Collective Uncertainty Entanglement Test
aa r X i v : . [ qu a n t - ph ] J un Collective Uncertainty Entanglement Test
Lukasz Rudnicki, ∗ Pawe l Horodecki,
2, 3 and Karol ˙Zyczkowski Center for Theoretical Physics, Polish Academy of Sciences,Aleja Lotnik´ow 32/46, PL-02-668 Warsaw, Poland Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk, PL-80-952 Gda´nsk, Poland National Quantum Information Centre of Gda´nsk, PL-81-824 Sopot, Poland Smoluchowski Institute of Physics, Jagiellonian University, ul. Reymonta 4, PL-30-059 Krak´ow, Poland
For a given pure state of a composite quantum system we analyze the product of its projectionsonto a set of locally orthogonal separable pure states. We derive a bound for this product analogousto the entropic uncertainty relations. For bipartite systems the bound is saturated for maximallyentangled states and it allows us to construct a family of entanglement measures, we shall call collectibility . As these quantities are experimentally accessible, the approach advocated contributesto the task of experimental quantification of quantum entanglement, while for a three–qubit systemit is capable to identify the genuine three-party entanglement.
PACS numbers: 03.67.Mn, 03.65.Ud
The phenomenon of quantum entanglement – non–classical correlations between individual subsystems – isa subject of an intense research interest [1–3]. Severalcriteria of detecting entanglement are known [2, 3], andsome of them can be implemented experimentally (see [4]for the review of specific experimental schemes). In par-ticular the issue of qualitative entanglement detection isquite well established including entanglement witnessesmethod (see [3]) and local uncertainty relations [5]. Onthe other hand, although various measures of quantumentanglement are analyzed [3, 6], in general they are moredifficult to be quantitatively measured in a physical ex-periment. To estimate experimentally the degree of en-tanglement of a given quantum state one usually relies[7] on quantum tomography or analogous techniques.The idea of entanglement detection and estimationwithout prior tomography [8, 9] involves the collectivemeasurement of two (or more) copies of the state asdemonstrated in [10]. Consequently recent attempts to-wards experimental quantification of entanglement arebased on finding collectively measurable quantities whichbound known entanglement measures from below and areexperimentally accessible [11, 12] (for review see [13]).The main aim of this work is to construct a familyof indicators, designed to quantify the entanglement of apure state of an arbitrary composite system, which can bemeasured in a coincidence experiment without attempt-ing for a complete reconstruction of the quantum state.Our approach, which leads to simple collective entan-glement test, is inspired by the entropic uncertainty rela-tions which are satisfied by any pure state. For instance,the sum of the Shannon entropies of the expansion co-efficients of a given pure state | ψ i ∈ H N expanded intwo mutually unbiased bases is bounded from below byln N [14]. This observation suggests to quantify the purestates entanglement by a function of the projections ofthe analyzed state | Ψ i of a composite system onto mutu-ally orthogonal separable pure states. The method we propose can be formulated in a rathergeneral case of a normalized pure state, h Ψ | Ψ i = 1, of acomposite system consisting of K subsystems. For sim-plicity we shall assume here that all their dimensions areequal, so we consider an element of a K-partite Hilbertspace H = H A ⊗ H B . . . ⊗ H K , where dim (cid:0) H A (cid:1) = . . . =dim (cid:0) H K (cid:1) = N . Let us select a set of N separable purestates of a K –quNit system, | χ sepj i = | a Aj i ⊗ . . . ⊗ (cid:12)(cid:12) a Kj (cid:11) .where | a Ij i ∈ H I with j = 1 , . . . , N and I = A, . . . , K .The key assumption is that all local states are mutuallyorthogonal, so that (cid:12)(cid:12) a I (cid:11) , . . . , (cid:12)(cid:12) a IN (cid:11) ∈ H I , (cid:10) a Ij (cid:12)(cid:12) a Ik (cid:11) = δ jk . (1) Entanglement detection —
