Mixture of entangled pure states with maximally mixed one-qudit reduced density matrices
aa r X i v : . [ qu a n t - ph ] D ec Mixture of entangled pure states with maximally mixed one-qudit reduced densitymatrices
Marvin M. Flores and Eric A. Galapon ∗ Theoretical Physics Group, National Institute of Physics,University of the Philippines, Diliman Quezon City, 1101 Philippines (Dated: January 5, 2018)We study the conditions when mixtures of entangled pure states with maximally mixed one-qudit reduced density matrices remain entangled. We found that the resulting mixed state remainsentangled when the number of entangled pure states to be mixed is less than or equal to thedimension of the pure states. For the latter case of mixing a number of pure states equal to theirdimension, we found that the mixed state is entangled provided that the entangled pure states tobe mixed are not equally weighted. We also found that one can restrict the set of pure states thatone can mix from in order to ensure that the resulting mixed state is genuinely entangled.
PACS numbers: 03.67.Mn
I. INTRODUCTION
One of the outstanding problems in the theory of quan-tum entanglement, which is of paramount importancewhen using entanglement as a resource for quantum in-formation processing is its detection [1]. In other words,given an arbitrary quantum state, the problem of detec-tion asks whether the state is entangled or not. Thedifficulty of the problem of entanglement detection is re-flected in the abundance of separability criteria in liter-ature [1, 2], some involving extremizations and workingonly for some special cases. The problem is even morepronounced in the case of mixed states, where the typi-cal recourse would be to construct convex roof extensionsof existing measures for pure states, something which ishard to compute in general [3, 4]. Hence, it would bevery helpful if we can come up with classes of mixedstates which are entangled, thus saving us from having toapply an entanglement detection scheme should a givenstate belong to these classes.In this paper, we introduce a class of entangled mixedstates by mixing distinct N -partite entangled pure stateshaving equal dimensions d and having maximally mixedone-qudit reduced density matrices. We are motivatedby the fact that although a convex combination of sepa-rable states is again separable, the mixture of entangledpure states is not necessarily entangled as can be eas-ily seen by mixing two distinct maximally entangled Bellstates of equal weights [5]. Our choice of mixing entan-gled pure states where all the one-qudit reduced den-sity matrices are maximally mixed is justified since weutilized an entropic-based separability criterion to deter-mine whether the resulting mixed states are entangled ornot. Entropic-based separability criteria in general de-tect entanglement by virtue of how much information ispresent in the correlations between subsystems, ratherthan in the subsystems themselves. ∗ Electronic address: [email protected]
We show that for the mixture to remain entangled, thedimension d of these entangled pure states must exceedthe number of states M in the convex combination. Also,we show that when mixing d number of these entangledpure states, the resulting mixed state is entangled pro-vided that their weights are not all equal. We also showthat we can construct mixed states which are genuinelyentangled at the expense of making the set of pure statesthat we can mix from smaller.The paper will begin by introducing the separabilitycriterion based on purity in Section 2 which we will beutilizing in determining which states are entangled or not.In Section 3, we study the conditions wherein the mixtureof entangled pure states with maximally mixed one-quditreduced density matrices remain entangled. Then we re-strict the set of these pure states that we can mix fromin Section 4 with the advantage of having our resultingmixed state to have an entanglement that we know tobe genuine. Section 5 discusses the consistency of ourresults with the measure of entangled mixed states forbipartite qubits known as concurrence. Finally, we drawour conclusions in Section 6. II. PURITY-BASED SEPARABILITYCRITERION
