Terhal, Quantum Data Hiding, arxiv :quant-ph/ A.W. Harrow, A. Montanaro, A.J. Short, Limitations on quantum dimensionality reduction, arxiv[quant-ph] : A. Ambainis, J. Emerson, Quantum t-designs : t-wise independence in the quantum world, arxiv :quant-ph/ W. Matthews, S. Wehner, A. Winter, Distinguishability of quantum states under restricted families of measurements with an application to data hiding, arxiv : [quant-ph]. F.G.S.L. Brandão, M. Christandl, J.T. Yard, Faithful Squashed Entanglement, arxiv[quant-ph] : H.N. Barnum, L. Gurvits, Separable balls around the maximally mixed multipartite quantum states, arxiv :quant-ph/ C. Lancien, A. Winter, Distinguishing multi-partite states by local measurements, arxiv[quant-ph] : Cécilia Lancien ( UCBL1 / UAB ) Distinguishing multi-partite states Madrid OA and QI - June 17 th / 19