1. Quantum Computers, Algorithms and Chaos - Varenna 2005 ENTANGLEMENT IN QUANTUM OPTICS Paolo Tombesi Department of Physics University of Camerino

2. Quantum Computers, Algorithms and Chaos - Varenna 2005 OUTLINE Introduction-definition of entangled states The Peres’ criterion for separability of bipartite states Experimental realization A general bipartite entanglement criterion Continuous variable case The Simon’s criterion for Gaussian bipartite states One example of continuous variable bipartite Gaussian states Tripartite continuous variable Gaussian states Example of tripartite Gaussian states

3. Quantum Computers, Algorithms and Chaos - Varenna 2005 entanglement: polarization of two photons (  H  1  H  2 ±  V  1  V  2 )/√2(  H  1  V  2 ±  H  2  V  1 )/√2 or In general, for a bipartite system, it is separable   =  i w i  i1  i2 w i ≥ 0  i w i =1 i.e. it can be prepared by means of local operations and classical communications acting on two uncorrelated subsystems 1 and 2 Simple criterion for inseparability or entanglement was derived by Peres (PRL 77, 1413 (1996) These are the so-called Bell states  (  ) and  (  )

4. Quantum Computers, Algorithms and Chaos - Varenna 2005 Given an orthonormal basis in H 12 = H 1 H 2 the arbitrary state of the bipartite state 1+2 is described by the density matrix (  12 ) m ,n (Latin indices for the first system and Greek indices for the second one). To have the transpose operation it means to invert row indices with column indices (  12 ) n,m  The partial transpose operation (PT) is given by the the inversion of Latin indices (Greek) PT : (  12 ) m ,n  (  12 ) n ,m  (  T 1 12 ) m ,n We ask if the operator  T 1 12 is yet a density operator i.e. Tr (  T 1 12 ) = 1 and  T 1 12 ≥ 0 It easy to prove this because the transposition does not change the diagonal elements, Thus the Trace remains invariant, and the positivity is connected with the positivity of the eigenvalues of the matrix, which do not change under transposition. Then the violation of the positivity of the partial transpose is a sufficient criterion for entanglement It easy to prove that the positivity of partial transpose of the state is a necessary condition for separability. i.e.  12 separable   T 1 12 ≥ 0  T 1 12 < 0   12 entangled In 2x2 and 2x3 dimension for the Hilbert space  12 separable   T 1 12 ≥ 0 Horodecki 3 Phys Lett A 223, 8 (1996) O x

5. Quantum Computers, Algorithms and Chaos - Varenna 2005 Non-linear crystal Pump laser Type IType II

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7. Dichroic Mirror Aperture (  (2) ) NLC Laser beam Phase Matching TYPE II 0.0 0.05 -0.05 0.050.0 -0.05 810nm Extraordinary Ordinary 810nm yy xx Pump Laser @405nm |  > = | H V > + e i  | V H >

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9. det  T 2 12 =  0.54

10. Quantum Computers, Algorithms and Chaos - Varenna 2005 We shall derive a general separability criterion valid for any state of any bipartite system. Let us consider a bipartite system whose subsystems, not necessarily identical, are labeled as 1 and 2, and a  sep separable state on the Hilbert space H tot = H 1 H 2. O x  sep =  i w i  i1  i2 w i ≥ 0  i w i =1 Let us now choose a generic couple of observables for each subsystem, say r j, s j on H j (j = 1, 2), C j = i [r j, s j ], j = 1, 2 C j is typically nontrivial Hermitian operator on the Hilbert subspaces. Let’s define two Hermitian operators on H tot u = a 1 r 1 + a 2 r 2, v = b 1 s 1 + b 2 s 2 where a j, b j are real parameters From the standard form of the uncertainty principle, it follows that every state  on H tot must satisfy ≥ | a 1 b 1 + a 2 b 2 | 2 4

11. Quantum Computers, Algorithms and Chaos - Varenna 2005 For separable states the following Theorem holds  sep  ≥ O 2 With O = ( | a 1 b 1 | + | a 2 b 2 | ) /2 =  k w k k k = Tr [ C j  k j ] i.e. the expectation value of operator C j onto  k j Proof : From the definition of and  sep it is easy to see that =  k w k [ a 1 2 k + a 2 2 k ] +  k w k 2 - (  k w k ) 2 With  r j (k) = r j - k the variance of r j onto the state  kj The same for For the Cauchy-Schwartz inequality  k w k 2 ≥ (  k w k ) 2 the last two terms in are bound below by zero Follows ≥  k w k [ a 1 2 k + a 2 2 k ] and ≥  k w k [ b 1 2 k + b 2 2 k ]

