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GEOMETRY OF QUANTUM ENTANGLEMENT CIRM, Marseille, January 2012 Selected results and open problems of quantum entanglement theory Michał Horodecki Institute of Theoretical Physics and Astrophysics and National Quantum Information Centre of Gdańsk University of Gdańsk PART I Preliminaries - states of single and composite systems - maps and quantum operations Characterization of entanglement - in terms of positive maps - some other criteria Distillation of entanglement - LOCC operations - bound entanglement - NPT bound entanglement - extended classes of operations PART II Private cryptographic key from entanglement - inequivalence between private key and maximal entanglement - drawing key from bound entanglement - low dimensional PPT states which are key distillable Entanglement and Local Hidden Variable models - examples of entangled states with LHV model - Peres conjecture Entanglement measures - locking entanglement Preliminaries Quantum system and quantum state Quantum system and quantum state Quantum composite system: entanglement Convex structure All states Entangled Separable states states Entanglement of pure states Maximally entangled state Resource for: - teleportation - dense coding - quantum cryptography - etc. For d=2 Positive and completely positive maps (positive maps: Jamiołkowski, Choi, Woronowicz 70-ties, completely positive maps: Stinespring, Kraus) Completely positive maps Examples of positive maps which are not CP Kraus operators Preserving of trace Quantum operations Characterization of entanglement Entanglement of pure bipartite states How to check that a given state is entangled? | a 00 b 01 c 10 d 11 a b X c d U A U B T U A XU B We find such U A , U B that X ' U A XU B T Schmidt is diagonal, with positive diagonal elements decomposition a' 0 X' ψ a' 00 b' 11 0 b' Fact: A pure state is entangled, if its matrix X is of rank greater than 1 Werner ’89 Entanglement of mixed states (also in mathematical literature in 70-ties: Stormer, Osaki) Two maximally entangled states: | 1 2 ( 00 11 ) | 1 2 ( 00 11 ) Their equal mixture: Reinhard Werner 1 1 | | 2 2 1 1 | 00 00 | 11 11 2 2 Is not entangled! Sandu Popescu Partial transpose Asher Peres Stanisław Woronowicz The Teleporters (Peres ‘96, see also Woronowicz ‘76) 11 12 13 14 11 21 13 23 21 22 23 24 12 22 14 24 32 33 34 41 33 43 31 31 42 43 44 42 34 44 41 32 (I T ) Fact: eigenvalues of a quantum state are non-negative (they are probabilities) Theorem: Separable (non-entangled) states remain positive after partial transpose Example: Bell diagonal states A 1 2 ( 00 11 ), I B 1 ( 00 11 ), E 2 4 C 1 2 ( 01 10 ), D 1 2 ( 01 10 ) Two-qubit and qubit-qutrit case [Peres & Horodeccy ’96, Osaki ’80-ties etc.] Theorem: For Hilbert space C2C2 and C2C3 a state is separable if and only if its partial transposition if a positive matrix: ρ is separable iff i.e. ρ is PPT Sufficiency follows from the following fact (Stormer/Woronowicz): Entangled Separable States = Arbitrary positive map States = PPT : 2 2 or = NPT : 2 3 is decomposable, i.e. CP CPT 1 2 Dimension greater than 2 x 3 (explicit exmaples for 4x2 and 3x3 given in P. Horodecki 1996) Separable PPT entangled NPT states states entangled states There is no easy way to determine whether a given state is separable or not - it is NP-hard problem (Gurvitz and Barnum) Separability versus extendability R. Werner, Lett Math. Phys 17, 359 (1989): Theorem: State is separable if and only if it is n-extendable for every n. (see also recent proof by D. Yang arXiv:quant-ph/0604168) A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. Phys. Rev. A, 69:022308, 2004. F. Brandao, M. Christandl and J. Yard arxiv 1011.2751. Realigment or cross norm criterion (Chen and