Quantification of entanglement via uncertainties

Transcription

1 Quantification of entanglement via uncertainties Barış Öztop Bilkent University Department of Physics September 2007

2 In blessed memory of Alexander Stanislaw Shumovsky

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5 Outline Introduction what can we say about entanglement? Measure of entanglement Two-qubit systems Higher dimensional bipartite systems Multi-qubit systems Dynamic symmetry approach Basic observables Variance/uncertainty New measure of entanglement Concluding remarks

6 Introduction Wikipedia: Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. Stanford Encyclopedia of Philosophy: Quantum entanglement is a physical resource, like energy, associated with the peculiar nonclassical correlations that are possible between separated quantum systems. Entanglement can be measured, transformed, and purified. A pair of quantum systems in an entangled state can be used as a quantum information channel to perform computational and cryptographic tasks that are impossible for classical systems.

7 Entanglement By these two and many other definitions: Entanglement non-classical correlations for two or more systems that are separated spatially. Non-classical correlations non-locality. Non-locality Bell s conditions. But, unentangled states can violate Bell s conditions or violation of Bell s conditions can not be interpreted as an indisputable sign of quantum non-locality. H. Barnum, E. Knill, G. Ortiz, and L. Viola, Phys. Rev. A (2003) A.A. Klyachko, J. Phys.: Conf. Series 36, 87 (2006)

8 Entanglement Entanglement is a physical phenomenon representing the characteristic trait of quantum mechanics (Schödinger 1935). We can all agree that Entanglement is one of the basic resources for quantum computing and quantum information technologies. Quantum teleportation, critical ingredient for quantum computation networks, uses entangled states. Secure communications based on quantum key distribution has been realized by using the properties of the entangled states. = It is needed to find reliable methods of detection of the amount of entanglement carried by the states.

9 Measure of entanglement Two qubits The most basic part of a quantum information system is a qubit two level quantum system (e.g. spin-1/2 particle). A state of two qubits: ψ H = H A H B, ψ = 1 l,l =0 ψ ll l l, dimh A = dimh B = 2. For pure states of two qubits, ρ = ψ ψ, entanglement (entanglement of formation) is entropy of either of the two subsystems E(ψ) = Tr(ρ A log 2 ρ A ) = Tr(ρ B log 2 ρ B ). ρ A(B) is the partial trace of ρ over subsystem A(B).

10 Two qubits This can be recast into E(C) = h C2 (ρ) 2 h(x) = xlog 2 x (1 x)log 2 (1 x) is the binary entropy function and, C(ρ) = max{0, λ 1 λ 2 λ 3 λ 4 } is called concurrence, where λ i s are the square roots of the eigenvalues of the non-hermitian matrix ρ ρ = ρ(σ A y σ B y ρ σ A y σ B y ) and σ A j = σ j 1, σ B j = 1 σ j. E(C) is a monotonically increasing function of C and ranges from 0 to 1 as C goes from 0 to 1 = Concurrence itself is a measure of entanglement. C.H. Bennett et al, Phys. Rev. A 54, 3824 (1996) S. Hill and W.K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).

11 Higher dimensional bipartite systems C(ψ) = 2 det[ψ] = 2 ψ 00 ψ 11 ψ 10 ψ 01 Concurrence can be recast into the form C(ρ) = ν[1 Tr(ρ 2 r)]. Here ν = d/(d 1) where d is the dimensionality of one of the subsystems. ρ r = ρ A = ρ B since reduced operators are isospectral. This general form of concurrence is the measure of the amount of entanglement that is valid for pure bipartite states or arbitrary dimensions. P. Rungta, V. Bužek, C.M. Caves, M. Hillery, and G.J. Milburn, Phys. Rev. A, 64, (2001).

12 Multi-qubit systems Example: Three qubit system An arbitrary normalized state of three qubits ψ = 1 l,m,n=0 ψ lmn lmn. The amount of entanglement in three qubit system is measured by 3-tangle τ(ψ) = 4 D(ψ). Here D(ψ) = det[ψ] the Cayley s hyperdeterminant (in this specific case 3-dimensional determinant) of the 3-dimensional coefficient matrix [ψ]. V. Coffman, J. Kundu and W.K. Wootters, Phys. Rev. A 61, (2000) A. Miyake, Phys. Rev. A 67, (2003). Problem: This only measures all party (in this case 3) correlations.

13 Multi-qubit systems Example: Three qubit system Consider the well-known GHZ-state GHZ = 1 2 ( ). 3-qubit correlations = τ(ghz) = 1, maximally entangled and no pairwise correlations. Consider the W -state W = 1 3 ( No 3-qubit correlations, but obviously there are pair correlations between all three qubits = entanglement, but τ(w ) = 0. What is the amount of entanglement? Consider biseparable state Bi = ( ). Again no 3-qubit correlations, but there are pair correlations between last two qubits = there is entanglement, but τ(bi) = 0. Let s consider another approach to entanglement and a possible measure for its degree.

