Detecting Quantum Entanglement: Positive Maps and Entanglement Witnesses

The interest on quantum entanglement has dramatically increased during the last 2 decades due to the emerging field of quantum information theory. It turns out that quantum entanglement may be used as basic resources in quantum information processing and communication. The prominent examples are quantum cryptography, quantum teleportation, quantum error correction codes, and quantum computation. Since the quantum entanglement is the basic resource for the new quantum information technologies, it is therefore clear that there is a considerable interest in efficient theoretical and experimental methods of entanglement detection.


Introduction
The interest on quantum entanglement has dramatically increased during the last 2 decades due to the emerging field of quantum information theory.It turns out that quantum entanglement may be used as basic resources in quantum information processing and communication.The prominent examples are quantum cryptography, quantum teleportation, quantum error correction codes, and quantum computation.Since the quantum entanglement is the basic resource for the new quantum information technologies, it is therefore clear that there is a considerable interest in efficient theoretical and experimental methods of entanglement detection.
One of the most important problems of quantum information theory [1][2][3] is the characterization of mixed states of composed quantum systems.In particular it is of primary importance to test whether a given quantum state exhibits quantum correlation, i.e. whether it is separable or entangled.For low-dimensional systems there exists simple necessary and sufficient condition for separability.The celebrated Peres-Horodecki criterion states that a state of a bipartite system living in C 2 ⊗ C 2 or C 2 ⊗ C 3 is separable iff its partial transpose is positive.Unfortunately, for higher-dimensional systems there is no single universal separability condition.
It turns out that the above problem may be reformulated in terms of positive linear maps in operator algebras [4]: a state ρ in H 1 ⊗ H 2 is separable iff (id ⊗ ϕ)ρ is positive for any positive map ϕ which sends positive operators on H 2 into positive operators on H 1 .Therefore, a classification of positive linear maps between operator algebras B(H 1 ) and B(H 2 ) is of primary importance.Unfortunately, in spite of the considerable effort, the structure of positive maps is still poorly understood (see "classical" papers on positive maps [5]- [17] and some recent papers [18]- [63]).
In this paper we provide characterization of important classes of positive maps in finite dimensional matrix algebras.Equivalently, due to the Choi-Jamiołkowski isomorphism, we characterized the corresponding classes of entanglement witnesses.Concerning the application in quantum entanglement theory the key role is played by indecomposable witnesses which can detect PPT entangled states, that is, a PPT state ρ AB is entangled iff there exists an indecomposable entanglement witness W such that Tr(Wρ) < 0. We illustrate the general presentation with several examples of indecomposable positive maps/entanglement witnesses: the Choi-like maps in M 3 (C), its generalizations in M d (C), and the Robertson map in M 4 (C) together with its generalizations in M 2k (C).These examples enables one to discuss several properties like optimality and/or exposedness which are crucial in entanglement theory.

Positive maps and entanglement witnesses
In this paper we restrict our analysis to linear maps where is positive.Finally, Λ is completely positive (CP) if it is k-positive for all k.
Let P k denotes a convex set of k-positive maps in M d (C).One has P k ⊂ P l for k > l.
Actually, due to the Choi theorem any d-positive map in M d (C) is CP, and hence P CP = P d .Therefore, one has the following chain of proper inclusions where P 1 denotes a set of all positive maps in M d (C).Let {e 1 , . . ., e d } denotes an orthonormal basis in C d , and let "T" denotes a transposition map with respect to this basis, i.e. for any a = ∑ ij a ij e ij one has T(a) = ∑ ij a ij e ji , where e ij := |e i e j |.
Let P k denotes a convex set of k-copositive maps.One has where P 1 denotes a set of all copositive maps in M d (C), and P CP stands for a set of completely copositive maps (CcP).Let P k l denotes a set of maps which are l-positive and k-copositive.One has the following relations where Λ 1 ∈ P CP and Λ 2 ∈ P CP .A map which is not decomposable is called indecomposable.A positive map Λ ∈ P 1 is called atomic if it cannot be written as in (6), where Λ 1 ∈ P 2 and Λ 2 ∈ P 2 .
It is clear that each atomic map is indecomposable but the converse is not true.Since P 1 is a convex set it is fully characterized by its extreme elements.Clearly a positive map Λ is extremal if for any Ψ ∈ P 1 , a map Λ − Ψ is not positive.Finally, a positive map Λ is optimal if for any Ψ ∈ P CP , a map Λ − Ψ is not positive.It is evident that each extremal map is optimal but the converse is not true.
where P + d denotes a canonical maximally entangled state in The inverse formula reads Actually, it is inner product isomorphism, that is, where A|B = Tr(A † B) , and for all ψ, φ ∈ C d .Moreover, if Λ is CP, then W Λ ≥ 0 [10], that is, where W 1 , W 2 ≥ 0 and A Γ = (id ⊗ T)A denotes partial transposition.
Let D be a subset of density operators of a composite quantum system living in C n ⊗ C n detected by a given EW W, that is, D = {ρ | Tr(Wρ) < 0}.Given two EWs W 1 and W 2 one says that W 2 is finer than that is, all states detected by W 1 are also detected by W 2 .A witness W is optimal if there is no other EW which is finer than W. It means that W detects quantum entanglement in the 'optimal way'.It is clear that the knowledge of optimal EWS is crucial to classify quantum states of composite systems.One proves the following Proposition 1. W is an optimal EW if and only if W − Q is no longer EW for arbitrary positive operator Q.
Authors of Ref. [32] formulated the following criterion for the optimality of W.
span the total Hilbert space C n ⊗ C n , then W is optimal.
The further classification of entanglement witnesses would be provided in the next section.
Open Systems, Entanglement and Quantum Optics Detecting Quantum Entanglement: Positive Maps and Entanglement Witnesses 5 10.5772/55829

