aa r X i v : . [ qu a n t - ph ] M a y Spectral properties of entanglement witnesses
G Sarbicki
Insitute of Physics, Nicolaus Copernicus UniversityGrudziadzka 5, 87–100 Toru´n, PolandE-mail: [email protected]
Abstract.
Entanglement witnesses are observables which when measured,detect entanglement in a measured composed system. It is shown what kindof relations between eigenvectors of an observable should be fulfilled, to allow anobservable to be an entanglement witness. Some restrictions on the signature ofentaglement witnesses, based on an algebraic-geometrical theorem will be given.The set of entanglement witnesses is linearly isomorphic to the set of maps betweenmatrix algebras which are positive, but not completely positive. A translationof the results to the language of positive maps is also given. The propertiesof entanglement witnesses and positive maps express as special cases of generaltheorems for k -Schmidt witnesses and k -positive maps. The results are thereforepresented in a general framework.
1. Introduction
The crucial resource in quantum-informational science is quantum entanglement. Itprovides essential difference between quantum and classical information theory. It istherefore of special importance, to know the techniques of measuring and quantyfyingentanglement. Despite of enormous efforts in the last decade, and a number ofsignificant partial results, the problem how to determine that a given density matrixpossesses this desired resource or not, is not solved in general. One of the mostimportant methods of detecting entanglement exploits a class of special observablescalled entanglement witnesses . In this paper the entanglement witnesses are consideredin the language of their spectral decomposition.Only finite-level quantum systems, which are composed of two subsystems, will beconsidered. The first one is a d -level system, and the second one is a d -level system.We assume, that d ≤ d . The Hilbert space of the system is then a tensor product ofthe Hilbert spaces of its subsystems: H = C d ⊗ C d . By fixing the orthogonal basis { e i } d i =1 as the basis of C d , and the orthogonal basis { f j } d j =1 as the basis of C d , onegets the linear isomorphism A : H → M C ( d × d )between vectors in H = C d ⊗ C d and d × d complex matrices of their coordinates,defined by the formula[ A (Ψ)] i,j = h e i ⊗ f j | Ψ i (1)The vector Ψ can be therefore expressed by the matrix A (Ψ) asΨ = d X i =1 d X j =1 [ A (Ψ)] ij | i i| j i pectral properties of entanglement witnesses Schmidt rank of a vector Ψ in H is defined as the rank of its coordinatematrix A (Ψ). Denote the set of vectors of the Schmidt rank non-greater than k as S k .We then have the ascending sequence of closures of orbits of the group GL ( d ) × GL ( d ) S ( . . . ( S d = C d ⊗ C d . Any matrix A ∈ M C ( d × d ) of rank one can be written as v T u , where v and u arevectors respectively in C d and C d . Taking a ∈ C ∗ one gets the new pair u ′ = a · u , v ′ = v/a representing the same vector. Aliasing such pairs, we get the isomorphism C d × C d / C ∼ = S .Next define a projection P : | Ψ i → | Ψ ih Ψ |h Ψ | Ψ i from C d ⊗ C d to the projective space C P d × d − . The projective space C P d × d − = C d ⊗ C d / C is the space of rays in C d ⊗ C d , i.e. the space C d ⊗ C d \ { } wherethe vectors of the same direction (equal up to multiplication by a non-zero scalar) arealiased (for a detailed discusion about projective geometry see [1], [2]). The space ofrays of the Hilbert space of a given quantum system is in one-to-one correspondencewith the set of pure quantum states of the system.Now projecting the above sequence of inclusions, one gets the following newascending sequence of sets of projectors onto one-dimensional subspaces S ( . . . ( S d = C d ⊗ C d ↓ P ↓ P ↓ P C P d − × C P d − ∼ = P ( S ) ( . . . ( P ( S d ) = C P d × d − , where the isomorphism in the lower sequence constitutes the Segre embeding C P d − × C P d − ֒ → C P d × d − (see [1], [2] for details). The elements of the set P ( S ) arecalled pure separable states . We define mixed separable states in accord with the workof Werner [3] as convex combinations of pure separable states.Following Terhal et al. [5], we extend this definition to higher k and define densityoperators of Schmidt number k ([5], [6]): Definition 1.
The state ρ is called to be of the Schmidt number k , when it can bedecomposed as a convex combination of projectors ρ = X i p i | ψ i ih ψ i | , where the Schmidt ranks of the vectors ψ i is less or equal k , and cannot be decomposedas a convex sum of projectors onto vectors of Schmidt rank less than k . The set of all states of Schmidt number less or equal to k is convex — it isthe convex roof of the set P ( S k ). The problem of separability is then a problemof membership in a convex set, whith given extremal points. One of the ways tohandle this problem is by using the concept of entanglement witnesses introduced in[4]. Taking any entangled state ρ , we have two compact, convex, non-empty subsetsof the linear coset of operators of rank 1 - the set of separable states and the singletonof the chosen entangled state. Now according to Banach separation theorem, thereexists an affine subspace ˆ V of codimension 1 (in the considered coset), which separates pectral properties of entanglement witnesses V ofcodimension 1 in the Hilbert space of Hermitian operators. Now using the self dualityof this Hilbert space, one can assign to the subspace V of codimension 1 the unique(up to multiplication by non-zero scalar) observable W , such that V is its orthogonalcomplement. Such an observable is called entanglement witness . We easly extend thisdefinition to k -Schmidt witnesses (see [6]). Definition 2.
