ENTANGLEMENT IN QUANTUM OPTICS Paolo Tombesi Department of

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

entanglement: polarization of two photons ( H 1 H 2 ± V 1 V 2)/√ 2 or ( H 1 V 2 ± H 2 V 1 )/√ 2 These are the so-called Bell states ( ) and ( ) In general, for a bipartite system, it is separable 12 = i wi i 1 i 2 wi ≥ 0 i wi =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) Quantum Computers, Algorithms and Chaos - Varenna 2005

x H 2 the arbitrary state of the bipartite state 1+2 is described Given an orthonormal basis in H 12 = H 1 O 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 inversion of Latin indices (Greek) PT : ( 12 )m , n ( 12 )n , m ( T 1 12 )m , n 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 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 T 1 12 < 0 12 entangled In 2 x 2 and 2 x 3 dimension for the Hilbert space 12 separable T 1 12 ≥ 0 Quantum Computers, Algorithms and Chaos - Varenna 2005 Horodecki 3 Phys Lett A 223, 8 (1996)

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

Quantum Computers, Algorithms and Chaos - Varenna 2005

Phase Matching TYPE II Laser beam Aperture NLC( (2)) Dichroic Mirror Pump Laser @405 nm 0. 05 810 nm Extraordinary y 0. 0 | > = | H V > + e i | V H > Ordinary 810 nm -0. 05 2005 Quantum Computers, Algorithms and Chaos - Varenna -0. 05 0. 0 x 0. 05

Quantum Computers, Algorithms and Chaos - Varenna 2005

det T 212 = 0. 54 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 separable state on the Hilbert space Htot = H 1 Ox H 2. sep = i wi i 1 i 2 wi ≥ 0 i wi =1 Let us now choose a generic couple of observables for each subsystem, say rj , sj on Hj (j = 1, 2), Cj = i [rj , sj ] , j = 1, 2 Cj is typically nontrivial Hermitian operator on the Hilbert subspaces. Let’s define two Hermitian operators on Htot From the standard form u = a 1 r 1 + a 2 r 2 , of the uncertainty principle, it follows that v = b 1 s 1 + b 2 s 2 every state on Htot must satisfy where aj , bj are real parameters < ( u Quantum Computers, Algorithms and Chaos - Varenna 2005 )2 > < ( v )2 > ≥ | a 1 b 1< C 1> + a 2 b 2< C 2>|2 4

For separable states the following Theorem holds sep < ( u )2> < ( v )2> ≥ O 2 With O = ( | a 1 b 1| < O 1> + | a 2 b 2| < O 2> ) /2 < Oj> = kwk < Cj >k = Tr [Cj k j] i. e. the expectation value of operator Cj onto k j Proof : From the definition of < ( u )2> and sep it is easy to see that < ( u )2> = kwk [ a 12 < ( r 1(k) )2>k + a 22 < ( r 2(k) )2>k ] + kwk < uk>2 - ( kwk < uk>)2 With rj(k) = rj - < rj>k the variance of rj onto the state kj For the Cauchy-Schwartz inequality < ( u )2> are bound below by zero The same for < ( v )2> kwk < uk>2 ≥( kwk < uk>)2 the last two terms in Follows < ( u )2> ≥ kwk [ a 12< ( r 1(k) )2>k + a 22 < ( r 2(k) )2>k ] and < ( v )2> ≥ kwk [ b 12 < ( s 1(k) )2>k + b 22 < ( s 2(k) )2>k ] Quantum Computers, Algorithms and Chaos - Varenna 2005

Given any two real non negative numbers and < ( u )2> + < ( v )2> ≥ kwk [ a 1 2< ( r 1(k) )2>k + b 12 < ( s 1(k) )2>k + + kwk [ a 2 2< ( r 2(k) )2>k ] + b 22 < ( s 2(k) )2>k ] Furthermore, by applying the uncertainty principle to the operators rj and sj on the state k j , it follows aj 2< ( rj(k) )2>k+ bj 2 < ( sj(k) )2>k ≥ aj 2< ( rj(k) )2>k + bj 2 |< Cj>k |2 ≥ 4 < ( rj(k) )2>k √ |ajbj| |< Cj>k | (min f(x) = 1 x + 2 /x fmin = 2√ 1 2) Finally < ( u )2> + < ( v )2> ≥ 2 √ O That has to be satisfied for any and positive, thus maximizing g(x) = 2 x O - x 2 < ( v )2> for x > 0 < ( u Quantum Computers, Algorithms and Chaos - Varenna 2005 )2 > ≥ O 2 < ( v )2>

Connection with other criteria The Duan et al. criterion (the sum criterion) Phys. Rev. Lett. 84, 2722 (2000) is a particular case of sep < ( u )2> < ( v )2> ≥ O 2 Indeed, if we pose = = 1 in < ( u )2> + < ( v )2> ≥ 2 √ O We get < ( u )2> + < ( v )2> ≥ 2 O however Then < ( u )2 > ≥ O 2 < ( v )2> 2 O 2 2 < ( u ) > + < ( v ) > ≥ + < ( v )2> ≥ 2 O 2 < ( v ) > The last inequality holds because of the function (min f(x) = 1 x + 2 /x fmin = 2√ 1 2) Giovannetti et al. Phys. Rev. A 67, 022320 (2003) 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 ) [X( ) , P( ) ] = i With 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 ) ] Quantum Computers, Algorithms and Chaos - Varenna 2005 in phase out of phase

