Detecting genuine multipartite entanglement in higher dimensional systems

Transcription

1 University of Vienna March 16, 2011 (University of Vienna) March 16, / 19

2 (University of Vienna) March 16, / 19

3 To start with some easy basics (University of Vienna) March 16, / 19

4 To start with some easy basics (University of Vienna) March 16, / 19

5 To start with some easy basics Just joking!! ;) (University of Vienna) March 16, / 19

6 Dicke- are entangled (University of Vienna) March 16, / 19

7 Dicke- are entangled First investigated with respect to light emission of a cloud of atoms (by Robert H. Dicke in 1954) (University of Vienna) March 16, / 19

8 Dicke- are entangled First investigated with respect to light emission of a cloud of atoms (by Robert H. Dicke in 1954) They can easily be converted to GHZ- or W- (University of Vienna) March 16, / 19

9 Dicke- are entangled First investigated with respect to light emission of a cloud of atoms (by Robert H. Dicke in 1954) They can easily be converted to GHZ- or W- Providing a rich resource for quantum information tasks (University of Vienna) March 16, / 19

10 Dicke- are entangled First investigated with respect to light emission of a cloud of atoms (by Robert H. Dicke in 1954) They can easily be converted to GHZ- or W- Providing a rich resource for quantum information tasks For experimentalists: Remarkable robustness to particle loss! (University of Vienna) March 16, / 19

11 We assume ψ H = C d1 C d2... C dn. If a state can be writen as ψ = φ 1 φ 2... φ k we call it k-seperable. We call a mixed state k-seperable if there exists a decomposition of the form ρ = i p i ψ i k ψi k with i p i = 1, p i > 0 (University of Vienna) March 16, / 19

12 Conceptual structur of in terms of seperability (University of Vienna) March 16, / 19

13 We introduce Π that stands for the global permutation operator which performs simultaneous permutations on all sub. P i = Π Ai 1 Bi permutes in the fictitious sub we labeled A i and B i (i = 1,..., 2 n 1 1), while the sum runs over all inequivalent bipartitions. (University of Vienna) March 16, / 19

14 We introduce Π that stands for the global permutation operator which performs simultaneous permutations on all sub. P i = Π Ai 1 Bi permutes in the fictitious sub we labeled A i and B i (i = 1,..., 2 n 1 1), while the sum runs over all inequivalent bipartitions. Theorem φ ρ 2 Π φ i φ P i ρ 2 P i φ 0 holds biseperable ρ S 2 if φ is fully seperable. (University of Vienna) March 16, / 19

15 Note that the potential to detect the character of a given entangled state will depend on a suitable choice of φ. (University of Vienna) March 16, / 19

16 Note that the potential to detect the character of a given entangled state will depend on a suitable choice of φ. This however is far less difficult then for comparable problems, e.g optimization of witnesses. (University of Vienna) March 16, / 19

17 Dicke To get a feeling for how Dicke look like, we have a look on the simplest case i.e. n = 4 and m = 2, D2 4 = 1 ( ). 6 For our citeria we need a generalized definition of the Dicke. (University of Vienna) March 16, / 19

18 We define the generalized Dicke as following Dm n = 1 d N {α} {α} with N = d {α} = i / {α} ( ) 1 n 2 m 0 i i {α} 1 i where i = [1, n] N {α} = m N where the sum is taken over all inequivalent permutations. (University of Vienna) March 16, / 19

19 For the simplest case, i.e. n = 4 and m = 2 we get {{12}, {13}, {14}, {23}, {24}, {34}} for the inequivalent permutations of {α}. Now we have d {12} = 1100 d {13} = and as we sum them up we have D2 4. (University of Vienna) March 16, / 19

20 We define Π {α} as the cyclic permutation operator acting on the twofold copy Hilbert space, e.g. Π {1} φ 1 φ 2 ψ 1 ψ 2 = ψ 1 φ 2 φ 1 ψ 2 Also {γ} = {({α}, {β}) : {α} {β} = 1} and ( ) n m 1 N D = m. m 1 (University of Vienna) March 16, / 19