The k -Schmidt witness is an observable which fulfils the conditions: • ∀ Ψ ∈ S k h Ψ | W | Ψ i ≥ (all states of Schmidt number less or equal to k are on thesame side of V ) • for some ρ h ρ | W i HS = Tr( ρW ) < (the singleton of ρ lies on the other side of V ) The problem of classification of k -Schmidt witnesses remains unsolved in general.In low dimensions (2 ×
2, 2 ×
3) we have such a classification. Any entanglementwitness (1-Schmidt witness) is of the form W = A Γ + B, A, B ≥ , where Γ denotes the partial transposition in one of the subsystems (see [12]). Suchwitnesses are called decomposable , and states which can be detected by witnesses fromthis class are called NPT ( negative partial transposition ) entangled states. In higherdimensions this class of entanglement witnesses are a proper subset of the set of allwitnesses. Entangled states not detected by this class are called PPT ( positive partialtransposition ) entangled states. We have a simple citerion to check, whether a givenstate is NPT. The most interesting are then non-decomposable witnesses and tools todetect PPT entanglement based on them.
2. The main theorems
Having a Hermitian observable W , we define a decomposition of its domain due tothe spectral decomposition of W C d × C d = V + ⊕ V − ⊕ V where the positive subspace V + is spaned by eigenvectors corresponding to positiveeigenvalues, the negative subspace V − is spaned by eigenvectors corresponding tonegative eigenvalues, and V is a kernel of W . Next due to the spectral decompositionone can represent W as a difference of two positive operators W = W + − W − , the firstone supported on V + , and the second one on V − . Their kernels are ker W ± = V ⊕ V ∓ .Having a vector Ψ given by it’s Schmidt decomposition Ψ = P i λ i α i ⊗ β i , wedefine a subspace˜ V Ψ = span { span { α i } ⊗ C d ∪ C d ⊗ span { β i } } . It is also possible to define this subspace independently of the basis as˜ V Ψ = span { ImTr | Ψ ih Ψ | ⊗ C d ∪ C d ⊗ ImTr | Ψ ih Ψ | } Here Im denotes the image or range of the operator. Partial traces Tr and Tr aredefined as follows: Tr ρ = P ik δ ik ρ ij,kl , respectively Tr ρ = P jl δ jl ρ ij,kl , where ρ ij,kl denotes the matrix element of ρ in standart basis, i.e. h e i ⊗ f j | ρ | e k ⊗ f l i . Theorem 1.
Let W be a k -Schmidt witness. Its eigenvectors satisfy the three followingconditions:pectral properties of entanglement witnesses (i) V − ) { } (ii) ∀ Ψ ∈ S k , Ψ ∈ V ⊕ V − ⇒ Ψ ∈ V .(iii) ∀ Ψ ∈ S k , ∩ V ˜ V Ψ ∩ ( V − ⊕ V ) ⊂ V . Proof: (i) A k − entanglement witness should detect something, so there aresome states, for which the mean value of the witness on them is negative (the secondcondition in the definition 2).(ii) If Ψ ∈ S k ∩ V ⊕ V − , then Ψ is a combination of eigenvectors only fromthe kernel and V − . Due to the first condition in Definion 2, the mean value of theobservable on the vector Ψ should be nonnegative, so Ψ can be only a combination ofthe eigenvectors from the kernel.(iii) To prove the neccesity of the third condition, observe that when an observable W is a k -SW, then ∀ Ψ ∈ S k h Ψ | W + | Ψ i − h Ψ | W + | Ψ i ≥
