Distinguishing multi-partite states by local measurements

Transcription

1 Distinguishing multi-partite states by local measurements arxiv[quant-ph]: Ecole Polytechnique Paris / University of Bristol Cécilia Lancien / Andreas Winter AQIS 12 August 24 th 2012 Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

2 Outline 1 Introduction 2 Technical preliminary results 3 Main results 4 Conclusion and open questions Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

3 Outline 1 Introduction 2 Technical preliminary results 3 Main results 4 Conclusion and open questions Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

4 Distinguishability norms (1) Situation 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 / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

5 Distinguishability norms (1) Situation 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 / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

6 Distinguishability norms (1) Situation 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 / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

7 Distinguishability norms (1) Situation 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 / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

8 Distinguishability norms (2) Question 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 and M is often defined by locality constraints (e.g. LOCC, SEP, PPT). Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

9 Distinguishability norms (2) Question 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 and M is often defined by locality constraints (e.g. LOCC, SEP, PPT). Example 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 d 2 (1 + F) and α = + d d 2 (1 F). d := 1 2 σ 1 2 α is s.t. PPT = 2 d = 1. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

10 Outline 1 Introduction 2 Technical preliminary results 3 Main results 4 Conclusion and open questions Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

11 Measurements on a multi-partite quantum system Multi-partite quantum system : H = H 1 H K. dimh j := d j < +, dimh = d 1 d K := d. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

12 Measurements on a multi-partite quantum system Multi-partite quantum system : H = H 1 H K. dimh j := d j < +, dimh = d 1 d K := d. Set M of locally restricted POVMs on H. Comparison of M with : 1 most natural? 2 most relevant! (equivalence with relating constants that are dimension-independent) Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

13 Measurements on a multi-partite quantum system Multi-partite quantum system : H = H 1 H K. dimh j := d j < +, dimh = d 1 d K := d. Set M of locally restricted POVMs on H. Comparison of M with : 1 most natural? 2 most relevant! (equivalence with relating constants that are dimension-independent) Definition Multi-partite generalization of the 2-norm : 2(K ) := ( Tr Tr I ) 2 I [K ] Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

14 Measurements on a multi-partite quantum system Multi-partite quantum system : H = H 1 H K. dimh j := d j < +, dimh = d 1 d K := d. Set M of locally restricted POVMs on H. Comparison of M with : 1 most natural? 2 most relevant! (equivalence with relating constants that are dimension-independent) Definition Multi-partite generalization of the 2-norm : 2(K ) := ( Tr Tr I ) 2 I [K ] Remark : On a single system, 2(1) = Tr 2 + Tr 2 reduces to the 2-norm on traceless Hermitians. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

15 One single symmetric local POVM : Tensor product of t-design POVMs M = M (1) M (K ) with M (j) := ( d j p xj P xj )x j X j a t-design POVM on H j : (p xj ) xj X j is a probability distribution and (P xj ) xj X j are rank-1 projectors s.t. p xj P t x j = ψ ψ t dψ. x j X ψ ψ =1 j Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

16 One single symmetric local POVM : Tensor product of t-design POVMs M = M (1) M (K ) with M (j) := ( d j p xj P xj )x j X j a t-design POVM on H j : (p xj ) xj X j is a probability distribution and (P xj ) xj X j are rank-1 projectors s.t. p xj P t x j = ψ ψ t dψ. x j X ψ ψ =1 j Technical Lemma (cf Berger and Harrow/Montanaro/Short) For ( an Hermitian ) on H, let S be the random variable taking value Tr P x1 P xk with probability px1 p xk. Then : M = d E S E S E(S 2 (E(S ) (Jensen) and E S 2 )) 3 (Hölder) So : d (E(S 2 )) 3 E(S 4 ) M d E(S 2 ). E(S 4 ) Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

17 One single symmetric local POVM : Tensor product of 4-design POVMs Theorem 1 (L/W) M = M (1) M (K ) with M (j) 4-design (hence also 2-design) POVM on H j. K E(S 2 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 / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