States of composite quantum systems
Let Ψ ∈ C d ⊗ C d such that Ψ|Ψ = 1, and consider the corresponding Schmidt decomposition where µ k > 0 and ∑ r k=1 µ 2 k = 1.In the above formula {e i } and { f j } are two mutually orthogonal normalized vectors in C d .One calls the number r the Schmidt rank of where the minimum is taken over all possible pure states decompositions Let us introduce the following family of positive cones: One has the following chain of inclusions Clearly, V 1 is a cone of separable (unnormalized) states and V d V 1 stands for a set of entangled states.Note, that a partial transposition (id ⊗ T) gives rise to another family of cones: , together with the following hierarchy of inclusions: Λ ∈ P k if and only if Finally, Λ ∈ P k l if and only if Let us denote by W a space of entanglement witnesses, i.e. a space of non-positive Hermitian operators One has Clearly Consider now the following class of witnesses that is, W ∈ W s r iff with P ∈ W r and Q ∈ W s .Note, that Tr(Wρ) ≥ 0 for all ρ ∈ V r ∩ V s .Hence such W can detect PPT states ρ such that SN(ρ) ≥ r or SN(ρ Γ ) ≥ s.It is clear that decomposable entanglement witnesses cannot detect PPT states.One has the following chain of inclusions: The 'weakest' entanglement can be detected by elements from W 1 1 W 2 2 .We shall call them atomic entanglement witnesses.

Let P •
1 denote a dual cone [23,64] to the convex cone P 1 of positive maps where P x = |x x| and P y = |y y|.It is clear that P •• 1 = P 1 , that is, one may consider P 1 as a dual cone to the convex cone of separable operators in H ⊗ H. Recall that a face of P 1 is a convex subset F ⊂ P 1 such that if the convex combination If a ray {λΦ : λ > 0} is a face of P 1 then it is called an extreme ray, and we say that Φ generates an extreme ray.For simplicity we call such Φ an extremal positive map.A face F is exposed if there exists a supporting hyperplane H for a convex cone P such that F = H ∩ P 1 .
A positive map Φ ∈ P 1 is exposed if it generates 1-dimensional exposed face.Let us denote by Ext(P 1 ) the set of extremal points and Exp(P 1 ) the set of exposed points of P 1 .Due to Straszewicz theorem [64] Exp(P 1 ) is a dense subset of Ext(P 1 ).Thus every extreme map is the limit of some sequence of exposed maps meaning that each entangled state may be detected by some exposed positive map.Hence, a knowledge of exposed maps is crucial for the full characterization of separable/entangled states of bi-partite quantum systems.

Choi-like maps in M 3 (C)
It is well known that all positive maps Λ : [5,12].The first example of an indecomposable positive linear map in M 3 (C) was found by Choi [10].The (normalized) Choi map reads as follows where ||A C ij || is the following doubly stochastic matrix: Let us consider a class of positive maps in M 4 (C) defined as follows [20] Φ Detecting Quantum Entanglement: Positive Maps and Entanglement Witnesses http://dx.doi.org/10.5772/55829 where D[a, b, c] is a completely positive linear map defined by and Moreover, being positive it is indecomposable if and only if 4bc < (2 − a) 2 .
Actually, Φ[a, b, c] is indecomposable if and only if it is atomic, i.e. it cannot be decomposed into the sum of 2-positive and 2-copositive maps.The corresponding entanglement witness reads as follows with In a recent paper [54] we analyzed a special case corresponding to a + b + c = 2.It turns out [54] that Φ[a, b, c] is parameterized by the ellipse on the bc-plane.Moreover, one proves the following Interestingly Interestingly, indecomposability of these maps may be proved by using the following family of PPT states in C 3 ⊗ C 3 : with ǫ > 0 and