12. Quantum Computers, Algorithms and Chaos - Varenna 2005 Given any two real non negative numbers  and   +  ≥  k w k [  a 1 2 k +  b 1 2 k + +  k w k [  a 2 2 k ] +  b 2 2 k ] Furthermore, by applying the uncertainty principle to the operators r j and s j on the state  k j, it follows  a j 2 k +  b j 2 k ≥  a j 2 k +  b j 2 | k | 2 4 k ≥ √   | a j b j | | k | (min f(x) =  1 x +  2 /x f min = 2 √  1  2 ) Finally  +  ≥ 2 √   O That has to be satisfied for any  and  positive, thus maximizing g(x) = 2 x O - x 2 for x > 0 ≥ O2O2

13. Quantum Computers, Algorithms and Chaos - Varenna 2005 Connection with other criteria The Duan et al. criterion (the sum criterion) Phys. Rev. Lett. 84, 2722 (2000) is a particular case of  sep  ≥ O 2 Indeed, if we pose  =  = 1 in  +  ≥ 2 √   O We get + ≥ 2 O ≥ O2O2 however Then + ≥ + ≥ 2 O O2O2 The last inequality holds because of the function (min f(x) =  1 x +  2 /x f min = 2 √  1  2 ) Giovannetti et al. Phys. Rev. A 67, 022320 (2003)

14. Quantum Computers, Algorithms and Chaos - Varenna 2005 Continuous variables A single qubit forms a 2-dim Hilbert space, a single quantum mechanical oscillator (i.e. single mode of the electromagnetic field or an acoustic vibrational mode) forms an ∞ - dim Hilbert space. This system can be described by observables (position and momentum), which have a continuous spectrum of eigenvalues. We will refer to this system as a continuous variable system (CV).One can introduce the so-called rotated quadratures for the CV system X(  ) = ( a e - i  + a + e i  )/√ 2 P(  ) = ( a e - i  - a + e i  )/i √ 2 = X(  +  /2 ) With [X(  ), P(  ) ] = i A single mode of the e.m. field in free space can be written as E (r,t) = E 0 [X cos (  t - k r ) + P sin (  t - k r ) ] in phaseout of phase

15. Quantum Computers, Algorithms and Chaos - Varenna 2005 An arbitrary single mode state  can be associated, by a 1 to 1 correspondence, to a symmetrically ordered characteristic function  ( ) = Tr [  D( ) ] = R +i I D( ) = exp( a + - * a ) = exp i √2( I x - R p ) With x and p position and momentum, or quadratures, of the CV mode The Wigner function W(  ) of the state is related to  ( ) by an inverse Fourier Transform W(  ) =  - 2 ∫d 2 exp( -  * + *  )  ( )  ( ) = ∫d 2  exp(  * - *  ) W(  ) Where the complex amplitude  =  R + i  I represents the coherent state |  > in the phase space. Passing to N modes the state   H O x N The W-function becomes W(  1 …  N ) =  - 2N [  k ∫d 2 k exp( - k  k * + k *  k ) ]  ( 1.. N )

16. Quantum Computers, Algorithms and Chaos - Varenna 2005 We shall consider N modes Gaussian states  with the characteristic function  ( 1,…., N ) Gaussian as it is the W-function W(  1,…,  N ) To the k-th mode we associate the complex variable k = k R +i k I which is Represented by the vector ( ) = k real  R 2 (k = 1,..,N) k I - k R In terms of the real variables ( T 1,….., T N )   T  R 2N the arbitrary N-mode Gaussian characteristic function takes the form  (  ) = exp ( -  T V  + i d T  ) Where d  R 2N and V is a 2N x 2N real, symmetric strictly positive matrix V = V T V > 0 The corresponding N-mode Gaussian W-function by the inverse F-T is: W(  ) = exp( -1/4 d T V -1 d)  N √ det V exp ( -  T V -1  + d T V -1  ) 2 √2√2  T  (x 1,p 1,….,x N,p N )  R 2N Define a 2N-dim phase space

17. Quantum Computers, Algorithms and Chaos - Varenna 2005 V is the correlation matrix (CM) of the N-mode state  V nm =  (  n  m +  m  n )  /2 (n,m = 1,2,….,2N)  m =  m - d = is the displacement of the state Both d and V are measurable quantities defined for every N-mode state When the state is Gaussian it is fully characterized by these two quantities  Gaussian  ( V, d ) The CM expresses the covariance between the position and momentum quadratures (in-phase and out-of-phase) of the state . It must respect the uncertainty principle  1/4  k k’ (k,k’ = 1,..,N) The uncertainty principle reads V + i I (N) /2  o ( -1 0 ) 0 1 Introducing the N-mode symplectic matrix I (N) = I i I i = 1 N