14 Dynamic symmetry approach Q: What is not entanglement? A: It is a property inherit in quantum systems only. = Any attempt to explain entanglement requires the definition of a quantum system. Quantum entanglement manifests itself via measurement of physical observables (Bell 1966). von Neumann picture: all Hermitian operators represent measurable physical quantities and all of them are supposed to be equally accessible. Physical nature of the system often imposes inevitable constraints. E.g. components of the composite system H AB = H A H B may be spatially separated by tens of kilometers = only local local observations X A and X B are available.

15 Dynamic symmetry approach So, available observables should be included in description of any quantum system from the outset. The basic principles of quantum mechanics seem to require the postulation of a Lie algebra of observables and a representation of this algebra by skew-hermitian operators. (Robert Hermann 1966) Lie algebra of observables L. We choose orthogonal basis X i of L as basic observables, whose measurement give us the whole allowed information about a given state of the system. The corresponding Lie group G = exp(il) determines the dynamic symmetry of the system, dynamic symmetry group. Unitary representation of G in the state space H S is quantum dynamical system. A.A. Klyachko, E-print quant-ph/

16 Dynamic symmetry approach Examples: A qubit (state in two-dimensional Hilbert space H 2 ). Dynamic symmetry groupg = SU(2) Basic observables = 3 Pauli operators X i = σ i, (i = x, y, z). N qubits (state in H = N j=1 H 2 ) Dynamic symmetry group G = N j=1 SU(2) Basic observables = 3N pauli operators (3 Pauli operators for each part) Xi α = σi α (e.g. σi A = σ i 1 1)). }{{} N 1

17 Dynamic symmetry approach Variance The level of quantum fluctuations of a basic observable Xi α ψ H S os a system S is given by the variance V (X α i, ψ) = ψ (X α i ) 2 ψ ψ X α i ψ 2 0. in state Summation over all basic observables of the quantum dynamic system V (ψ) = V α i (Xα i, ψ) = i ψ (Xα i ) 2 ψ ψ Xi α ψ 2 is total uncertainty peculiar to the state ψ. For a compact Lie algebra L, i Xi 2 = C HS 1 is Casimir operator. = V (ψ) = NC HS α N is the number of parties. i ψ Xα i ψ 2,

18 Dynamic symmetry approach Definition of completely entangled states Necessary and sufficient condition of complete entanglement i, α ψ CE X i ψ CE = 0, for ψ CE H S. A.A. Klyachko, E-print quant-ph/ A.A. Klyachko and A.S. Shumovsky 2004 J. Opt. B: Quant. and Semiclas. Optics 6 S29. = V (ψ CE ) = max ψ H S V (ψ) = C HS. Very similar to the maximum of entropy principle, defining the equilibrium states in quantum statistical mechanics (Landau and Lifshitz 1980). These definitions of basic observables and equations of complete entanglement do not assume the composite nature of the system S.

19 Dynamic symmetry approach New measure of entanglement So, one can see entanglement as a manifestation of quantum fluctuations in a state where they come to their extreme! Q: Can total uncertainty for a pure state of a quantum system tell us anything about the amount of entanglement? A: YES! States opposite to entangled ones, those with minimal total level of quantum fluctuations are known as coherent states. For multicomponent systems like H A H B ψ coh = ψ A ψ B. For two component systems C 2 (ψ) = V (ψ) V coh V CE V coh. A.A. Klyachko, B. Öztop and A.S. Shumovsky, Appl. Phys. Lett. 88, (2006).

20 Dynamic symmetry approach New measure of entanglement This clarifies physical meaning of the concurrence as a measure of overall quantum fluctuations in the system and leads us to the natural measure of entanglement of pure states µ(ψ) = V (ψ) V coh V CE V coh. A.A. Klyachko, B. Öztop and A.S. Shumovsky, Phys. Rev. A 75, (2007). This measure is shown to be valid to measure the amount of entanglement for pure states of any dimensional multi-component systems. For bipartite systems of arbitrary dimensions it coincides with the earlier definition C(ψ) = d d 1 (1 Tr ρ2 r).

21 Dynamic symmetry approach New measure of entanglement In the case of multipartite system, it gives the total amount of entanglement carried by all types of inter-party correlations. Examples GHZ-type state of three qubits G = x x 2 111, x [0, 1] carries only 3-party entanglement and its amount is τ(g) = 4 x 2 (1 x 2 ). Easy to check τ(g) = µ 2 (G). W -state of three qubits W = 1 3 ( ) is a nonseparable state, but it does not manifest 3-party entanglement, τ(w ) = 0. The measure µ(w ) = is nonzero since there is 2-qubit entanglement. Similar results can be obtained for the so-called biseparable states of three qubits (e.g. Bi = ( )) since there is also 2-qubit entanglement.

22 Conclusion Successful measures of entanglement for some specific type of systems are present, and they work well within their boundaries. We need a precise definition of a quantum system to define and measure a quantum-only property = dynamic symmetry approach. A new measure of entanglement for pure states of arbitrary systems, it works for the known examples and measures entanglement carried by all types of inter-party correlations. It cannot be directly applied to calculation of entanglement of mixed states because it is incapable of separation of classical and quantum contributions into the total variance. Therefore, µ(ρ) always gives estimation from above for the entanglement of mixed states.