Entanglement Detection via Direct-Sum Majorization Uncertainty Relations

In this paper we investigate the relationship between direct-sum majorization formulation of uncertainty relations and entanglement, for the case of two observables. Our primary results are entanglement detection methods based on direct-sum majorization uncertainty relations. These detectors provide a set of sufficient conditions for detecting entanglement whose number grows linearly with the dimension of the state being detected.


I. INTRODUCTION
Uncertainty relations form a central part of quantum mechanics.They impose fundamental limitations on our ability to simultaneously predict the outcomes of noncommuting observables.Different approaches have been proposed to quantify these relations.The original formulation is given by Heisenberg [1] in terms of standard deviations for momentum and position operators.His result is then generalized to two arbitrary observables [2].Later it is recognized that one can express uncertainty relations in terms of entropies [3][4][5].In this approach, entropy functions like Shannon and Rényi entropies are used to quantify uncertainty (Ref.[6] is a nice survey on this topic).However, entropies are by no reason the most adequate to use.With this motivation, majorization is used to study uncertainty relations [7].This line of research is further investigated in [8][9][10].
Entanglement is another appealing feature of quantum mechanics and has been extensively investigated in the past decades [11].Entangled states play important roles in quantum information processing, such as quantum teleportation [12] and dense coding [13].Deciding whether a given quantum state is entangled is a key problem of quantum information theory and known to be computationally intractable in general [14].Therefore, computationally tractable necessary conditions for entanglement detection, which provide a partial solution, have been the subject of active research in recent years [15].
Refs. [16,17] present several methods for detecting entanglement via variance based uncertainty relations.Similar methods have been designed using entropy based uncertainty relations [18,19].One may wonder whether there exists a relationship between the majorization based uncertainty relation and entanglement.The answer is affirmative.In [20], the author applies the tensorproduct majorization formulation of uncertainty [7] to the problem of entanglement detection.In this paper we use the direct-sum majorization uncertainty relation, developed in [10], to design an entanglement detection method.As the direct-sum majorization bound has analytical solution while the tensor-product majorization bound does not, our direct-sum majorization based detection method is more practical than the tensor-product majorization based method.
The rest of this paper is organized as follows.In Sec.II, we establish the notation and briefly review the directsum majorization formulation of uncertainty.In Sec.III, we present our central result -an entanglement detection method based on the direct-sum majorization uncertainty.In Sec.IV, we generalize our result to the case of many observables.We conclude in Sec.V. Some proofs are given in the Appendix.

II. DIRECT-SUM MAJORIZATION UNCERTAINTY RELATIONS
This section presents a basic review of the majorization theory and the formulation of direct-sum majorization approach to uncertainty relations.

A. Majorization
Let R + = [0, ∞) be the set of non-negative real numbers, be the set of ddimensional real vectors with non-negative components.We denote by p ∈ R d + a d-dimensional vector and by p i the i-th element of p.For any vector p ∈ R d + , let p ↓ be the vector obtained from p by arranging the components of the latter in descending order.Given two vectors p, q ∈ R d + , p is said to be majorized by q and written p q ↓ i , and where [d] = {1, • • • , d}.Intuitively, p ≺ q means that the sum of largest k components of p is no larger than the sum of k largest components of q.The majorization order is a partial order, i.e., not every two vectors are comparable under majorization.When studying majorization among two vectors of different dimensions, we append 0(s) to the vector with smaller dimension so that two vectors have the same dimension.A related concept is the supremum of a set of N vectors, defined as the vector that majorizes every element of the set and, is majorized by any vector that has the same property.We now briefly describe how to construct the supremum vector, more details can be found in [7,21].Let S = {p (1) + } a set of N vectors.To construct the supremum for S, we define a (d + 1)dimensional vector Ω with components The desired supremum ω sup is then given by The construction given in Eq. ( 1) guarantees that ω sup majorizes every element of the set S, but ω sup does not necessarily appear in a descending order and may, therefore, fails to be majorized by other vectors with the same property.In such case, we must perform a "flattening" process.This process starts with ω sup obtained in Eq. ( 1), and for every pair of components violating the descending order, say, ω sup k < ω sup k+1 , replaces the pair by their mean such that the updated two elements are This process continues until a descending vector corresponding to the supremum is obtained.