In order to construct mea-surable indicators of quantum entanglement and findpractical entanglement criteria valid for any analyzedstate | Ψ i we define now the following quantity Y max [ | Ψ i ] = max | χ sep i N Y j =1 (cid:12)(cid:12) h Ψ | χ sepj (cid:11)(cid:12)(cid:12) . (2)This product of the projections of the state onto theset of N separable states, optimized over all possi-ble sets of mutually locally orthogonal states, | χ sep i = {| χ sep i , . . . , | χ sepN i} , will be called maximal collectibility .Note the difference with respect to the geometric mea-sure of entanglement [15], to define which one takes themaximum over a single separable state, | χ sep i . In thiscase this maximum, denoted in [15] by Λ max , is equalto unity if the analyzed state | Ψ i is separable and it issmaller for any entangled state, so to define the geometricmeasure of entanglement one takes 1 − Λ max . In contrast,taking in (2) the maximum of the product of the projec-tions of | Ψ i onto N ≥ | χ sepj i we facean inverse situation: we show below that Y max is thelargest for maximally entangled states, so this quantitycan serve directly as a quantificator of entanglement.To this end we shall start with a variational equation δδ | Ψ i N Y j =1 (cid:12)(cid:12) h Ψ | χ sepj (cid:11)(cid:12)(cid:12) − λ h Ψ | Ψ i = 0 , (3)where λ plays the role of a Lagrange multiplier associ-ated with the normalization constraint. This idea wasdeveloped by Deutsch in order to obtain the entropic un-certainty relation [16]. Equation (3) implies N Y j =1 (cid:12)(cid:12) h Ψ | χ sepj (cid:11)(cid:12)(cid:12) N X i =1 ( h χ sepi | Ψ i ) − h χ sepi | = λ h Ψ | . (4)Multiplying (4) by | Ψ i we find out that λ = N · Q Nj =1 (cid:12)(cid:12) h Ψ | χ sepj (cid:11)(cid:12)(cid:12) . Moreover, the contraction of (4) with | χ sepm i leads to |h Ψ | χ sepm i| = 1 /N for all values of m .From this result we havemax | Ψ i N Y j =1 (cid:12)(cid:12) h Ψ | χ sepj (cid:11)(cid:12)(cid:12) = N Y j =1 N = N − N , (5)which after formal optimization over | χ sep i implies thedesired inequality Y max [ | Ψ i ] ≤ N − N . (6)Using an auxiliary variable, Z max = − ln Y max this rela-tion takes the from Z max [Ψ] ≥ N ln N , analogous to theentropic uncertainty relation. Interestingly, for a bipar-tite system this inequality is saturated for the maximallyentangled state, | Ψ + i = √ N P i | i, i i . while in the caseof K –quNit system it is saturated for a generalized GHZstate, | GHZ i K = √ N P i | i i A ⊗ · · · ⊗ | i i K .Consider now the other limiting case of a separablestate | Ψ sep i = | Ψ A i ⊗ . . . ⊗ | Ψ K i . In this case the pro-jections factorize, h Ψ sep | χ sepj (cid:11) = K Y I = A h Ψ I | a Ij (cid:11) . (7)Furthermore, for each value of the index ( I = A, . . . , K )we can independently apply the result (5) and obtain N Y j =1 (cid:12)(cid:12) h Ψ I | a Ij (cid:11)(cid:12)(cid:12) ≤ N − N . (8)Thus, for any separable state we have N Y j =1 (cid:12)(cid:12) h Ψ sep | χ sepj (cid:11)(cid:12)(cid:12) ≤ K Y I = A max {| Ψ I i} N Y j =1 (cid:12)(cid:12) h Ψ I | a Ij (cid:11)(cid:12)(cid:12) = N − N · K , (9)so that Y max [ | Ψ sep i ] ≤ N − N · K . (10)This observation leads to the following separability cri-teria based on the maximal collectibility:( Y max [ | Ψ i ] > α K,N ) ⇒ ( | Ψ i - entangled) . (11)Here α K,N = N − N · K is the discrimination parameter. Multi qubit systems —
In the definition (2) of themaximal collectibility one performs a maximization overthe set of all N mutually orthogonal separable states | χ sepj i . The maximal collectibility Y max can be consid-ered as a pure state entanglement measure, and we derivebelow its explicit expression in the simplest case of a twoqubit system. However, it is also convenient to performthe optimization procedure stepwise and to consider firstan optimization over a single separable state.Let us then define a one-step maximum over the sepa-rable states belonging to the first subspace H A , Y a [ | Ψ i ] = max | a A i N Y j =1 (cid:12)(cid:12) h Ψ | χ sepj (cid:11)(cid:12)(cid:12) . (12)Note that the collectibility Y a , a function of the analyzedstate | Ψ i , is parameterized by the set a of N productstates | a Bj i ⊗ . . . ⊗ | a Kj i , with j = 1 , . . . , N . By construc-tion one has max a Y a [ | Ψ i ] = Y max [ | Ψ i ].Consider now the case of a K –qubit system ( N = 2).Writing an equation analogous to (3) and following thestandard variational approach we obtain an analyticalformula for the collectibility, Y a [ | Ψ i ] = 14 (cid:18)p G G + q G G − | G | (cid:19) , (13)expressed in terms of elements of the Gram matrix de-fined