18 One single symmetric local POVM : Tensor product of 4-design POVMs Theorem 1 (L/W) M = M (1) M (K ) with M (j) 4-design (hence also 2-design) POVM on H j. K E(S 2 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, M is essentially equivalent to 2(K ). ] 2 Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

19 One single symmetric local POVM : Tensor product of 4-design POVMs Theorem 1 (L/W) M = M (1) M (K ) with M (j) 4-design (hence also 2-design) POVM on H j. K E(S 2 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, M is essentially equivalent to 2(K ). Remark : Other 4 th vs 2 nd order moments inequality obtained from hypercontractive inequality (Montanaro). ] 2 Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

20 Outline 1 Introduction 2 Technical preliminary results 3 Main results 4 Conclusion and open questions Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

21 Comparison between distinguishability norms and 2-norm Theorem 2 (L/W) Let an Hermitian and M a tensor product of 4-design POVMs on H K /2 2 M LOCC. 2 2 K /2 2 SEP. 2 PPT. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

22 Comparison between distinguishability norms and 2-norm Theorem 2 (L/W) Let an Hermitian and M a tensor product of 4-design POVMs on H K /2 2 M LOCC. 2 2 K /2 2 SEP. 2 PPT. Previously known results : Let a traceless Hermitian on H. K = 1 : M for M a 4-design POVM (Ambainis/Emerson) Applications in quantum algorithms. 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 larger K and non-necessarily traceless useful too? Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

23 Comparison between distinguishability norms and 1-norm Theorem 3 (L/W) Let an Hermitian and M a tensor product of 4-design POVMs on H. 1 1 d 18 K /2 1 M LOCC d 2 K /2 1 SEP PPT 1. d Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

24 Comparison between distinguishability norms and 1-norm Theorem 3 (L/W) Let an Hermitian and M a tensor product of 4-design POVMs on H. 1 1 d 18 K /2 1 M LOCC d 2 K /2 1 SEP PPT 1. d Tightness of the lower bounds? U H = U H1 U HK tensor product of the uniform ( -design) POVMs. /2 δk 0 : UH d 1 with 2 π < δ < 1 Dependence on K and d of the constant relating the norms is real. If I [K ] : dim i I H i = dim i / I H i = d, then 0 : PPT 2 d+1 1 Dependence on d of the constant relating the norms is real. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

25 Outline 1 Introduction 2 Technical preliminary results 3 Main results 4 Conclusion and open questions Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

26 Conclusion and open questions 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 parties but not on the local dimensions. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

27 Conclusion and open questions 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 parties but not on the local dimensions. Comparison between distinguishability norms and 1-norm : Existence of Data-Hiding states Optimality of the constants of domination in both their number of parties and their dimensional dependence. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

28 Conclusion and open questions 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 parties but not on the local dimensions. Comparison between distinguishability norms and 1-norm : Existence of Data-Hiding states Optimality of the constants of domination in both their number of parties and their dimensional dependence. Question : 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 / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15

29 References D.P. DiVincenzo, D. Leung, B.M. Terhal, Quantum Data Hiding, arxiv :quant-ph/ v1. A.W. Harrow, A. Montanaro, A.J. Short, Limitations on quantum dimensionality reduction, arxiv[quant-ph] : v2. A. Ambainis, J. Emerson, Quantum t-designs : t-wise independence in the quantum world, arxiv :quant-ph/ v2. W. Matthews, S. Wehner, A. Winter, Distinguishability of quantum states under restricted families of measurements with an application to data hiding, arxiv : v2[quant-ph]. F.G.S.L. Brandão, M. Christandl, J.T. Yard, Faithful Squashed Entanglement, arxiv[quant-ph] : v5. Cécilia Lancien / Andreas Winter (AQIS 12) Distinguishing multi-partite states August 24 th / 15