18. Quantum Computers, Algorithms and Chaos - Varenna 2005 CV entanglement Simon ( Phys. Rev. Lett. 84, 2726 (2000) showed how to extend the Peres criterion for separability to the CV case. Consider a bipartite CV state  AB and introduce its phase space representation through the W-function W(  )  T  ( x A, p A, x B, p B ) the PT operation on the state  AB in H AB is equivalent to a partial mirror reflection of W(  ) in the phase space PT :  AB  PT(  AB )  W(  )  W(  ) with   diag (1,1,1,-1) PT is a local time-reversal which inverts the momentum of only one subsystem (B in our case) The extension to CV of the Peres criterion is:  AB separable  W(  ) is a genuine W-function genuine means corresponding to a physical state A genuine W-function implies a genuine V CM i.e.  AB separable  V is a genuine CM where V is the CM of W(  ) Then the PT implies V  V   AB separable   V  + i I (2) /2  o  V  > 0

19. Quantum Computers, Algorithms and Chaos - Varenna 2005 The condition  AB separable   V  + i I (2) /2  o if V is of the form ( ) A C C T B implies In the case of two-mode Gaussian states, the positivity of PT represents a necessary and sufficient condition for separability, i.e.  AB(Gaussian) separable  PT(  AB )  o This is the Simon’s criterion Where J = ( ) 0 1 -1 0 is the one-mode symplectic matrix det A det B + ( 1/4 - |det C|) 2 - Tr (AJCJBJC T J) - det A + det B 4  o

20. Quantum Computers, Algorithms and Chaos - Varenna 2005 Generation of CV bipartite entangled state a1a1 a2a2 b1b1 b2b2 Consider a beam splitter described by the operator B( ,  ) = exp  /2(a + 1 a 2 e i  - a 1 a + 2 e -i  ) where the transmission and reflection coefficients are t = cos  /2 r = sin  /2 while  is the phase difference between the reflected and transmitted fields If the input fields are coherent states we have B( ,  )|  1 > 1 )|  2 > 2 = B( ,  )D 1 (  1 )D 2 (  2 )|0 > 1 )|0 > 2 = D(  ) = exp(  a + -  *a ) D 1 (  1 cos  /2 +  2 e i  sin  /2) D 2 (  2 cos  /2 -  1 e -i  sin  /2)| 0 > 1 |0 > 2 = |  1 t +  2 r e i  > 1 |  2 t -  1 r e -i  > 2 And are not entangled. When the input states are squeezed states we get: ( with  j = r j e -2i  ) B( ,  ) S 1 (  1 )S 2 (  2 )|0 > 1 )|0 > 2 = B( ,  ) S 1 (  1 ) B + ( ,  ) B( ,  ) S 2 (  2 ) B + ( ,  ) B( ,  ) | 0 > 1 )|0 > 2 = S 1 (r 1 + r 2 e 2i  ) S 2 (r 1 e -2i  + r 2 ) S 12 (r 1 e -i  - r 2 e i  ) | 0 > 1 )|0 > 2 S 12 (  ) =exp( -  a 1 a 2 +  * a + 1 a + 2 ) is the two-mode squeezing operator

21. Quantum Computers, Algorithms and Chaos - Varenna 2005 Applying the two-mode squeeze operator S(  ) = exp - r( a + 1 a + 2 - a 1 a 2 ) (  = r e 2i  with  = 0 is the squeezing parameter) to two vacuum modes | 0 > 1 | 0 > 2 And using the disentangling theorem of Collet ( Phys. Rev. A 38, 2233 (1988)). S(  ) | 0 > 1 | 0 > 2 = exp(a + 1 a + 2 tanh r ) ( ) 1 cosh r a + 1 a 1 + a + 2 a 2 + 1 X exp - (a 1 a 2 tanh r ) | 0 > 1 | 0 > 2 = √ 1 - ∑ n n/2 | n > 1 | n > 2 = tanh 2 r The two-mode squeezed vacuum state is the quantum optical representative for bipartite continuous-variable entanglement.

22. Quantum Computers, Algorithms and Chaos - Varenna 2005 An arbitrary two-mode Gaussian state  AB can be associated to its displacement d and CM V ( such association is a 1-1 correspondence in the case of a Gaussian state). Since its separability properties do not vary under LOCCs, we may, first, cancel its displacement d via local displacement operators D k ( k = A,B), and then reduce its CM V to the normal form For the two-mode vacuum squeezed state we have

23. Quantum Computers, Algorithms and Chaos - Varenna 2005 The best way to see that it is really entangled is to consider the Simon criterion. Applied to the previous CM gives 1/2  (cosh 4r)/2 that is never satisfied for  r  0. With the V matrix given in normal form the Simon’s separability criterion  V  + iI (2) /2  0 reads 4(ab - c 2 ) (ab - c’ 2 )  (a 2 + b 2 ) + 2 |c c’| - 1/4