An arbitrary single mode state can be associated, by a 1 to 1 correspondence, to a symmetrically ordered characteristic function = R +i I ( ) = Tr [ D( ) ] 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 Ox N The W-function becomes Quantum Computers, Algorithms and Chaos - Varenna 2005 W( 1… N) = - 2 N [ k ∫d 2 k exp( - k k* + k* k )] ( 1. . N)

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 R)= k real R 2 (k = 1, . . , N) I - k In terms of the real variables ( T 1, …. . , TN) T R 2 N the arbitrary N-mode Gaussian characteristic function takes the form ( ) = exp ( - TV + i d. T ) Where d R 2 N and V is a 2 N x 2 N real, symmetric strictly positive matrix V=VT V >0 The corresponding N-mode Gaussian W-function by the inverse F-T is: exp( -1/4 d. TV -1 d) W( ) = N √ det V exp ( - T V-1 + d. T V-1 ) 2 √ 2 T (x 1, p 1, …. , x. N, p. N) R 2 N Quantum Computers, Algorithms and Chaos - Varenna 2005 Define a 2 N-dim phase space

V is the correlation matrix (CM) of the N-mode state Vnm = ( n m + m n) /2 (n, m = 1, 2, …. , 2 N) m= 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 < ( xk ) 2 > < ( pk’ ) 2 > 1/4 k k’ (k, k’ = 1, . . , N) 0 1 N Introducing the N-mode symplectic matrix I(N)= Ii Ii = -1 0 1 ( ) The uncertainty principle reads V + i I(N)/2 o 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 HAB is equivalent to a partial mirror reflection of W( ) in the phase space PT : AB PT( AB ) 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 Quantum Computers, Algorithms and Chaos - Varenna 2005 V >0

The condition AB separable V + i I(2)/2 o if V is of the form ( A C CT B ) implies det A det B + ( 1/4 - |det C|)2 - Tr (AJCJBJCTJ) - det A + det B 4 0 1 Where J = is the one-mode symplectic matrix -1 0 ( o ) 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 Quantum Computers, Algorithms and Chaos - Varenna 2005

Generation of CV bipartite entangled state b 2 Consider a beam splitter described by the operator B( , ) = exp /2(a+1 a 2 e i - a 1 a+2 e-i ) a 1 where the transmission and reflection coefficients are t = cos /2 r = sin /2 while is the phase difference between the reflected and transmitted fields b 1 a 2 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= rj e -2 i ) 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 2 i ) S 2(r 1 e-2 i + r 2) S 12(r 1 e-i - r 2 ei ) | 0 >1)|0 >2 S 12( )=exp( - a 1 a 2+ * a+1 a+2) is the two-mode squeezing operator 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 2 i 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. 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 Dk ( k = A, B), and then reduce its CM V to the normal form For the two-mode vacuum squeezed state we have Quantum Computers, Algorithms and Chaos - Varenna 2005

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

CV TRIPARTITE entangled states A tripartite state is composed by three distinguishable parties A, B and C We’ll consider the scheme introduced by Dür et al. PRL 83, 3562 (1999) According with their classification one has five entanglement classes 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 1 N , 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 1 N 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 1 N 1 + i. I (N+1)/2 0 The classification is given then as: class 1 V’A 0, class 2 V’A 0, class 3 V’A 0, class 4 or 5 V’A 0, (V’K = K VABC K+ i. I (3)/2 K = A, B, C ) V’B 0, V’C 0 (permutation of A, B, C) V’B 0, V’C 0 Quantum Computers, Algorithms and Chaos - Varenna 2005

Pirandola et al. J. Mod. Opt. 51, 901 (2004) For small mirror displacements, in interaction picture (Loudon et al. 1995) 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) Quantum Computers, Algorithms and Chaos - Varenna 2005 BOB

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

Quantum Computers, Algorithms and Chaos - Varenna 2005

RWA (neglecting all terms oscillating faster than W) quant-ph/02/07094 - JOSA B 2003 Heff = - i (a 1 b - a 1+b+) - i (a 2 b+ - a 2+b) a 1 @ 0 -W and a 2 @ 0+W parametric interaction generates EPR-like entangled states used in continuous variable teleportation Quantum Computers, Algorithms and Chaos - Varenna 2005 rotation (BS) it might degrade entanglement

Dynamics is studied with the normally ordered characteristic function ( , , , t=0) = e -nth| |2 , , ������ corresponding to a 1, b, a 2 nth = 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 Quantum Computers, Algorithms and Chaos - Varenna 2005

After an interaction time t the state is 1 b 2 = ∫ d 2 ( , , , t) e -(| |2+ | |2)D 1(- ) Db(- ) D 2(- ) Di = normally ordered displacement operators Di ( ) = e- ci e *ci ( , , , t) is the evolution of ( , , , ) which is still Gaussian ( , , , t) = e- V T T = 1, 2, 1, 2 ) We can now study the class of entanglement of the tripartite state 1 b 2 Werner & Wolf PRL 86, 3653 (2001) Quantum Computers, Algorithms and Chaos - Varenna 2005

It turns out that 1 b 2 is bi-separable with respect to b at interaction times t = , 3 , 5 , …. 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!! Quantum Computers, Algorithms and Chaos - Varenna 2005

At all other times the tripartite state 1 b 2 is fully entangled (class 1) at any nth b a 1 By tracing out one mode of the three we study the entanglement of a bipartite subsystem a 2 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 nth) Modes a 1 and b are entangled even though the region of entanglement is small and depends on nth Quantum Computers, Algorithms and Chaos - Varenna 2005

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