for a set of projected states. Here G jk = h ϕ j | ϕ k i ,while | ϕ j i ∈ H A denotes the state | Ψ i projected onto the j –th separable state living in K − B, . . . K , so that | ϕ j i = [ h a Bj | ⊗ . . . ⊗ h a Kj | ] | Ψ i .Because of (5) and (9) the collectibility Y a satisfies thesame uncertainty relations (6) and (10) as the maximalcollectibility Y max . This approach can be generalized tothe case of Hilbert spaces with different dimensions. Itcan be especially useful when dim (cid:0) H A (cid:1) is much largerthan the dimensions of remaining Hilbert spaces. Thiscase may for instance describe the entanglement with an environment . Two qubits —
Let us now investigate in more detailthe simplest case of a two–qubit system for which K = N = 2 and H = H A ⊗ H B . Any pure state | Ψ i AB can bethen written in its Schmidt form [2], | Ψ i AB = ( U A ⊗ U B ) (cid:20) cos (cid:18) ψ (cid:19) | i + sin (cid:18) ψ (cid:19) | i (cid:21) , (14)where U A ⊗ U B is a local unitary. The Schmidt angle ψ ∈ [0 , π ] is equal to zero for the separable state and to π/ Y a [ | Ψ i AB ] ≤ /
4. Moreover,if the state (14) is separable we have (10) Y a [ | Ψ sep i ] ≤ / Π Π Π Π Π Π Ψ - - FIG. 1: Color online. Parameters describing entanglement ofa two–qubit pure state | Ψ i AB as a function of the Schmidtangle ψ . We plot the minimal (blue/dotted), the average(green/dashed–dotted) and the maximal (red) values of therescaled collectibility [16 Y θ ( ψ ) − /
3. Positive values iden-tify entanglement. The black, dashed line shows the prob-ability P Y that the entanglement of | Ψ i AB is detected in aparticular random measurement. detector basis spanned in the second subspace H B , (cid:12)(cid:12) a B (cid:11) = cos (cid:18) θ (cid:19) | i + e ıφ sin (cid:18) θ (cid:19) | i , (cid:12)(cid:12) a B (cid:11) = sin (cid:18) θ (cid:19) | i − e ıφ cos (cid:18) θ (cid:19) | i , (15)where θ ∈ [0 , π ] and φ ∈ [0 , π ]. Due to this general form,our analysis becomes independent of the local unitary U B in (14). Note also that the expression (13) is independentof U A , thus our approach works universally for any two-qubit pure state. Using (15) we shall calculate the entriesof the Gram matrix and find Y θ ( ψ ) = (cid:16) ψ ) + p − θ ) cos ( ψ ) − cos (2 ψ ) (cid:17) . The collectibility Y θ ( ψ ) depends on the analyzed state( ψ ) and the detector parameters a = ( θ, φ ). The depen-dence on the azimuthal angle φ is trivial. If the state (14)is maximally entangled ( ψ = π/ Y θ ( π/
2) = 1 / θ, φ ).In order to characterize various possibilities to detectthe entanglement we analyze four quantities. Considerfirst the minimal ( θ = 0) and the maximal ( θ = π/ Y θ ( ψ ) with respect to the de-tector parameters ( θ, φ ), Y min ( ψ ) = sin ( ψ )4 , Y max ( ψ ) = (1 + sin ( ψ )) . Then define the mean collectibility Y = h Y a i a , averagedover the set of the detector parameters a = ( θ, φ ) with themeasure d Ω = sin ( θ ) dθ dφ/ (4 π ). This case, correspond-ing to the average over a random choice of the detectorparameters, Y ( ψ ) = ´ S d Ω Y θ ( ψ ), yields the result Y ( ψ ) = 11 − ψ ) + 3( π − ψ ) tan ( ψ )96 . (16) FIG. 2: Determination of the Gram matrix via conditionaloverlapping in the case of two polarization–entangled photonpairs. Each source produces a pair of photons in a polarizationstate | Ψ i AB . On the left side B the statistics of pairs of clicksafter two PBS–elements are measured. On the right side A the Hong–Ou–Mandel interference is performed. The number | G | is equal to the probability of the pair of the clicks at B multiplied by that of double click at A . Furthermore, we study the probability that the entan-glement is detected in a measurement with a randomchoice of the detector angle θ , P Y ( ψ ) = ´ P d Ω, where P = (cid:8) ( θ, φ ) ∈ S : Y θ ( ψ ) > / (cid:9) : P Y ( ψ ) = ( √ ψ ) − sin ( ψ ) | cos( ψ ) | for ψ ∈ (cid:2) , π (cid:3) ∪ (cid:2) π , π (cid:3) ψ ∈ (cid:2) π , π (cid:3) . (17)Analytical results for a pure state of the 2 × ψ ∈ [ π/ , π/ Y corresponds to an average obtainedby a sequence of measurements with a random choice ofthe detector parameters. Looking at the expression (13)we see that to compute the collectibility Y a it is enoughto determine the elements of the Gram matrix. Assumefirst that we analyze a two–photon polarization entangledstate. The diagonal element G jj represents