Detection of photonic Bell states

Transcription

1 LECTURE 3

2 Detection of photonic Bell states d a c Beam-splitter transformation: b ˆB ˆB EXERCISE 10: Derive these three relations V a H a ˆB Detection: or V b H b or Two photons detected in H a, H b, V a, or V b Cannot distinguish between Φ + and Φ

3 Three-qubits state: TELEPORTATION II ( α + β ) ( ) 1 2 I Z 1 = Ψ Φ Ψ ( ) ( ) α β Ψ α β ( ) ( ) α β Φ α β + + Φ + + ± ± X 1 ( ) = ± 2 1 ( ) = ± 2 Alice measures her Bell states and informs Bob, who applies appropriate transformations to his qubit XZ Bell states Just two bits!

4 Teleportation with photons: recall of the principle EXPERIMENT: D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger. Nature 390, 575 (1997)

5 Teleportation with photons: the experiment EXPERIMENT: D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger. Nature 390, 575 (1997) d1 d

6 Teleportation with photons: the experiment EXPERIMENT: D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger. Nature 390, 575 (1997) Four-fold coincidences reduce spurious three-fold coincidences

7 ENTANGLED MIXED STATES A mixed state is called separable if it can be prepared by all the parties in a classical way, that is, agreeing over the phone on the local preparation of their states. This state contains only classical correlations. Mathematically: A state represented by the density matrix is said to be entangled if, and only if, it cannot be expressed as the convex combination of product states of all constituent subsystems, i.e., iff where i µ, 1 apple i apple N, is the density matrix corresponding to system i, and the subindex µ refers to the µ -th member of the ensemble of product-state realizations of. Otherwise, the state is said to be separable. It is said to have genuine `-partite entanglement if it is not separable such that each one of the subsets contains less than 6= P µ p µ 1 µ... Nµ, ` parts. Example of genuine N-partite state: GHZ state P µ p µ =1, p µ 0, - there is no partition into non-empty subsets A and B for which the state is unentangled. 1 p 2 ( )

8 It is very difficult to show directly that a state cannot be expressed in a separable form. Since for pure states one has entanglement measures, like the entropic measures, and since a density matrix can always be expressed as a convex sum of pure-state projectors (for instance, its eigenvectors), ENTANGLEMENT CRITERIA = P i=0 p i i i, one might try to define M( )= X i p i M ( i i ) But this yields different values for different decompositions. Example: = = with ± =(1/ 2) ( 01 ± 10 ) separable. A general definition would then be =inf pi, i X i, which shows that this state is actually p i M ( i i ) where one takes the minimum over all possible decompositions. But this is usually very difficult to calculate. So one needs simpler criteria!

9 THE POSITIVE PARTIAL TRANSPOSE CRITERION The positive partial transpose (PPT) criterion (Peres, 1996, Horodecki family, 1996) is capable of detecting a large family of entangled states, and requires a simple algebraic calculation. This criterion is based on the operation of transposition of a matrix, which is a positive operation that converts density matrices in other density matrices, that is, bounded, positive-semidefinite, trace-1 operators. However, the operation of partial transposition (on system A or B) is not positive. Take for instance the Bell state ± =(1/ 2) ( 00 ± 11 ); the partial transposition of the corresponding density matrix yields: ± ± ±1 0 0 ± which has a negative eigenvalue! ( 1/2)

10 THE POSITIVE PARTIAL TRANSPOSE CRITERION - SUITE Consider now a bipartite system, with parts A and B, described by a density operator AB = X i p i A i B i The partial transpose with respect to subsystem B, for instance, yields T B AB = X i p i A i ( B i ) T which is also a density matrix, since it involves the (total) transposition of B. Therefore: If AB is separable, then its partially transposed matrix T B AB is also a density matrix. Is the converse true? As shown by the Horodecki family (Physics Letters A, 1996), this is also a sufficient condition only for 2X2 or 2X3 systems. It follows therefore that there are PPT entangled states ( bound entangled states - to be discussed later). States with negative partial transpose = NPT states.