21 We define Π {α} as the cyclic permutation operator acting on the twofold copy Hilbert space, e.g. Π {1} φ 1 φ 2 ψ 1 ψ 2 = ψ 1 φ 2 φ 1 ψ 2 Also {γ} = {({α}, {β}) : {α} {β} = 1} and ( ) n m 1 N D = m. m 1 Main result For 1 < m n 2 I n m[ρ] 0 holds for all biseparable. (University of Vienna) March 16, / 19

22 Maximal violation Note that the inequality is always maximally violated by the corresponding Dicke state D n m, with the number of excitations m equal to {α}, with a value of I n m[ D n m D n m ] = m((n m) N D ). (University of Vienna) March 16, / 19

23 Noise resistance We can now mix the Dicke state with white noise to figure out the noise resistance. Evaluating our criteria for the state ρ noise = (1 p) D n m D n m + p 1 2 n 1. we can derive a general expression for the noise resistance 1 m p < 1 m 1 ( n ). 2 n m (2n 3m + 1) (University of Vienna) March 16, / 19

24 Figure: The plot shows the white noise resistance parameter p in terms of n, where from left to right the resistance is plotted for m = 2 to m = 20 in ascending order. (University of Vienna) March 16, / 19

25 Figure: This plot shows the detection quality for the six qubit state ρ = pρ D2 + qρ D3 + 1 p q 2 1, where ρ 6 D2 = D2 6 D6 2 and ρ D3 = D3 6 D6 3. Region I : D 2 represents the parameter region for which the state is detected to be ly entangled corresponding to D2 6 and analogously I : D 3 corresponds in the same way using D3 6. The region F : D 3 depicts the parameter region detected by the fidelity witness used in Ref. 1 which clearly demonstrates the improvement provided by inequality. (University of Vienna) March 16, / 19

26 Experimental feasibility We can re-express any density matrix element in terms of local expectation values of Pauli operators. (University of Vienna) March 16, / 19

27 Experimental feasibility We can re-express any density matrix element in terms of local expectation values of Pauli operators. This is a crucial issue for experimental implementation. (University of Vienna) March 16, / 19

28 Experimental feasibility We can re-express any density matrix element in terms of local expectation values of Pauli operators. This is a crucial issue for experimental implementation. While a full state tomography becomes unachievable with growing system size, i.e. the number of local measurements required grows exponentially (with 2 2n for n qubits). (University of Vienna) March 16, / 19

29 Experimental feasibility We can re-express any density matrix element in terms of local expectation values of Pauli operators. This is a crucial issue for experimental implementation. While a full state tomography becomes unachievable with growing system size, i.e. the number of local measurements required grows exponentially (with 2 2n for n qubits). The number of density matrix elements which need to be ascertained for our inequalities grows only polynomially with system size (e.g. for m = 2 with n 3 ). (University of Vienna) March 16, / 19

30 Experimental feasibility We can re-express any density matrix element in terms of local expectation values of Pauli operators. This is a crucial issue for experimental implementation. While a full state tomography becomes unachievable with growing system size, i.e. the number of local measurements required grows exponentially (with 2 2n for n qubits). The number of density matrix elements which need to be ascertained for our inequalities grows only polynomially with system size (e.g. for m = 2 with n 3 ). E.g. for four qubits there are 39 local measurement settings required, which is a lot more feasible than the 255, required for a full sate tomography. (University of Vienna) March 16, / 19

31 and Outlook New inequalities for other classes of entangled? Inequalities as lower bounds for a measure? see arxiv: Outlook We are expecting that we are able to improve the structure of the presented criteria and make them even stronger. (University of Vienna) March 16, / 19

32 R. H. Dicke, Phys. Rev. 93, 99 (1954). M. Huber, F. Mintert, A. Gabriel and B.C. Hiesmayr, Phys. Rev. Lett. 104, (2010). M. Huber, P. Erker, H. Schimpf, A. Gabriel and B.C. Hiesmayr, arxiv/quant-ph: (University of Vienna) March 16, / 19