Wu, Rudolph 2002) Realignment: consider the following tranformation Usually it is not a state anymore. Permutation criteria Strong criterion: can detect entanglement of a tripartite state separable uner any cut! Necessary and sufficient conditions for more than 2 particles (M.P. and R Horodecki, quant-ph/0006071) Problem: provide necessary and sufficient condition of separability of three qubits Distillation of entanglement Distant labs paradigm Distant parties share some joint state Since the labs are distant, not all operations are easy, we can introduce - cheap operations (the easy ones) - expensive operations, which we count Restricting class of operations LOCC operations Distillability LOCC operations Distillable states All states Nondistillable Distillability Entanglement manipulations: pure states Suppose we have pair of particles in state 0 – photon in upper path a0 0 B b1 A 1 B 1 – photon in lower path A T 2 R2 1 T R Given that klick was not observed: normalize ~ Ta 0 A 0 B b1 A 1 B 1 2 (0 A 0 B 1 A 1 B) choose T b a Entanglement manipulations: pure states Entanglement manipulations: pure states a 00 b 11 1 2 ( 00 11 ) The largest probability of such transition is pm ax min{ 2b 2 ,2a 2 } What if we have TWO pairs? a 0 A 0 B b1 A 1 B a 0 A' 0 B' b 1 A' 1 B' AA' BB' a2 00 AA' 00 BB' ab 10 AA' 10 BB' 01 AA' 01 BB' b2 11 AA' 11 BB' Maximally entangled Many pairs of pure states n a0 A 0 B b1 A 1 B n Alice and Bob measure NUMBER of 1’s Example: n 4, k 2 4 pairs, number of 1’s = 2 n a 2b2 1100 A 1100 B 1010 A 1010 B 1001 A 1001 B 0110 A 0110 B 0101 A 0101 B 0011 A 0011 B After normalization: 1 6 ( 2) i 6 i 1 A i B equivalent to log 6 2.58 e - bits Many pairs of pure states n a0 A 0 B b1 A 1 B n Alice and Bob measure NUMBER of 1’s n k n (k ) 1 n i k i 1 A i B equivalent to log e - bits k Only terms with typical frequency will appear: k np where p | b |2 H ( x) x log x (1 x) log(1 x) n Obtained number of e-bits: log log(2nH ( p ) ) nH (| a |2 ) np Reversible transformation of pure states (Bennett, Bernstein, Popescu and Schumacher 1996) a0 A 0 B b1 A 1 B m n m … n … m H (| a |2 ) H ( x) x log x (1 x) log(1 x) n More generally: m E ( ) where E ( ) is entropy of Alice' s (or Bob' s) system n Pure states transformations: many copies no error allowed m n (cf. Nila Datta talk) m m … n Emin ( ) … n m n m m Emax( ) … n n … Pure states transformations: single copy case (Nielsen, 1998) Theorem: A state can be transformed into if and only if is majorized by k k i.e. | a i 1 k | | bk |2 for all k 1,..., d 2 i 1 where a k , bk are Schmidt coeffcients of states , Distillability: characterization for two qubit eggs (HHH1997) Separable Entangled states = states = Nondistillable Distillable Two qubit entanglement is always useful! Distillability: Higher dimension (HHH 1998) Entangled states Separable states Nondistillable Distillable states! states There exists a passive type of entanglement which does not allow for quantum communication: we have called it Bound entanglement Partial transpose NPT bound entanglement If a state has positive partial transpose (PPT) then it is not distillable NPT states Separable PPT states states Distillable states Is this set nonempty? Does NPT bound entanglement exist? Why is this question interesting? Existence of bound entangled states implies that P. Shor, J. A. Smolin, B. Terhal, arXiv:quant-ph/0010054 Phys. Rev. Lett. 86, 2681--2684 (2001) , K. G. H. Vollbrecht, Michael M. Wolf, arXiv:quant-ph/0201103, Phys. Rev. Lett. 88, 247901 (2002) NPT states PPT states Distillable Separable states states Papers related to the problem of existence of NPT bound entanglement: [1] D. P. Divincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, Phys. Rev. A 61, 062312 (2000). [2] W. Dür, J. I. Cirac, M. Lewenstein, D. Bruss Phys. Rev. A 61, 062313 (2000). [3] D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M. Lewenstein, and A. Sanpera, J. Mod. Opt. 49, 1399 (2002). [4] B. Kraus, M. Lewenstein, and J. I. Cirac, Phys. Rev. A 65, 042327 (2002). [5] S. Bandyopadhyay and V. Roychowdhury, Phys. Rev. A 68, 022319 (2003). [6] J. Watrous, Phys. Rev. Lett. 93, 010502 (2004). [7] L. Clarisse, Phys. Rev. A 71, 032332 (2005). [8] L. Clarisse, Quant. Inf. Comput. 