11 THE POSITIVE PARTIAL TRANSPOSE CRITERION - CONCLUSION Examples. 1. = p + + +(1 p) 00 00, + =(1/ 2)( ) ρ = 1 p p / 2 p / p / 2 p / , ρ T A = 1 p 0 0 p / 2 0 p / p / 2 0 p / The determinant of the partially transposed matrix is there is a negative eigenvalue for any p 6= 0 entangled for any. p>0, therefore, implying that the state is 2. Werner state. One shows in the W = p +(1 p) 1 4, 0 p 1 p>1/3 same way that it is entangled for. (p/2) 4 EXERCISE 11: Show this.

12 Negativity. [K. Zyczkowski et al., PRA 58, 883 (1998)]. Defined as the magnitude of the sum of all negative eigenvalues of as where T B AB SOME USEFUL ENTANGLEMENT QUANTIFIERS N( AB ) T B AB 1 2, T B AB T B AB. Can also be expressed is the trace norm (sum of the absolute values of all eigenvalues) of. For 2 X 2 and 2 X 3 systems, this is a complete characterization. For mixed states with d>6, it fails to quantify bound entanglement. Concurrence For pure 2X2 states, it is simply expressed in terms of the linear entropy: C ( Ψ ) AB = 2 1 Tr ( 2 ρ ) R = S ( L ρ ) R Relation between concurrence and negativity for two qubits. [Verstraete et al., J. Phys. A 34, (2001)]. C N p (1 C) 2 + C 2 (1 C) C coincides with N iff the eigenvector of T B AB corresponding to its negative eigenvalue is a Bell state (up to a local unitary transformation) Negativity Concurrence

13 Entanglement witnesses: examples 1. Bell operator witness: W CHSH 21 B CHSH

14 Entanglement witnesses: examples (2) 2. Energy: Consider the Hamiltonian Eigenstates: H = J( 1 X 2 X + 1 Y 2 Y + 1 Z 2 Z) = 1 2 ( ), 00, 11, + = 1 2 ( ), with respective eigenvalues -3J, J, J e J. For any product state, we have X 1 X 2 + Y 1 Y 2 + Z 1 Z 2 = X 1 X 2 + Y 1 Y 2 + Z 1 Z 2 q q 1X 2 + 1Y 2 + 1Z 2 X Y Z where we have used the Cauchy-Schwarz inequality and that h X i 2 + h Y i 2 + h Z i 2 apple 1. This remains true when we average over product states, so it is true for any separable state. Therefore, for any separable state H sep J. This implies that, if the energy of some state is outside this bound, then the state is entangled. This is the case of the ground state of H, the state, which has energy -3J. One should note however that the + entanglement of the state is not detected by this witness.

15 SEPARABILITY AND LOCAL REALISM If the state of a bipartite system is separable, AB = X i p i A i B i then the average of the product of two local observables (i.e, observables that act on system A or system B), A and B, is given by habi =Tr( AB AB) = X p i ha i ihb i i, i where ha i i =Tr(A A i ), hb i i =Tr(B B i ). This coincides with the expression that would be obtained by a local realistic theory: [ p i takes the role of P ( )] a i b j = P( λ)a( α i,λ)b( β i,λ) dλ Therefore, separable states must satisfy all possible Bell inequalities [Werner, PRA, 40, 4277 (1989)]. This is a necessary condition. It is also sufficient for pure states: any entangled pure state violates the CHSH inequality. But it is not sufficient for mixed states: there are entangled mixed states that do not violate any Bell inequality.

16 SEPARABILITY AND LOCAL REALISM Example: Werner states [Werner, PRA 40, 4277 (1989)]. W = p +(1 p) 1 4, 0 p 1 Entangled for p>1/3 Violates a CHSH inequality for p>1/ p Violates a Bell inequality with many more settings for p> (T. Vértesi, PRA 2008) Explicit LRT model built by Werner for p apple 1/2 Explicit LRT model that explains all projective measurements on W for p apple was found by Acín, Gisin, and Toner [PRA 73, (2006)] Therefore, for 1/3 <papple0.6595, state is entangled but can be described by LRT! In the gap <p<0.7056, state is entangled but up to now nothing is known about its non-locality properties!