24. Quantum Computers, Algorithms and Chaos - Varenna 2005 CV TRIPARTITE entangled states We’ll consider the scheme introduced by Dür et al. PRL 83, 3562 (1999) According with their classification one has five entanglement classes A tripartite state is composed by three distinguishable parties A,B and C

25. Quantum Computers, Algorithms and Chaos - Varenna 2005 The extension to more dimensions of the Simon’s criterion was proved by Werner and Wolf PRL 86, 3658 (2001) and is based on the positivity of the partial transpose Let us consider the Gaussian state  1N, which is a bipartite state of 1 X N ( i.e. 1 mode at Alice site and N on Bob’s site) and V 1,N be its CM, the partial transposition is given applying  1 =  I (the partial transpose at Alice). CRITERION A Gaussian state  1N of a 1 X N system is separable (with respect to the grouping A ={1} and B ={2,…..,N}) if and only if  1 V 1N  1 + iI (N+1) /2  0 The classification is given then as: (V’ K =  K V ABC  K + iI (3) /2 K = A,B,C ) class 1  V’ A  0, V’ B  0, V’ C  0 class 2  V’ A  0, V’ B  0, V’ C  0 (permutation of A,B,C) class 3  V’ A  0, V’ B  0, V’ C  0 (permutation of A,B,C) class 4 or 5  V’ A  0, V’ B  0, V’ C  0

26. Quantum Computers, Algorithms and Chaos - Varenna 2005 For small mirror displacements, in interaction picture H = –∫d 2 r P(r,t) x(r,t) P is the radiation pressure force and x is the mirror displacement (r is the coordinate on the mirror surface) x(r,t) BOB (Loudon et al. 1995) Pirandola et al. J. Mod. Opt. 51, 901 (2004)

27. Quantum Computers, Algorithms and Chaos - Varenna 2005 Pinard et al. Eur.Phys. J. D 7,107 (1999) Frequency  and mass M 00 0-0- 0+0+ x(r,t)  b e -i  t + b + e i  t )exp[-r 2 /w 2 ] fundamental Gaussian mode

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29. RWA (neglecting all terms oscillating faster than  ) H eff = - i  (a 1 b - a 1 + b + ) - i  (a 2 b + - a 2 + b) a 1 @  0 -  and a 2 @  0 +  parametric interaction generates EPR-like entangled states used in continuous variable teleportation rotation (BS) it might degrade entanglement quant-ph/02/07094 - JOSA B 2003

30. Quantum Computers, Algorithms and Chaos - Varenna 2005 Dynamics is studied with the normally ordered characteristic function  ( ,t=0) = e -n th | | 2   corresponding to a 1,b,a 2 n th = average number of thermal excitations for mode b i.e. initially vacuum states for a 1 and a 2 and b in a thermal state

31. Quantum Computers, Algorithms and Chaos - Varenna 2005 After an interaction time  the state is  1 b 2 = ∫ d 2  ∫ d 2 ∫ d 2  ( ,  ) e -(|  | 2+ | | 2+ |  | 2) D 1 (-  ) D b (- ) D 2 (-  )  ( ,  ) is the evolution of  ( ,  ) which is still Gaussian  ( ,  ) = e -  V  T               ) D i = normally ordered displacement operators D i (  ) = e -  c i  e  *c i We can now study the class of entanglement of the tripartite state  1 b 2 Werner & Wolf PRL 86, 3653 (2001)

32. Quantum Computers, Algorithms and Chaos - Varenna 2005 It turns out that  1 b 2 is bi-separable with respect to b at interaction times  =  …. i.e.  1 b 2 is a one-mode bi-separable state (class 2 entangled) In particular the tripartite state at these times can be written as the tensor product of the initial thermal state of the mirror and a pure EPR state for a 1 and a 2 with squeezing parameter depending on  /  !!Radiation pressure could be a source of two-mode entangled states!!

33. Quantum Computers, Algorithms and Chaos - Varenna 2005 At all other times the tripartite state  1 b 2 is fully entangled (class 1) at any n th By tracing out one mode of the three we study the entanglement of a bipartite subsystem We find that mode a 2 and b are never entangled Modes a 1 and a 2 are entangled (extremely robust with respect to the mirror temperature n th ) Modes a 1 and b are entangled even though the region of entanglement is small and depends on n th a1a1 a2a2 b

34. Quantum Computers, Algorithms and Chaos - Varenna 2005 Braunstein and van Loock Quantum Information with Continuous Variables to appear in Rev. Mod. Phys. quant-ph/0410100 REFERENCES