6, 539 (2006). [9] R. O. Vianna and A. C. Doherty, Phys. Rev. A 74, 052306 (2006). [10] R. Simon, quant-ph/0608250. [11] I. Chattopadhyay and D. Sarkar, quant-ph/0609050. [12] F. G. S. L. Brandao and J. Eisert, quant-ph/0709.3835. [13] Ł. Pankowski, M. Piani, M. Horodecki, and P. Horodecki, quant-ph/0711.2613. [14] Ł. Pankowski, F. Brandao, M.Horodecki and G. Smith quant-ph/11…. L. Clarisse, arXiv:quant-ph/0612072, PhD Thesis Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory What do we know State is distillable if and only if for some n (HHH1998) A state for which is it not possible, we call: Bound entangled n-copy nondistillable for all n Equivalent condition What do we know If there exist NPT bound entangled states, then there must exist Werner states which are NPT bound entangled NPT states Separable states PPT states Distillable states Recall: to find NPT bound, we need to show n-copy nondistillability for ALL n 1-copy non-distilability of Werner states 1-copy non-distilability of Werner states Numerically: the state on the edge seems to be 2-copy nondistillable We tried to prove this analytically. 2-copy nondistilability: Equivalent matrix analysis problem (Ł. Pankowski, M. Piani, MH, P. Horodecki arxiv: 0711.2613) We have shown it for A, B being diagonalizable matrices (i.e. normal matrices). Existence of NPT bound entanglement and two-positivity [D. P. Divincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, Phys. Rev. A 61, 062312 (2000).] NPT bound from hiding states? (DiVincenzo, Leung, Terhal 2001, Werner, Eggeling 2002) Private cryptographic key from entanglement Entanglement and privacy Recall: the goal of distillation of entanglement was actually purification – obtaining maximal entanglement in pure form. A new task: distill from a state private key. If one can purify entanglement – i.e. obtain maximally entangled states, then one can also obtain private key. (Deutsch et al. 1996) Namely, Alice and Bob distill maximally entangled states, and then measure it locally: Gisin and Wolf asked whether this is „if and only if” i.e. whether key may come only in this way. Two qubits Separable Entangled states = states = Nondistillable= Distillable= No key Private key Is private key equivalent to maximally entangled state? For ANY Alice’s basis, there is Only ONE basis gives key, Bob’s one which gives key other bases are insecure Thus obtaining private key seems to be less demanding than obtaining maximally entangled state. Distillability of private key: mathematical formulation Recall the formulation of distillability: Distillability of private key: mathematical formulation Distillability of private key: mathematical formulation Which states have perfect bit of key? Characterization was given in HHH and J.Oppenheim, PRL 2005 Is private key equivalent to maximally entangled state? Is private key equivalent to maximally entangled state? Is private key equivalent to maximally entangled state? Private key from bound entangled states Can private key be extremely inequivalent with maximally entangle states ? I.e. can we get private key from bound entangled states? There do not exist bound entangled states which have perfect bit of key, But: they may be arbitrary good bit of key. HHH + J. Oppenheim, arXiv:quant- ph/0309110, PRL 94, 160502 (2005) unconditional