17 Decoherence and the classical limit of quantum mechanics

18 Schrödinger on the classical limit of quantum mechanics!1926: At first sight it appears very strange to try to describe a process, which we previously regarded as belonging to particle mechanics, by a system of such proper vibrations.'' Demonstrates that a group of proper vibrations of high quantum number $n$ and of relatively small quantum number differences may represent a particle executing the motion expected from usual mechanics, i. e. oscillating with a constant frequency.

19 Schrödinger on the classical limit of quantum mechanics!1935: An uncertainty originally restricted to the atomic domain has become transformed into a macroscopic uncertainty, which can be resolved through direct observation... This inhibits us from accepting in a naive way a `blurred model' as an image of reality...there is a difference between a shaky or not sharply focused photograph and a photograph of clouds and fogbanks.

20 Quantum physics and localization! Let Ψ 1 and Ψ 2 be two solutions of the same Schrödinger equation. Then Ψ = Ψ 1 +Ψ 2 also represents a solution of the Schrödinger equation, with equal claim to describe a possible real state. When the system is a macrosystem, and when Ψ 1 and Ψ 2 are `narrow with respect to the macro-coordinates, then in by far the greater number of cases, this is no longer true for Ψ. Narrowness in regard to macro-coordinates is a requirement which is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them. Letter from Einstein to Born, January 1, 1954

21 Quantum measurement Linear evolution:

22 Why interference cannot be seen? Decoherence: entanglement with the environment - same process by which quantum computers become classical computers! Dynamics of decoherence: related to elusive boundary between quantum and classical world

23 DECOHERENCE AND THE CLASSICAL LIMIT OF QUANTUM MECHANICS Transforms quantum superpositions into mixtures and quantum computers into classical ones! Quantum computer is like a Schrödinger cat: a coherent superposition of classically distinguishable states Decoherence time decreases with size of system

24 DYNAMICS OF ENTANGLEMENT! Multiparticle system, initially entangled, with individual couplings of particles to independent environments: each particle undergoes decay, dephasing, diffusion.! How is local dynamics related to nonlocal loss of entanglement?! How does loss of entanglement scale with number of particles?

25 DYNAMICS OF ENTANGLEMENT

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27 Negativity as a measure of entanglement N ( AB ) 2 i Negative eigenvalues of partially transposed matrix =1 for a Bell state i Dimensions higher than 6: =0 does not imply separability! 97

28 A paradigmatic example: Atomic decay Qubit states: Amplitude channel : Our strategy: follow evolution as a function of p, not t Weisskopf and Wigner (1930)! Usual master equation for decay of two-level atom, upon tracing on environment (Markovian approximation) Apply evolution to two qubits, take trace with respect to environment degrees of freedom, find evolution of twoqubit reduced density matrix, calculate entanglement EXERCISE 12: Calculate this entanglement.

29 Realization of amplitude map with photons g 0 g 0 e 0 1 p e 0 + p g 1 p =sin 2 (2 ) H V V H 0 E Sagnac-like interferometer 1 E

30 Investigating the dynamics of entanglement

31 Sudden death of entanglement N (p = 0) = 2 Negativity Entanglement Sudden Death (Yu and Eberly)

32 Decay of entanglement for N qubits, other environments?! Independent individual environments

33 Results for amplitude damping! Bipartitions k : N k! Critical transition probability for which negativity vanishes (same for all partitions): p AD c (k) = / 2/N! Smaller than 1 if finite-time / < 1 disappearance of entanglement! Critical value approaches 1 when! Does entanglement become more robust when N EXERCISE 13: Show this (challenging!) State fully separable at this point! number of qubits increases? 103

34 Does entanglement become more robust with increasing N? 104

35 Is ESD relevant for many particles? 105

36 Role of environment! Usually one traces out environment, and one looks at irreversible evolution of system! As entanglement decays and eventually disappears, what is its imprint onto the environment? 106

37 Measuring the environment?

38 108

39 Collaborators: entanglement dynamics Leandro Aolita Fernando de Melo Rafel Chaves Malena Hor-Meyll Alejo Salles Osvaldo Jiménez-Farías Gabriel Aguillar Marcelo P. de Almeida Andrea Valdés-Hernandéz 109 Paulo Souto Ribeiro Stephen Walborn Daniel Cavalcanti Antonio Acín Joe Eberly Xiao-Feng Qian