B. Direct-sum majorization uncertainty
We now briefly introduce the uncertainty relation characterized by direct-sum majorization relation.We remark that the results summarized here is originally presented in [10].
Let H be a d-dimensional Hilbert space.Denote by D(H) the set of quantum states in H. Let X and Z be two rank-one projective observables, and ρ be a state on H. Assume the spectral decompositions of X and Z are given by where {|x i } and {|z j } are the eigenstates of X and Z, respectively.By measuring ρ, X induces a probability distribution given by Similarly, Z induces a probability distribution given by We are interested in the uncertainty relation induced by these two observables.In [10] the direct-sum majorization approach is used to is to characterize the uncertainty about p(X|ρ) and q(Z|ρ): where ω X⊕Z is a 2d-dimensional vector independent of ρ which can be explicitly calculated from observables X and Z. Intuitively, ω X⊕Z is the supremum vector of the set Now we show how to compute ω X⊕Z analytically.From the definitions of p and q, we can see that only the eigenstates of X and Z matter.We define a d × d unitary operator U whose elements are given by U ij = x i |z j .U is known as the overlapping matrix as it characterizes the overlap of the two orthonormal bases.For each k ∈ [d], let SUB(U, k) be the set of submatrices of class k of U defined as The symbols ♯ col(M ) and ♯ row(M ) denote the number of columns and rows of matrix M , respectively.Based on the concept of submatrices, we define the following set of coefficients, which is important in computing ω X⊕Z : where M ∞ is the operator norm of M , and the maximum is optimized over all submatrices of class k.By construction we have In [10] it is proved that ω X⊕Z = {1} ⊕ s, where s is given by We append d − 1 0s to make ω X⊕Z a 2d-dimensional vector.We remark that the vector s is not necessarily sorted in descending order, but we can use the "flattening" process described in Sec.II A to make it descending ordered.
In words, the direct-sum majorization uncertainty relation can be summarized in the following theorem.

III. ENTANGLEMENT DETECTION
An entanglement detector decides whether a given bipartite state is separable by providing a condition that is satisfied by all separable states, and if violated, witnesses entanglement.In this section, we design a detection method based on the direct-sum majorization bound described in Sec.II B. As majorization relations, our detector actually provides a set of conditions whose number will grow with the dimension of the state.We first describe a majorization bound for all separable states.Then we show how this bound serves as a detector.In the end, we illustrate by some examples how well the detector works.

A. Majorization bounds
If an observable X is degenerate, the definition of p(X|ρ), given in Eq. ( 2), is not unique, since the spectral decomposition is not unique.By combining eigenstates with the same eigenvalue, however, there exists a unique spectral decomposition of the form X = i λ i P i , with λ i = λ i ′ for i = i ′ and P i are orthogonal projectors of maximal rank [22].Under this convention, we define for degenerate observable X the distribution p i = Tr [P i ρ].
Our entanglement detection method relies on the degeneracy properties of the product observables on bipartite systems.It is possible that for two non-commuting observables X A and X B , their product X A ⊗ X B is degenerate.Consequently, it may happen that X A ⊗ X B and Z A ⊗Z B have a common eigenstate, and this eigenstate is an entangled pure state.In such cases, the probabilities p(X A ⊗ X B |ρ) and p(Z A ⊗ Z B |ρ) will reflect the stated difference and may be capable of detecting entanglement.As an example, consider the Pauli Z operator σ z on system A and B. The product observable on AB is given by σ z ⊗ σ z .The spectral decomposition of σ z ⊗ σ z is (under our convention) Similarly, we have σ x ⊗ σ x = P + − P − , where There exists no state ρ A that can result in certain outcomes for both σ x and σ z , because they do not commute.But there do exist an entangled state |Ψ that can give certain outcomes for both σ x ⊗ σ x and σ z ⊗ σ z , as they commute.By the Schmidt decomposition, they can be expressed in the same eigenbases which are possibly entangled.
Let X A and X B be two full rank observables on A and B, respectively.Assume their spectral decompositions are given by Performing the product observable X A ⊗ X B on a bipartite state ρ AB , we obtain a joint distribution As X A ⊗ X B might be degenerate, some elements p(i, j) are grouped together since they belong to the same eigenvalue.We denote by p(X A ⊗ X B |ρ) the joint distribution after grouping.If we perform local observables, we obtain marginal distributions p(X A |ρ A ) and p(X B |ρ B ).It is proved in [22] that the joint distribution of a product state is majorized by the distribution of its marginal.
Lemma 2 ([22], Lemma 1).Let ρ = ρ A ⊗ρ B be a product state and let X A and X B be two observables on A and B, respectively.Then Intuitively, this is because for the product observable X A ⊗ X B , its eigenstates are possibly entangled, and thus product state gives uncertain outcomes, however it is possible that the reduced state gives certain outcome for the corresponding local observable.Now we consider the effect of several product observables.Let X A and Z A be two observables on A, X B and Z B be two observables on B, respectively.For arbitrary product state ρ = ρ A ⊗ρ B , we obtain from Lemma 2 that p As the direct-sum operation preserves the majorization order [23], we have The RHS. of Eq. ( 6) is the direct-sum of two distributions.By the virtue of Thm. 1, it holds that Combining Eq. ( 6) and Eq. ( 7), we reach the following statement for arbitrary product states The majorization relation derived in Eq. ( 8) holds for product states.Now we show that this relation actually holds for arbitrary separable states.We are actually interested in the optimal state that majorizes all possible probability distributions p (X A ⊗ X B |ρ) ⊕ p (Z A ⊗ Z B |ρ) induced by performing X A ⊗X B and Z A ⊗Z B on separable states.Such a state can be defined as where SEP(H A :H B ) is the set of separable states of bipartite space H A ⊗ H B .In Appx.A we prove that ω (XAXB )⊕(ZAZB ) SEP can be achieved among pure product states, and thus we reduce the optimization over all separable states required in Eq. ( 9) to the optimization over all pure product states: For an arbitrary separable state (be it pure or not) ρ AB , it then holds that p The first inequality follows from the definition of ω SEP , while the second inequality follows from the fact that each element of ω SEP is achieved by some pure product state, and which in turn be majorized by ω XA⊕ZA as proved in Eq. ( 8).To summarize, we have the following theorem.
Theorem 3. Let X A ⊗ X B and Z A ⊗ Z B be two product observables.For arbitrary separable state ρ ∈ D( where ω XA⊕ZA is defined in Thm. 1.Similarly, one has