0. This implies the conditionthat the supremum of the function F : S k \ V ⊕ V − → R F (Ψ) = h Ψ | W − | Ψ ih Ψ | W + | Ψ i is less than 1. The function F plays a key role in this proof.Let’s take a vector P ki =1 x i ⊗ y i ∈ S k in V and some other vector P ki =1 ˜ x i ⊗ ˜ y i ∈ S k . Consider now the family of vectors form S k :Φ( t ) = k X i =1 ( x i + t ˜ x i ) ⊗ ( y i + t ˜ y i ) (2)This family forms an algebraic curve (over R ) of degree 2. The intersection { Φ( t ) : t ∈ R } ∩ V can be one of the folowing sets:(i) The whole curve - { Φ( t ) : t ∈ R } ⊂ V .(ii) { Φ( t ) : t ∈ R } meets the kernel in no more than two points. Because P ki =1 x i ⊗ y i ∈ V , one of these points is 0.To begin with, consider the second case. In this case the root of the denominatorof F Φ in 0 is separated, i.e. ∃ ǫ : { Φ( t ) : t ∈ ( − ǫ, ǫ ) } ∩ V = { Φ(0) } = k X i =1 x i ⊗ y i ∈ V . The restriction of the function F to the set Φ(( − ǫ, ǫ ) \ { } ) gives a fuction F Φ :( − ǫ, ǫ ) \ { } → R , given by the formula F Φ ( t ) = h Φ( t ) | W − | Φ( t ) ih Φ( t ) | W + | Φ( t ) i (3)Computing the numerator of F Φ and taking into account that W ± | Φ i = 0 onegets h Φ( t ) | W − | Φ( t ) i = h k X i =1 ( x i + t ˜ x i ) ⊗ ( y i + t ˜ y i ) | W − | k X i =1 ( x i + t ˜ x i ) ⊗ ( y i + t ˜ y i ) i = t ( t h k X i =1 ˜ x i ⊗ ˜ y i | W − | k X i =1 ˜ x i ⊗ ˜ y i i pectral properties of entanglement witnesses t Re h k X i =1 x i ⊗ ˜ y i + k X i =1 ˜ x i ⊗ y i | W − | k X i =1 ˜ x i ⊗ ˜ y i i + h k X i =1 x i ⊗ ˜ y i + k X i =1 ˜ x i ⊗ y i | W − | k X i =1 x i ⊗ ˜ y i + k X i =1 ˜ x i ⊗ y i i ) . (4)For the denominator we get h Φ( t ) | W + | Φ( t ) i = h k X i =1 ( x i + t ˜ x i ) ⊗ ( y i + t ˜ y i ) | W + | k X i =1 ( x i + t ˜ x i ) ⊗ ( y i + t ˜ y i ) i = t ( t h k X i =1 ˜ x i ⊗ ˜ y i | W + | k X i =1 ˜ x i ⊗ ˜ y i i +2 t Re h k X i =1 x i ⊗ ˜ y i + k X i =1 ˜ x i ⊗ y i | W + | k X i =1 ˜ x i ⊗ ˜ y i i + h k X i =1 x i ⊗ ˜ y i + k X i =1 ˜ x i ⊗ y i | W + | k X i =1 x i ⊗ ˜ y i + k X i =1 ˜ x i ⊗ y i i ) . (5)For a given { ˜ x i , ˜ y i } , we get a rational function F Φ : R → R . Such a function is thequotient of two quadratic polynomials F { ˜ x i , ˜ y i } ( t ) = at + bt + cdt + et + f . (6)Now the assumtion that the supremum of F on its domain is finite, implies thatfor any curve Φ, the limit of F Φ in 0 is finite.First consider the degenerated subcase, when P ki =1 ˜ x i ⊗ ˜ y i ∈ V . Then a = d = b = e = 0, and f is non-zero (because we assume at the moment, that the curve Φ isnot a subset of V ). The limit of F Φ in 0 is finite.We can then consider the generic case, when d = 0. At the beginning observe,that f = 0 ⇒ e = 0, c = 0 ⇒ b = 0. Suppose that the denominator has a root in 0(by the above observation, it is then a double root), which means that f = 0 ⇐⇒ P ki =1 x i ⊗ ˜ y i + P ki =1 ˜ x i ⊗ y i ∈ ker W + (by the formula (5) and the positivity of W + ).Finiteness of the limit of F Φ in 0 (according to the formula (4) and the postivity of W − ) now implies that also c = 0 ⇐⇒ P ki =1 x i ⊗ ˜ y i + P ki =1 ˜ x i ⊗ y i ∈ ker W − (thenalso b = 0 and the numerator has also a double root in 0). The limit is then equal a/d , which is finite. Using the definition of the subspace ˜ V P x i ⊗ y i and consideringthat ker W + ∩ W − = V , we can rewrite this implication in the following form ∀ φ ∈ ˜ V P x i ⊗ y i φ ∈ V ⊕ V − ⇒ φ ∈ V . (7)what gives the assertion of the theorem.It remains to consider the situation, when { Φ( t ) : t ∈ R } ⊂ ker W + . This impliesthat k X i =1 ˜ x ⊗ ˜ y ∈ ker W + k X i =1 (˜ x ⊗ y + x ⊗ ˜ y ) ∈ ker W + . pectral properties of entanglement witnesses { Φ( t ) : t ∈ R } ⊂ ker W − , so inparticular k X i =1 (˜ x ⊗ y + x ⊗ ˜ y ) ∈ ker W − . Again, identyfying the kernels and using the defnition of ˜ V P x i ⊗ y i we get the assertionof the theorem. (cid:3) Under some stronger assumptions about the subspaces V , V + and V − , one canprove the opposite theorem: Theorem 2.