Let ρ be a density matrix for a composite system. Wesay that ρ is a product state if there exist states ρ A forAlice and ρ B for Bob such that ρ = ρ A ⊗ ρ B . (1)The state is called separable , if there are convex weights p i and product states ρ Ai ⊗ ρ Bi such that ρ = X i p i ρ Ai ⊗ ρ Bi (2)holds. Otherwise, the state is called entangled . In gen-eral, for a mixed state of N systems, separability entailsthat we could write the state as ρ = X i p i ρ i ⊗ · · · ⊗ ρ Ni . (3)In this paper, we also distinguish between k -separablestates and genuinely multipartite entangled states. Astate is k -separable if it factorizes into k states ρ i , eachof which describes either one or several subsystems, thatis, the N -partite state ρ = X i p i ρ i ⊗ · · · ⊗ ρ ki (4)is k -separable with 1 < k ≤ N . On the other hand, an N -separable N -partite state is called fully separable anda state which is not 2-separable (or biseparable) is calledgenuinely multipartite entangled.This paper utilizes a separability criterion based onpurity due to its straightforwardness. Purity is definedas P ( ρ ) = tr ρ (5)whose value ranges from 1 /d ( d being the dimension ofthe system) for maximally mixed states and 1 for purestates. Taking the purity of Eq. (3), we obtaintr ρ = X i,i ′ p i p i ′ tr ( ρ i · ρ i ′ ) · · · tr ( ρ ni · ρ ni ′ ) (6)where orthogonality between states is not assumed.Now, if we trace out all the other subsystems exceptone, say the first subsystem ρ and obtain the purity, wethen have tr ( ρ ) = X i,i ′ p i p i ′ tr ( ρ i · ρ i ′ ) . (7)As can be easily seen, the purity in Eq. (7) is greaterthan Eq. (6) since the factors of the form tr ( ρ ji · ρ ji ′ ) isbounded by one. In general, we havetr ρ ≤ tr ( ρ i ) (8)for all subsystems i , the equality holding if we assumeorthogonality between the states. Since Eq. (3) is thegeneral form of separable states, we then have the fol-lowing proposition: Proposition 1. If ρ is separable, then tr ρ ≤ tr ( ρ i ) for all the subsystems i . It is important to note here that the above state-ment’s converse does not hold. In other words, havingtr ρ ≤ tr ( ρ i ) true for all the subsystems i does notguarantee that the state ρ is separable. Now, we can takethe contrapositive of the above statement as follows: Proposition 2.
If there exists a subsystem i such thattr ( ρ i ) < tr ρ , then ρ is entangled. Proposition 2 qualitatively encapsulates the physicalimplication of what it means to be entangled. To be en-tangled means that the maximal knowledge of the wholedoes not necessarily constitute the maximal knowledge ofits parts. The ignorance of the subsystem could be quali-tatively attributed to the fact that part of it is correlatedto the other subsystem, hence looking at the subsystemalone is not enough to gain the entire information aboutit. The inequality tr ( ρ i ) < tr ρ says just that, i.e., ifthe purity of the whole system (your knowledge of thewhole system) is greater than the purity of the subsys-tem (your knowledge of the subsystem), then this meansthat the subsystems must be correlated, hence entangled.In other words, you obtain lesser information when look-ing at the individual subsystems rather than the wholeand this missing information is attributed to the cor-relations of the subsystems with each other. The leapfrom purity to information (knowledge) is done abovesince the purity is related to the 2-Renyi entropy givenby S α =2 ( ρ ) = − log tr ( ρ ) [12, 13]. Since the 2-Renyientropy is a monotonic function of the purity, we consid-ered the latter instead of the former for the sake of sim-plicity. In fact, Proposition 2 is a simpler restatement ofthe 2-Renyi entropic inequality S α =2 ( ρ ) ≥ S α =2 ( ρ i ).Also, we use the above criterion as opposed to existingones since it is very straightforward to apply. Given astate ρ , obtain its purity then take the partial traces inorder to obtain the purity of the subsystems. If one ofthe purity of the subsystems turns out to be less thanthe purity of the whole, then the state ρ is entangled.Note also that the utility of our separability criterion isnot limited in any way by the number of systems or thedimensionality of the Hilbert space.Just like the Renyi entropy, there are certain stateswhere the purity-based separability criterion fails [9].Consider the Størmer state [10, 11] which is a 3 × α with values 2 ≤ α ≤ σ α = 27 | ψ + ih ψ + | + α σ + + 5 − α σ − (9)where σ + = 13 ( | i| ih |h | + | i| ih |h | + | i| ih |h | ) (10) σ − = 13 ( | i| ih |h | + | i| ih |h | + | i| ih |h | ) (11) | ψ + i = 1 √ | i + | i + | i ) . (12)It is known that the Størmer state is separable for 2 ≤ α ≤
3, bound entangled for 3 < α ≤ < α ≤
5. However, simple calculation will show thatour criterion detects the entanglement (that is, P ( σ α,i )