Indecomposable maps in M d (C) -generalized Choi maps
In this section we provide several examples of positive maps in M d (C) which generalize Choi map in M 3 (C).
Example 1.The Choi map in M 3 (C) may be generalized to a positive map in M d (C) as follows [24]: let S be a unitary shift defined by: S e i = e i+1 , i = 1, . . ., d , where the indices are understood mod d.One defines where ǫ(X) denotes the following projector

Entanglement witnesses based on spectral conditions
Any entanglement witness W can be represented as a difference W = W + − W − , where both W + and W − are semi-positive operators in C d ⊗ C d .However, there is no general method to recognize that W defined by W + − W − is indeed an EW.One particular method based on spectral properties of W was presented in [42].Let ψ α (α = 1, . . ., D = d 2 ) be an orthonormal Having defined eigenvectors of W one needs the corresponding eigenvalues: let λ − α ≤ 0, for α = 1, . . ., L < D, and λ + α > 0 for α = L + 1, . . ., D, that is, Let us analyze the condition for the spectrum {λ − α , λ + α } which guarantees that W is block positive.Consider a normalized vector ψ ∈ C d ⊗ C d and let s 1 (ψ) ≥ . . .≥ s d (ψ) denote its Schmidt coefficients.For any 1 ≤ k ≤ d one defines k-norm of ψ by the following formula It is clear that ||ψ|| 1 ≤ ||ψ|| 2 ≤ . . .≤ ||ψ|| d .Note that ||ψ|| 1 gives the maximal Schmidt coefficient of ψ, whereas due to the normalization, ||ψ|| 2 d = ψ|ψ = 1.In particular, if ψ is maximally entangled then Equivalently one may define k-norm of ψ by where the maximum runs over all normalized vectors φ such that SR(ψ) ≤ k (such φ is usually called k-separable).Recall that a Schmidt rank of ψ -SR(ψ) -is the number of non-vanishing Schmidt coefficients of ψ.One calls entanglement witness W a k-EW if ψ|W|ψ ≥ 0 for all ψ such that SR(ψ) ≤ k.One has the the following If the following spectral conditions are satisfied where then W being k-EW is not (k + 1)-EW.
Interestingly, one has the following The proof is easy [43]: note that W = A + B , where and Now, since λ + α ≥ µ 1 , for α = L + 1, . . ., D, it is clear that A ≥ 0. The partial transposition of B reads as follows Let us recall that the spectrum of the partial transposition of rank-1 projector |ψ ψ| is well know: the nonvanishing eigenvalues of |ψ ψ| Γ are given by s 2 α (ψ) and ±s α (ψ)s β (ψ), where s 1 (ψ) ≥ . . .≥ s d (ψ) are Schmidt coefficients of ψ.Therefore, the smallest eigenvalue of B Γ (call it b min ) satisfies and using the definition of µ 1 (cf.Eq. ( 48)) one gets b min ≥ 0 which implies B Γ ≥ 0. Hence, the entanglement witness W is decomposable.
Remark 1.Interestingly, saturating the bound (47), i.e. taking one has A = 0 and hence Detecting Quantum Entanglement: Positive Maps and Entanglement Witnesses 13 10.5772/55829 is completely co-positive.Note that where F α is a linear operator F α : C d → C d defined by and {e 1 , . . ., e d } denotes an orthonormal basis in C d .In particular, if L = 1, i.e. there is only one negative eigenvalue, then formula (56) (up to trivial rescaling) gives with It reproduces a positive map (or equivalently an EW W = κ I d ⊗ I d − P 1 ) which is known to be completely co-positive [3,39,43].If ψ 1 is maximally entangled, that is, Its spectral decomposition has the following form together with the corresponding eigenvalues One finds µ 1 = 1 and hence condition (47) is trivially satisfied λ + α ≥ µ 1 for α = 2, 3, 4. We stress that our construction does not recover flip operator in d > 2. It has d(d − 1)/2 negative eigenvalues.Our construction leads to at most d − 1 negative eigenvalues.