Consider the decomposition of C d ⊗ C d into direct sum of mutualorthogonal subspaces: C d ⊗ C d = V + ⊕ V ⊕ V − . Let W ± be arbitrary positiveoperators supported on V ± .Define ˆ V = T Ψ ∈S k ∩ V ˜ V ⊥ Ψ . If(i) V − ) { } (ii) ∀ Ψ ∈ S k , Ψ ∈ V ⊕ V − ⇒ Ψ ∈ V (iii) V − ⊂ ˆ V then for large enough λ the observable λW + − W − is an k -SW. Proof: (i) The first condition guarantees the existence of detected states, so thefirst condition in the definition (2) is fulfilled.One can decompose any k − separable vector Ψ asΨ = ˆΨ + ˜Ψ , where ˆΨ ∈ ˆ V and ˜Ψ ∈ ˆ V ⊥ . Now using the decomposition W = W + − W − we canwrite the second condition in the definition (2) as ∀ ˆΨ ∈ ˆ V ∀ ˜Ψ ∈ ˆ V ⊥ : ˆΨ + ˜Ψ ∈ S k h ˆΨ + ˜Ψ | W + | ˆΨ + ˜Ψ i≥ h ˆΨ + ˜Ψ | W − | ˆΨ + ˜Ψ i = h ˆΨ | W − | ˆΨ i . (8)The last equality holds because of the third assumption. When ˆΨ = 0, then condition(8) is fulfilled. It is then enough to focus on vectors Ψ for which ˆΨ = 0. When thecondition (8) is fulfilled for a vector Ψ, then also for a vector α Ψ, where α ∈ C ∗ . Itis then sufficient to restrict the quantificated set to the set of vectors Ψ, for which || ˆΨ || = 1. Condition (8) now takes the form ∀ ˆΨ ∈ ˆ V : || ˆΨ || = 1 ∀ ˜Ψ ∈ ˆ V ⊥ : ˆΨ + ˜Ψ ∈ S k h ˆΨ + ˜Ψ | W + | ˆΨ + ˜Ψ i ≥ h ˆΨ | W − | ˆΨ i . (9)ˆ V is a product space. To see it, let’s introduce the notations A Ψ = ImTr | Ψ ih Ψ | B Ψ = ImTr | Ψ ih Ψ | and write the subspace ˜ V Ψ using them:˜ V Ψ = span { A Ψ ⊗ C d ∪ C d ⊗ B Ψ } . It’s orthogonal complement is then equal:˜ V ⊥ Ψ = ( A Ψ ⊗ C d ) ⊥ ∩ ( C d ⊗ B Ψ ) ⊥ = A ⊥ Ψ ⊗ C d ∩ C d ⊗ B ⊥ Ψ = A ⊥ Ψ ⊗ B ⊥ Ψ pectral properties of entanglement witnesses V is an intersection of such subspaces, so it’s also a product subspace:ˆ V = T Ψ ∈S k ∩ V ˜ V ⊥ Ψ = T Ψ ∈S k ∩ V ( A ⊥ Ψ ⊗ B ⊥ Ψ )= ( T Ψ ∈S k ∩ V A ⊥ Ψ ) ⊗ ( T Ψ ∈S k ∩ V B ⊥ Ψ ) . Denote ˆ V = T Ψ ∈S k ∩ V A ⊥ Ψ and ˆ V = T Ψ ∈S k ∩ V B ⊥ Ψ .Because ˆ V = ˆ V ⊗ ˆ V is a product subspace, after the apriopriate change of basisof C d and C d the matrix A ( ˆΨ) of an element ˆΨ ∈ ˆ V is a matrix, which has zeros outof some subblock. To see it, set such basis { e } d i =1 , { f } d j =1 of C d and C d respectively,that ˆ V = span { e , . . . , e dim ˆ V } and ˆ V = span { f , . . . , f dim ˆ V } . Now the matrix A ( ˆΨ)of any element ˆΨ ∈ ˆ V has zeros outside the left-upper corner of the size dim ˆ V × dim ˆ V .From now, we use such basis. The matrix A ( ˆΨ) is therefore non-zero only in itsleft-upper subblock of size dim ˆ V × dim ˆ V , and the matrix A ( ˜Ψ) is therefore non-zeroonly outside its left-upper subblock of size dim ˆ V × dim ˆ V . This means that a givenvector Ψ = ˆΨ + ˜Ψ is k − separable, only if the vector ˆΨ is k − separable (If a matrix hasthe rank lower than k , then any its subblock has the rank lower than k ). Now for agiven normalized vector ˆΨ ∈ S k ∩ ˆ V define the set U ˆΨ = { ˜Ψ ∈ ˆ V ⊥ : ˆΨ + ˜Ψ ∈ S k } . (10)Using this observation, one can rewrite the condition (9) in the following form ∀ ˆΨ ∈ ˆ V ∩ S k : || ˆΨ || = 1 ∀ ˜Ψ ∈ U ˆΨ h ˆΨ + ˜Ψ | W + | ˆΨ + ˜Ψ i ≥ h ˆΨ | W − | ˆΨ i . (11)The sets U ˆΨ are: • non-empty: Zero is always an element. • closed (in the metric topology of ˆ V ⊥ ):By the isomorphism A defined by (1), the set A ( U ˆΨ ) is a set of matrices havingtheir left-upper block of the size dim ˆ V × dim ˆ V fixed and the rank less or equal k (direct reformulation of definition (10) ). It is a set of common zeros of allminors of rank k + 1 (the matrix entries from left-upper block we treat as fixedconstants). The minors are polynomial functions, and hence continous, so the set U ˆΨ is the inverse image of a closed set { } , and therefore closed.Now define a continous function f ˆΨ : U ˆΨ → R ∪ { } by the left side of theinequality (9) f ˆΨ = h ˆΨ + ˜Ψ | W + | ˆΨ + ˜Ψ i . (12)By the second and the third assumption, we have: ∀ Ψ ∈ S k Ψ ∈ V ⊕ V − ⇒ Ψ ∈ V ⇒ Π ˆ V Ψ = 0(here Π ˆ V denotes an orthogonal