an amplitudeof the state | ϕ j i in the first subspace H A , under the as-sumption that the second photon was measured by thedetector in the state | a Bj i . To determine the absolutevalue of the off diagonal element, | G | = |h ϕ | ϕ i| , ofthe two–photon state | Ψ i AB one projects the H B partof the first copy onto the state | a B i , the same part ofthe second copy onto | a B i , and performs a kind of theHong–Ou–Mandel interference experiment [17] with theremaining two photons of the first subsystem H A . Aspecific scheme of this kind is depicted in Fig.2.Apart from two sources of pure entanglement (whichmay base on type–I PDS sources modified by dumping FIG. 3: Quantum network exploiting two copies of an ana-lyzed state | Ψ i AB , a control qubit | c i initially in state | i ,controlled SWAP gate (cf. [8]) and two Hadamard gates.The mean value of Pauli σ z matrix of | c i is measured underthe condition that the chosen pair ( i, j ) of results is obtainedin measurement of the same observables performed on bothqubits at the bottom of the scheme. one of the polarization components) it involves the 50:50beamsplitter (BS), two polarization rotators R † ( θ, φ ) inthe same setting and the polarized beamsplitters (PBS).If by p ij (+ , +) we denote the probability of double clickafter the beamsplitter, and by p i ≡ p (cid:0) ( − i +1 (cid:1) ( p i ≡ p (cid:0) ( − i +1 (cid:1) ) the probability of click in the D ,i -th de-tector ( D ,i -th detector) i.e. one of the detectors locatedafter upper PBS (lower PBS) then all the Gram matrixelements are: | G ij | = p i p j (1 − p ij (+ , +)) . (18)Alternatively one can apply the following network de-signed to measure all three quantities (see Fig.3).Measuring the σ z component of the first qubit, condi-tioned by pair of the results ( i, j ) (coming with proba-bilities p (( − i +1 ), p (( − j +1 )) of the measurementsof the same ( σ z ) observable on the last two qubitsone gets an estimation of the parameter | G ij | = p (( − i +1 ) p (( − j +1 ) h σ z i ij . Without going into de-tailed analysis here we only mention that the purityassumption may be dropped at a price of performingtwo variants of the experiment each with one of twocomplementary (in Heisenberg sense) settings R † ( θ, φ ), R † ( θ ′ , φ ′ ). Then the discrimination parameter α K,N = α , = 1 /
16 in the inequality (11) may be successfullycorrected to take into account noise in Hong–Ou–Mandelinterference occurring in both variants. Explicit deriva-tion of such correction is rather complicated and will beconsidered in details elsewhere.
Three qubits —
Now let us investigate the case of athree qubit state ( K = 3). In this case the separabilitydiscrimination parameter is equal to α , = 1 /
64. Wecompare a bi–separable state | BS i = | Ψ i AB ⊗ | φ i C andtwo the most important representatives, the GHZ -stateand the W -state: | GHZ i = | i + | i√ , | W i = | i + | i + | i√ . entanglement test GHZ -state W -state BS -stateminimal Y min Y max .
250 0 .
141 0 . Y .
053 0 .
049 0 . P Y .
807 0 .
807 0 . GHZ -state the W -stateand the bi–separable state | BS i . We present numerical valuesfor the minimal, maximal and average collectibilities and theprobabilities P Y of entanglement detection in a particular,random measurement. If the values of the collectibility arelarger than 1 / ≈ .
016 then the entanglement is detected.
Numerical results for the collectibility are compared inTable I. We can see that the maximal and averagecollectibilities detect entanglement of all three states.The maximum value is attained for the
GHZ -state, Y max [ | GHZ i ] = 16 /
64 while Y max [ | W i ] = 9 /
64. As thisquantity for the bi–separable state reads Y max [ | BS i ] =4 /
64 and Y max [ | Ψ sep i ] = 1 /
64, the collectibility offers anexperimentally accessible measure capable to distinguishthe genuine three–parties entanglement.
Acknowledgements.-
It is a pleasure to thank K. Ba-naszek, O. G¨uhne, M. Ku´s and M. ˙Zukowski for fruitfuldiscussions and helpful remarks. Financial support bythe grant number N N202 174039, N N202 090239 andN N202 261938 of Polish Ministry of Science and HigherEducation is gratefully acknowledged. ∗ Electronic address: [email protected][1] F. Mintert et al. , Phys. Rep. , 207 (2005).[2] I. Bengtsson and K. ˙Zyczkowski,
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