ρ is entangled, then it is not neces-sarily true that there exists a subsystem i such thattr ( ρ i ) < tr ρ , as is the case when 3 < α ≤
5, wherebytr ( ρ i ) ≥ tr ρ even though σ α is entangled for these val-ues of α . What this entails is that we can actually thinkof two types of entanglement: (I) entangled states whichare detected by Proposition 2 and (II) entangled stateswhich are not detected by Proposition 2. Hence, any en-tanglement detected using Proposition 2 will fall undertype I. These are actually entangled states which are use-ful in specific nonclassical tasks such as the reduction ofcommunication complexity or quantum cryptography aspointed out in [6]. In particular, under the context ofoptimal state merging protocol, the same reference men-tioned two regimes which they called the classical (typeII in our case where the purity of the whole is less thanthe purity of its parts) whose partial information S ( A | B )is positive and the quantum (type I in our case where thepurity of the whole is greater than the purity of its parts)whose partial information S ( A | B ) is negative. The par-tial information [2, 6] is defined as S ( A | B ) = S ( ρ AB ) − S ( ρ B ) (13)where S ( ρ ) is entropy of the state ρ .Another important thing to note about Proposition 2is that the entanglement it detects may not be genuine.It is easy to see this by considering the following example.We construct the biseparable 4-partite state ρ = ρ ⊗ ρ where ρ = ρ = | ψ ih ψ | , and | ψ i could be any of thebipartite entangled Bell states | ψ ± i = √ ( | i ± | i )and | φ ± i = √ ( | i ± | i ). Then it is easy to see thattr ( ρ i ) < tr ρ for all i . In other words, Proposition 2detects ρ to be entangled even though this state is ob-viously biseparable. The reason is that our propositiondetects all the entanglements within the system, and hav-ing an i such that tr ( ρ i ) < tr ρ only means that thissubsystem is entangled with some other particle, but notnecessarily with all of them. III. ENTANGLEMENT OF THE MIXTURE OFENTANGLED PURE STATES HAVINGMAXIMALLY MIXED ONE-QUDIT DENSITYMATRICES
We define the class of entangled pure states, M N ( d ),in the Hilbert space H = ⊗ Nk =1 H k = H ⊗ H ⊗ · · · ⊗ H N ,where d = dim H k . Elements of M N ( d ), | Φ i , has thedefining property tr H / H k | Φ ih Φ | = I k d (14)where I k is the identity matrix in H k and d is the di-mension H k . Note that Eq. (14) means that the di-mensions of the subsystems must all be equal, that is, d = · · · = d N = d . The property also implies the nor-malization condition h Φ | Φ i = 1. For N = 2 and d = 2, the states satisfying Eqs. (14) are the four Bell states.Some examples of states belonging to M N ( d ) for arbi-trary N and d include the N -qudit states ( d N − − | Φ j d i = 1 √ d d − X k =0 | k i ⊗ j − | k + 1 i| k i ⊗ N − j (15)where j ∈ { , , , . . . , d N − − } as well as the common N -qudit GHZ states given by | Φ GHZ i = 1 √ d ( | i ⊗ N + | i ⊗ N + · · · + | d i ⊗ N ) . (16)We are interested in the mixture of the elements of M N ( d ). We construct the mixed state ρ ( ~λ M , ~ Φ) = M X k =1 λ k | Φ k ih Φ k | (17)where ~λ M = { λ , . . . , λ M } with P Mk =1 λ k = 1, λ k > ~ Φ = {| Φ i , . . . , | Φ M i } with | Φ k i ∈ M N ( d ), for somepositive integer M ≥
2. We refer to the collection of thesestates as c M M,N ( d ). When it happens that the vectorsin ~ Φ are pairwise orthogonal, that is, h Φ k | Φ l i = δ kl , wedenote the subset by c M ort M,N ( d ). The mixed state ρ isa mixture of entangled pure states. However, it is notnecessary that the mixture is entangled as can be readilydemonstrated. As a trivial example, the concurrence ofthe mixed state ρ = | ψ ih ψ | + | φ ih φ | is zero, with | ψ i and | φ i being distinct bipartite entangled Bell states.We now wish to investigate this class of mixed statesby means of the separability criterion given in this paper.We can readily establish the following equalitiestr ρ = M X k,l =1 λ k λ l |h Φ k | Φ l i| (18)tr k ρ k = 1 d . (19)Then ρ is entangled provided M X k,l =1 λ k λ l | h Φ k | Φ l i | > d . (20)Also central to the development to follow is the func-tion P ( ~λ M , ~ Φ) = M X k,l =1 λ k λ l |h Φ k | Φ l i| (21)which is just the purity of the given mixed state. Theability of the separability criterion to detect entangle-ment will depend on the minimum of this function thatoccurs when the involved states are mutually orthogonaland we denote this minimum by P ort( ~λ M , ~ Φ) = M X k =1 λ k = P ort( ~λ M ) . (22) Definition 1.