projection). Now, by the contraposition rule ∀ Ψ ∈ S k Π ˆ V Ψ = 0 ⇒ Ψ V ⊕ V − ⇒ h Ψ | W + | Ψ i > . Therefore on any ˜Ψ ∈ U ˆΨ the function f ˆΨ takes stricty non-negative values. Theimage of the domain of f ˆΨ is then a closed subset of R + . The infimum of the function f ˆΨ on its domain is then stricty positive. Denote it by c ˆΨ .Consider now the continous function c : ˆΨ → c ˆΨ , defined on the set { ˆΨ : ˆΨ ∈ ˆ V ∧ || Ψ || = 1 } . The domain is compact, so the function c is bounded from below, pectral properties of entanglement witnesses C is reached at some point of the domain and then is positive. Thisresult allows to bound the left-hand side of the inequality in (9) from below as h ˆΨ + ˜Ψ | W + | ˆΨ + ˜Ψ i ≥ C On the same set { ˆΨ : ˆΨ ∈ ˆ V ∩ S k ∧ || ˆΨ || = 1 } we have also given a real-valuedcontinous function by the right-hand side of the inequality (9) g ( ˆΨ) = h ˆΨ | W − | ˆΨ i . Again, because of compactness of the domain, this function is bounded from above.Denote its maximum by G .Now by rescaling the positive part W + one obtains C ≥ G . This ensures, thatthe inequality in (8) is fulfilled for any vector of the Schmidt rank k , and then theobservable is a k -SW. (cid:3) The next theorem will allow us to give some restrictions on the signature of k − Schmidt witnesses when the dimensions of the subsystems are given.
Lemma 1.
The set S k ⊂ C d ⊗ C d is an affine variety of dimension d × d − ( d − k ) × ( d − k ) . Proof:
By the isomorphisms A defined by (1) we can treat the set S k as a set of d × d matrices of rank k . It is an affine variety generated by the ideal of all minorsof rank k + 1. There are (cid:18) d k + 1 (cid:19) × (cid:18) d k + 1 (cid:19) such a minors, but locally only( d − k ) × ( d − k ) of them are indepentent.For any regular point of this variety A (a matrix of the rank equal k ) we canfind basis in C d and C d , respectively { f } d i =1 and { e } d i =1 such that A = P ki =1 f Ti e i .In this basis a minor built from the first k columns and the first k rows is non-zero.There exists an open neighbourhood of this regular point (in the space M C ( d × d )),such that any matrix in this neighbourhood has its first k columns and first k rowslinearly independent. To check whether a given matrix from this neighbourhood hasrank less or equal k , one has to check ( d − k ) × ( d − k ) independent conditionsdet b . . . b k b i ... . . . ... ... b k . . . b kk b ki b j . . . b jk b ji = 0 . (in the new basis). Dimension of the tangent space in a regular point, and hence thedimension of the variety is then equal d × d − ( d − k ) × ( d − k ). (cid:3) For detailed discusion and proofs of the facts about manifolds of matrices of afixed rank see [7] or [8]
Theorem 3.
Any subspace V ∈ C d ⊗ C d of dimension dim V > ( d − k ) × ( d − k ) contains a k − separable vector. Proof:
Because the variety S k is defined by a zero of an ideal of homogenouspolynomials, one can consider its projectivisation of the dimension d × d − ( d − k ) × ( d − k ) −
1. It is a subvariety of C P d × d − . Consider now another subvarietyof C P d × d − - a projectivisation of linear subspace V ⊂ C d ⊗ C d . Its dimension isdim V −
1. Projective Dimension Theorem [2] says that if the sum of dimensions oftwo subvarieties of C P N is greater or equal N , then their intersection is a nonemptyset. Using this for the above subvarieties, we find than if d × d − ( d − k ) × ( d − k ) − V − ≥ d × d − , pectral properties of entanglement witnesses V consists of some k − separable vectors. (cid:3) The theorem used is a generalization of the well-known fact, that two projectivelines on a projective plane always have an intersection. To show that there is nostricter restriction on this dimension, I give an example of basis (non-orthogonal),which spans the ( d − k ) × ( d − k )-dimensional subspace, which does not contain any k − separable vector. Theorem 4.