B. The detection framework
Thm. 3 states that ω XA⊕ZA is a necessary condition for separability and its violation signals the existence of entanglement.This statement provides an operational method of entanglement detection.Given a bipartite state ρ ∈ D(H A ⊗ H B ), we first calculate the direct-sum probability distribution p (X A ⊗ X B |ρ) ⊕ p (Z A ⊗ Z B |ρ) induced by the product observables X A ⊗ X B and Z A ⊗ Z B .Then we investigate the majorization relation between it and ω XA⊕ZA .If ω XA⊕ZA does not majorize the direct-sum distribution, then we conclude that ρ is entangled.However, if ω XA⊕ZA majorizes the distribution, we can say nothing about ρ: it can be separable, it can also be entangled.
The proposed method is a collection of linear detectors.Indeed, Thm. 3 states the following fact.For arbitrary As the first and the last d inequalities are trivial, we have d − 1 effective inequalities in total, thus d − 1 linear detectors.States that violate any of these inequalities will necessarily be entangled.

IV. ENTANGLEMENT DETECTION VIA MANY OBSERVABLES
The entanglement detection method described in Sec.II makes use of two incompatible observables on each part.In this section, we generalize this method to the case of many incompatible observables.
Tensor-product majorization based uncertainty relations for many observables was first studied in [24].Here we show their results can be extended to the direct-sum majorization based uncertainty relations.Let ρ be a quantum state and {X (l) } l∈[L] be a set of N observables on H, where [L] = {1, • • • , L}. Assume the spectral decomposition of X (l) is given by where {|x (l) i } are the eigenstates of X (l) .By measuring ρ, X (l) induces a probability distribution given by The direct-sum majorization based uncertainty relations for this set of observables has the following form: where ω is a N d-dimensional vector independent of ρ which can be explicitly calculated from observables X (l) .
To compute ω, we define the following coefficients where λ 1 (A) denotes the maximal singular value of A, and the terms S l , U (S 1 , • • • , S L ) are defined in [24].The main differences between our definition of s k in Eq. (10) and the s k defined in Eq. 15 of [24] lie in that 1.In our definition 10, S l ≥ 0; while in their definition, S l is strictly positive.
2. In our definition 10, the optimization is over all {S l } such that L x=1 S l = k; while in their definition, the optimization is over all {S x } such that L x=1 S l = k + L − 1.These two differences guarantee that we can use s k to give upper bounds on the sum of the first d terms of ω.With coefficients {s k }, we can derive a direct-sum majorization bound for many observables.