Bell-diagonal entanglement witnesses
Let us define a generalized Bell states [65] in where U mn are unitary matrices defined as follows with λ = e 2πi/d .The matrices U mn define an orthonormal basis in the space M d (C) of complex d × d matrices.One easily shows Some authors [66] call U mn generalized spin matrices since for d = 2 they reproduce standard Pauli matrices: One calls a Hermitian operator p mn P mn , (65) with p mn ∈ R , and

Optimal maps in M 2k (C)
In this section we provide several examples of optimal indecomposable maps in M 2k (C).
Interestingly, some of them turn out to be extremal and even exposed [60,61].Consider where In what follows we shall use the following notation to display the block structure of X. Robertson map [13] in M 4 (C) is defined as follows: where Theorem 8. Φ 4 defines positive indecomposable map.Moreover, it is extremal in the convex set of positive maps in M 4 (C).
Following [35] and [34] one defines where U is an antisymmetric unitary matrix in M 2k (C).The above normalization guaranties that Φ U 2k is unital.The characteristic feature of these maps is that for any rank-1 projector P its image under Φ U 2k reads as follows: where Q = UP T U † is a rank-1 projector orthogonal to P. Hence Φ U 2k (P) is a projector which proves positivity of Φ U 2k .Denote by U 0 the following "canonical" antisymmetric unitary matrix in M 2k (C) where J is a symplectic matrix in M 2 (R), that is, is antisymmetric and unitary.Interestingly, the map Φ 0 2k corresponding to U = U 0 has the following block structure where and and hence it reduces for k = 2 to the Robertson map (74).
In the recent paper [44] we proposed another construction of maps in M 2k (C).Now, instead of treating a 2k × 2k matrix X as a k × k matrix with 2 × 2 blocks X ij we consider alternative possibility, i.e. we consider X as a 2 × 2 with k × k blocks and define  44]).Ψ 2k defines a linear positive map in M 2k (C).Moreover, it is an atomic and optimal map.In Ref. [51] we proposed the following generalization of the Robertson map Φ 2k : for any collection of k(k − 1)/2 complex numbers z ij , with i < j, satisfying |z ij | ≤ 1 we define It is clear that form z ij = 1 the map Φ 2k is optimal and indecomposable iff |z ij | = 1.

Conclusions
We provide characterization of several classes of positive maps in M d (C).Equivalently, due to the Choi-Jamiołkowski isomorphism, we characterized the corresponding classes of entanglement witnesses.Concerning the application in quantum entanglement theory the key role is played by indecomposable maps which can detect PPT entangled states.The presentation was illustrated with several examples of indecomposable positive maps/entanglement witnesses: the Choi-like maps in M 3 (C) and its generalizations in M d (C).It was shown that several maps from these families are optimal and even exposed.Similarly, the Robertson map in M 4 (C) and its generalizations in M 2k (C) turn out to be optimal maps [original Robertson map is even exposed].It should be stressed that there is no general method enabling one to construct all indecomposable positive maps and hence this subject deserves further studies.

Proposition 4 .
Elements from W d d are decomposable entanglement witnesses.

1 2Theorem 2 .
The normalization factor N abc = (a + b + c) −1 guarantees that Φ[a, b, c] is unital, i.e.Φ[a, b, c](I 3 ) = I 3 .Note, that Φ[a, b, c] gives rise to the following doubly stochastic circulant matrix D = N abc   well known examples of positive maps: note that Φ[0, 1, 1](X) = (TrX I 3 − X) which reproduces the reduction map.Moreover, Φ[1, 1, 0] and Φ[1, 0, 1] reproduce Choi map and its dual, respectively.One proves the following result [20] A map Φ[a, b, c] is positive but not completely positive if and only if kk .The map τ d,0 defined is completely positive and the map τ d,d−1 reproduces the reduction map in M d (C) (and hence it is completely copositive).Note that τ d,k (I d ) = (d − 1)I d , and Tr τ d,k (X) = (d − 1)Tr X, hence the normalized maps

Example 3 .
for some unitary U ∈ U(d), then one finds for κ = 1/d and the map (58) is unitary equivalent to the reduction map Λ(X) = UR(X)U † , where R d (X) = I d TrX − X.Consider an EW corresponding to the flip operator in d = 2: W = e 11 ⊗ e 11 + e 22 ⊗ e 22 + e 12 ⊗ e 21 + e 21 ⊗ e 12 .
2k reduces to Φ 2k .One proves Theorem 11([51]).Φ (z) 2k defines a positive map.Moreover, Φ The vectors spaces of linear maps in M d (C) and linear operators in M d (C) ⊗ M d (C) have the same dimensions d 2 , and hence they are isomorphic.Fixing an orthonormal basis {e 1 , . . ., e d } in C d one may establish the following isomorphism known in the quantum information community as the Choi-Jamiołkowski isomorphism.
11 = p 0 e 11 + p 3 e 33 , W 22 = p 0 e 22 + p 1 e 11 , W 33 = p 0 e 33 + p 2 e 22 , W ij = −e ij , i = j , and the normalization factor reads N[p] Open Systems, Entanglement and Quantum Optics Detecting Quantum Entanglement: Positive Maps and Entanglement Witnesses 11 10.5772/55829 basis in C d ⊗ C d and denote by P α the corresponding projector P α = |ψ α ψ α |.It leads therefore to the following spectral resolution of identity