The set { P ki =1 e m + i ⊗ f n + i } m ≤ d − k,n ≤ d − k spans the subspace V kmax which does not contain any non-zero k − separable vector. Proof:
Again using the isomorphism A defined by (1), we will prove that anynon-zero matrix in the subspace V kmax ⊂ M C ( d × d ) spanned by the set of matrices { P ki =1 e Tm + i f n + i } m ≤ d − k,n ≤ d − k has rank greater than k .We call any subset of matrix elements with a constant difference between indicesa diagonal of a matrix (in general rectangular). This difference will be called the number of a diagonal. The number of a diagonal varies between 1 − d and d − V kmax has the first k and the last k of diagonals equal zero(any diagonal which has less than k elements is equal zero). Now assume, that thereexists a matrix of the rank k in V kmax .We will prove that if all diagonals of numbers less than p are zero, then also the p -th diagonal is zero.Any minor of rank k + 1 is equal zero, in particular the minors whose diagonal (instandard sense) is created from elements of the considered p -th diagonal. Because weassume that all diagonals with their numbers less than p are equal zero, such minorsare determinats of upper-triangular matrices, and therefore the products of k + 1elements from the p -th diagonal. If all such minors are equal zero, then all products of k + 1 elements from the considered diagonal are equal zero. All basis elements whichgive a non-zero contribution to p − th diagonal are v m = P ki =1 e m + i ⊗ f n + i , where n − m = p . In the same way let us number the elements of diagonal b m . Teh relationbetween elements of diagonal and the coefficients of combination of v m is given by thefollowing system of linear equations: . . . . . . α ... α k +1 ... = b ... b k +1 ... (13)Because any product of k + 1 elements from the considered diagonal is equal zero, atmost k equations can have the right-hand side different from 0. Removing any k rowsfrom the matrix in (13), we have a nonsingular square matrix. Because we alreadyremoved all equations with non-zero right-hand side, the system (13) has only zerosolution. By induction, we then zero all diagonals, so the only matrix in V kmax withrank less than k is the matrix with all coefficients equal 0. (cid:3) pectral properties of entanglement witnesses
3. Propositions and examplesProposition 1.
Any eigenvector of a k -SW related to a negative eigenvalue is not k − separable Proof:
Use the second condition in the theorem 1. (cid:3)
The theorem 3 leads to proposition:
Proposition 2.
For k -SW, ≤ dim V − ≤ ( d − k ) × ( d − k ) . If in V there are no k − separable vectors, then the third condition in the theorem1 is fulfilled, and then we have the following: Proposition 3. If ker W ∩ S k = ∅ and W is an k -SW, then dim V + ≥ k ( d + d ) − k Proof:
By the second condition in Theorem 1, if there is a k − separable vector in V ⊕ V − then it belongs to V . Because by the assumption no k − separable vectors arein V , no k − separable vectors are in V ⊕ V − , and therefore the dimension of V ⊕ V − is bounded from above by ( d − k ) × ( d − k ). Because dim V + = d d − dim( V − ⊕ V ),we get the above inequality. (cid:3) It is not possible to establish any non-trivial upper bound for positive subspaceof an arbitrary k -Schmidt witness, because we can always get a new k -SW by addinga positive observable, with its support contained in the kernel of the witness. Aninteresting question would be, whether there exist such bounds for optimal witnesses,i.e. such witnesses, from which no positive observable cannot be subtracted withoutdestroying the property of being k -Schmidt witness.We can construct in practice examples of subspaces, which do not contain anyseparable vector, by means of unextendible product basis (denoted further as UPB,see [9], [10]), as the orthogonal complement of a subspace spanned by UPB. Such asubspace by the definion of UPB contains no separable vectors. The maximal number n of vectors in UPB in C d ⊗ C d is bounded from below by the number n ≥ X i ( d i −
1) + 1(see Lemma 1 in [9]). Subtracting it from the dimension of the space, we get an upperbound for the dimension of the constructed subspace, which does not contain anyseparable vectorsdim V ≤ ( d − × ( d − . Then, for subspaces constructed by means of UPB, Proposition 2 recovers the previousresults.Now for a subspace V with no non-negative separable vectors, Theorem 2 assuresus that an observable: W = I − (1 + ǫ ) P V is an EW for small enough positive ǫ . ( P V denotes a projection onto subspace V ).For subspaces constructed by means of UPB, such a construction was made in [11].Moreover, the nondecomposability of such W was proven. Example 1.