security: HHH + D. Leung + J. Oppenheim, arXiv:quant-ph/0702077, arXiv:quant-ph/0608195 PRL, IEEE accepted Basic open question: can we distill private key from all entangled states? Separable states PPT states Other question The smallest PPT state, from which we can obtain key acts on 4x4 system Does there exist PPT state on 3x3 system, from which one can obtain key? (Note that for 2x2 and 2x3 there is no such problem: all PPT states do not give private key, as they are separable) Entanglement versus Local Hidden Variable models Detecting entanglement via Bell inequalities Bell 60-ties, Clauser, Horne, Shimony and Holt 60-ties, Werner 89 A, A' B, B' A, A' , B, B' are observables with two outcomes +1 and -1 | AB A' B AB' A' B' | 2 Proof of Bell-CHSH inequality: ( ') B ( ') B ' AA AA ( ') B ( ') B ' AA AA 2 0 0 2 For maximally etangled state, there exist obervables such that AB A' B AB' A' B' 2 2 2 Example: Detecting entanglement via CHSH inequality Fact: if for a given state there exist observables that violate Bell inequalities, then the state is entangled Example: Max. entangled state mixed with white noise I p | (1 p) 4 Max. entangled state: Max. chaotic state: CHSH value = 2 2 CHSH value =0 CHSH ( ) p 2 2 Violation of CHSH, must be entangled I 4 | p0 1/ 2 p 1 Local Hidden Variable model Werner states vs Bell inequalities I p | (1 p) 4 CHSH ( ) p 2 2 0.7056 0.7071 Vertesi, 465 CHSH, 2 observables observables Existence of local-realistic model Violation of CHSH I ? 4 | p0 1/ 3 0.416 p 1 proj. meas. 0.6595 gen. meas. 1 / 2 0.7071 separable entangled states states Peres conjecture Bound entangled states satisfy all Bell inequalities Progress: Acin et al. and Toth: multipartite bound entangled states which violate Bell inequalities Augusiak and P. Horodecki: Four-partite states that violate Bell inequalities Vertesi: Tripartite bound entangled state which is separable under any bipartite cut that violates Bell inequalities Open: does there exist PPT bipartite state, which violates Bell Inequalies? Entanglement measures Entanglement measures Main division: 1) operational measures – which come from some task 2) axiomatic measures – functions which satisfy some natural postulates Examples of operational measures: - distillable entanglement - distillable key Basic postulates for axiomatic measures: - nonnegative function defined on states - nonincreasing under LOCC operations - zero on separable states LOCC operations Digression No fixed domain and co-domain. Given entanglement measure is usually defined on all possible tensor products Hence more precisely, entanglement measure is a collection of functions: satisfying compatibility conditions: Digression The monotonicity condition writes then as follows: Usually we use shorthand notation: Just two examples of axiomatic entanglement measures Entanglement of formation (Bennett, DiVincenzo, Smolin and Wooters 1996) Relative entropy of entanglement (Plenio, Vedral, Rippin and Knight 1997) Locking If a quantity drops by a lot when we remove single qubit, we say it is lockable. (DiVincenzo, M.H., Leung, Smolin and Terhal 2002) Entanglement can be locked (HHH and Oppenheim 2005) Lockable entanglement measures include: entanglement of formation / entanglement cost squashed entanglement Nonlockable entanglement measure: Relative entropy of entanglement. It can drop at most by 2 (proved in HHHO, example that 2 is possible - Andrzej Grudka) Open problem: is distillable entanglement lockable? Open problem: locking down to product state Typically, in examples of locking, removal of a qubit causes the state to be separable (or PPT). Suppose that after removing a qubit, state becomes product. Is it possible to lock entanglement in such case? (J. Oppenheim, A. Winter, MH)