Any EW for 2 qubits has exactly one negative eigenvalue and exactlythree positive eigenvalues.pectral properties of entanglement witnesses Proof:
At least one negative eigenvalue is needed, to ensure that the witnessdetects anything (first condition in Definition 2). The eigenvector related to thiseigenvalue is of Schmidt rank two. It will be denoted as Ψ − . The negative spacecannot consist of any separable vector, so Proposition 2 bounds its dimension by 1.No vector of Schmidt rank greater than one can be in the kernel. If there weresuch a vector Ψ, then by Nullstellensatz there exists such α : β , thatrank( α A (Ψ − ) + β A (Ψ)) = 1 , but then β = 0, so there would be a separable vector α Ψ + β Ψ − ∈ V ⊕ V − , but α Ψ + β Ψ − ∈ V ⊕ V − V , which would be in contradiction with the second conditionin Theorem 1. Thus there can be only separable vectors in the kernel.Write such a vector in its Schmidt decomposition A (Ψ ) = (cid:20) (cid:21) . The vector Ψ − is of Schmidt rank greater than one and orthogonal to Ψ and can bewritten as A (Ψ − ) = (cid:20) ab c (cid:21) , where ab = 0. Now let us identify all separable vectors in V ⊕ V − rank( α A (Ψ − ) + β A (Ψ )) = 1 ⇐⇒ β ( αc + βab ) = 0 . If c = 0, then there exists a separable vector, which is in V ⊕ V − , but not in V , whichis in contradiction with the second condition in Theorem 1. Thus c = 0.If there is any other vector in the kernel, let us say Ψ , then by the conditionthat no entangled vector is in V , the subspace V consists of only separable vectors.We then conclude that A (Ψ ) has the second column or the second row equal 0, butno such a vector (other than Ψ ) is orthogonal to Ψ − . Therefore there is at most onevector in the kernel (up to scaling by nonzero scalar) and it is separable.Let us identify now the subspace ˜ V Ψ , which will be needed to check the thirdcondition in Theorem 1. It is the subspace of all vectors Ψ, for which [ A (Ψ)] = 0.Observe that Ψ − ∈ ˜ V Ψ , which is in contradiction with the third condition in Theorem1. No non-zero vector thus can be in the kernel. (cid:3) Because we know that any EW in C ⊗ C is decomposable [12], we conclude thatthe partial transposition of an entangled positive matrix in C ⊗ C has exactly onenegative and 3 positive eigenvalues. For another proof of this fact, see [13]
4. Conditions on eigenvalues of entanglement witnesses
Theorem 2 assures, that given two semipositive observables W ± which have othogonalsupports, when some conditions on the support of W − are fulfilled, one can constructa witness λW + − W − for sufficiently large alpha. There is still a question of how big λ should be. We restrict ourselves in this subsection to entanglement witnesses, i.e.we fix k = 1. In some cases, for k = 1 parameter λ can be calculated or at least itacquires some new interpretation.In the beginning, take W − = P V − and W + = I − P V − . Because V = ( V + ⊕ V − ) ⊥ = { } , the third condition of Theorem 2 is fulfilled in a trivial way. The second one inthis situation reads V − ∩ S k = ∅ . Let us take therefore a subspace V − which does notcontain any vectors of Schmidt rank one. pectral properties of entanglement witnesses λW + − W − is equal to W = ǫI − P V − , ǫ = λλ + 1 . Because the observable is an entanglement witness, it has positive eigenvalues whenrestricted to separable states. For any normalized Ψ ∈ S we have therefore ǫ ≥ h Ψ | P V − | Ψ i = || P V − Ψ || = (sup {|h Φ | Ψ i| : Φ ∈ V − ∧ || Φ || = 1 } ) . The smallest ǫ which fulfils the above condition is a supremum of the right hand sidewith respect to all normalized Ψ = φ ⊗ χ , ǫ min = (sup {|h Φ | φ ⊗ χ i| : || φ || = 1 ∧ || χ || = 1 ∧ || Φ || = 1 ∧ Φ ∈ V − } ) . It is easy to observe, that h Φ | φ ⊗ χ i = φ T A (Φ) χ . Using it, we find the final formulafor ǫ : ǫ min = (sup {| φ T A (Φ) χ | : || φ || = 1 ∧ || χ || = 1 ∧ || Φ || = 1 ∧ Φ ∈ V − } ) = (sup {|| A (Φ) || : || Φ || = 1 ∧ Φ ∈ V − } ) = sup {|| A (Φ) || : || Φ || = 1 ∧ Φ ∈ V − } . The supremum norm of A (Ψ) is the maximal Schmidt coefficient of the vector Ψ.Translating by the isomorphism A the subspace V − into the corresponding subspace A ( V − ) in the space of matrices of coefficients, we are looking for the supremum of thesup norm on the unit sphere in this subspace. The quantity ǫ min can be thereforeinterpreted as the supremum norm of the subspace of matrices . We will thereforedenote it further as ǫ min = || A ( V − ) || sup . When the subspace V − is one-dimensionaland spanned by a vector Ψ, ǫ min can be calculated as || A (Ψ) || sup . We can use thisspecial class of witnesses to find the conditions for the eigenvalues of entanglementwitnesses. One can estimate any witness W = W + − W − from above as W + − W − ≤ λ max + P V + − λ min − P V − ≤ λ max + P V + ⊕ V − λ min − P V − = λ max + I − ( λ max + + λ min − ) P V − , (14)and from below as W + − W − ≥ λ + min P V + − λ − max P V − ≥ λ min + P V + − λ max − P V − ⊕ V = λ min + I − ( λ min + + λ max − ) P V − ⊕ V . (15)The upper bound is EW iff λ max + λ max + + λ min − ≤ || A ( V − ) || sup . (16)We conclude therefore, that (16) is a necessary condition for W to be entanglementwitness.The lower bound is EW iff λ min + λ min + + λ max − ≤ || A ( V − ⊕ V ) || sup . (17) pectral properties of entanglement witnesses W to be an entanglementwitness. Observe, that this sufficient condition can be fulfilled iff there are no vectorsof Schmidt rank 1 in the kernel.When the negative subspace of W is one-dimensional and the kernel is trivial,then one gets the sufficient condition λ min + λ min + + λ − ≤ || A (Ψ) || sup One can find this condition for related positive maps in [14].It rests to show how Theorem 2 works when there exist vectors of Schmidt rank1 in the kernel. In this example, consider the space C ⊗ C . Denote by { e i } i =1 thebasis of C and by { f i } i =1 the basis of C Let us span the kernel by two vectors: e ⊗ f and e ⊗ f . The space ˆ V is nowˆ V = span { e , e } ⊗ span { f , f } . The subspace V − can be now embeded in a two-qubit space. It is therefore spannedby one normalized vector of Schmidt rank two. Denote it by ψ − . Now we choosearbitrary orthonormal basis of V ⊥− ∩ span { e , e } ⊗ span { f , f } . Denote its elementsby k , k and k . Finaly, we complete the set e ⊗ f , e ⊗ f , ψ − , k , k , k } to theorthonormal basis of C ⊗ C by six vectors l , ..., l . It will be the basis of our witness.Now Theorem 2 assures, that fixing the negative eigenvalue, say −
1, and choosinglarge enough positive eigenvalues, one can get an entanglement witness. To do it, letus fix the eigenvalues related to the vectors k , k and k to be equal ǫ/ (1 − ǫ ), where ǫ = || A ( ψ − ) || sup . Suppose, that the rest of eigenvalues are equal zero — we havetwo-qubit witness whose domain is embeded in C ⊗ C . We can leave the eigenvaluesrelated to the eigenvectors l , . . . , l unchanged, or increase them to some positivevalues — it only destroys the optimality of the witness.
5. Translation to some properties of positivity-preserving mappingsbetween the matrix algebras
The set of linear maps between the matrix algebras B ( C d ) and ( B ( C d )) and theset of bilinear forms on C d ⊗ C d is isomorphic by the well-known Jamio lkowskiisomorphisms [15] J : B ( C d ⊗ C d ) → L ( B ( C d ) , B ( C d )), W Λ = J (Λ) = [ I ⊗ Λ] | Ψ + ih Ψ + | , (18)where A (Ψ + ) = 1 /d Id d (a projector onto Ψ + is the maximally entangled state).The subset of linear maps, which preserve the hermiticity of a matrix, isisomorphic by (18) to the subset of Hermitian bilinear forms. Any Hermitian matrixhas spectral decomposition W = p X i =1 λ + i | Ψ + i ih Ψ + i | − q X i =1 | λ − i || Ψ − i ih Ψ − i | , (19)where ( p, q ) is the signature of W . Such a matrix is related by the isomorphism (18)to a hermiticity-preserving linear map:Λ ρ = p X i =1 A i ρA † i − q X i =1 B i ρB † i , (20) pectral properties of entanglement witnesses A i = q λ + i A (Ψ + i ) and B i = q | λ − i | A (Ψ − i ). For more details and other factsabout this correspondence see [16] and the references therein.The matrices A i and B i are orthogonal with respect to the Hilbert-Schmidt innerproduct. The observable W admits many decompositions into linear combinationsof projectors, but the decomposition into a combination of orthogonal projectors isunique up to degeneracy of eigenvalues. Similarly, one hermiticity-preserving map Λadmits many decompositions into the form (20) (such a form is called the Kraus-Choiform of hermiticity preserving map due to Kraus-Choi representation theorem [17]),but if we assume the ortogonality of matrices A i and B i , then the decompositionbecomes unique.In the set of maps, which preserve the hermiticity we are allowed to define asubset of maps which preserve positivity. Such maps are called positive maps . We cangeneralize the definition of a positive map to k -positive map such that Λ is k - positivewhen the map Id k ⊗ Λ is positive. The isomorphism (18) relates a k -positive map to a k -Schmidt witness. The isomorphism (18) allows to reformulate the propositions aboutproperties of k -SWs to propositions describing properties of k − positive maps. Observethat if matrices A i , B i in (20) are not orthogonal, but still lineary independent, then( p, q ) remains unchanged.Cosider now a hermiticity-preserving map in its Kraus-Choi form:Λ ρ = p X i =1 A i ρA † i − q X i =1 B i ρB † i . (21)Assume that matrices A i , B i are linearly independent. We have then the following: Proposition 4. If Λ is a k -positive map, then ≤ q ≤ ( d − k ) × ( d − k ) . Proposition 5. If Λ is a k -positive map and does not map any state of rank less orequal k to zero, then p ≥ d d − ( d − k ) × ( d − k ) .
6. Conclusions
The necessary and sufficient condition for k -Schmidt witness has been presented andproved. Moreover, in the system of two qubits, the necessary condition fully determinesthe spectral type of any entanglement witness. The neccesary condition provides alsoan upper bound for the dimension of negative subspace of a k -Schmidt witness inarbitrary dimensions, which can be translated to analogous properties of k − positivemaps. Also the sufficient condition for a k -Schmidt witness (up to rescaling of thepositive part of an observable) has been proved. Acknowledgments
This work was partially supported by the Polish Ministry of Science and HigherEducation Grant No 3004/B/H03/2007/33. I would like to thank prof. Ma´ckowiakand dr Michalski for language corrections and to prof. Zwara for fruitfull discussionsabout algebraic geometry.
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