We refer to a subset, M , of c M M,N ( d ) asentangled if every density matrix ρ of M is entangled. Lemma 1. If c M ort M,N ( d ) is entangled, then c M M,N ( d ) isentangled.Proof. For every ~λ M we have the inequality P ( ~λ M , ~ Φ) ≥ P ort( ~λ M ), with equality only when ~ Φ is mutually orthog-onal. This follows from the positivity of all the termsin Eq. (21) and the fact that P ort( ~λ M ) is just the di-agonal of P ( ~λ M , ~ Φ). Since c M ort M,N ( d ) is, by hypothesis,entangled, we have P ort( ~λ M ) > /d for all ~λ M . Thenfrom the inequality P ( ~λ M , ~ Φ) ≥ P ort( ~λ M ), we have also P ( ~λ M , ~ Φ) > /d for all ~λ M and ~ Φ so that c M M,N ( d ) isentangled. Theorem 1. c M M,N ( d ) is entangled for all d > M .Proof. Using the above Lemma it is sufficient to establishthat c M ort M,N ( d ) is entangled when d > M . Imposing theconstraint P Mk =1 λ k = 1, the purity function P ort( ~λ M )assumes the form P ort( ~λ M ) = M − X k =1 λ k + (1 − M − X k =1 λ k ) . (23)The minimum is obtained by taking its first derivativeand equating to zero, ∂P ort /∂λ l = 0, λ l + M − X k =1 λ k = 1 . (24)The solution is λ k = 1 /M for all k = 1 , . . . , M , whichwe will denote as ~λ M, . Then we have the inequal-ity P ort( ~λ M ) ≥ /M for all ~λ M . Since d > M , then P ort( ~λ M ) > /d . Hence c M ort M,N ( d ) is entangled and c M M,N ( d ) is likewise entangled by the above Lemma. Theorem 2. c M d,N ( d ) / { ρ ( ~λ M, , ~ Φ ort ) } is entangled,where ~λ M, = { /M, . . . , /M } .Proof. The purity of the state ρ ( ~λ M, , ~ Φort) is P ort( ~λ M, ) = 1 /d so that the criteria fails on this state.However, for any ~λ M = ~λ M, the purity is necessar-ily P ort( ~λ M ) > /d . This, together with the fact that P ( ~λ M , ~ Φ) > P ort( ~λ M, ) for all ~ Φ = ~ Φort imply that c M d,N ( d ) / { ρ ( ~λ M, , ~ Φort) } is entangled. IV. GENUINE ENTANGLEMENT OF THEMIXTURE OF ENTANGLED PURE STATESWITH SEPARABLE REDUCED DENSITYMATRICES
We go back to the caveat at the end of Section 2 thatour criterion detects entanglement which may or may notbe genuine. In particular, although the class of mixedstates that were constructed following the theorems inthe previous section are entangled, we could not be surewhether the entanglement is genuine or not.Here we define a class of states N N ( d ) ⊂ M N ( d ). Itselements, which are entangled pure states | Φ i , has theproperty ρ s = tr H i | Φ ih Φ | (25)is a separable state. In other words, tracing out a sin-gle subsystem leaves a state which is separable. As ithappens, this is also one of the criteria for maximal en-tanglement [7, 19], i.e., a measurement on any one of thequdits destroys the entanglement between the remainingones. This property is meaningful only for states with N ≥ N = 3 and d = 2, these are the states of subtype 2-0 using the classification in [8]. For an explicit example,the 4-qubit GHZ state | GHZ ± i = 1 √ | i ± | i ) (26)belongs to N N ( d ) while the 4-qubit Dicke state | D i = 1 √ | i + | i + | i (27)+ | i + | i + | i )and the 4-qubit W state | W i = 1 √ | i + | i + | i + | i ) (28)do not.As with the previous section, we construct the mixedstate ρ ( ~λ M , ~ Φ) = M X k =1 λ k | Φ k ih Φ k | (29)where ~λ M = { λ , . . . , λ M } with P Mk =1 λ k = 1, λ k > ~ Φ = {| Φ i , . . . , | Φ M i } with | Φ k i ∈ N N ( d ), for somepositive integer M ≥
2. We refer to the collection ofthese states as ˜ N M,N ( d ). We then show that the followingtheorem holds: Theorem 3.
Suppose that ρ ∈ ˜ N M,N ( d ) . If tr ( ρ i ) < tr ρ for all i , then ρ is genuinely entangled. Proof.
To prove the theorem above, we show that allpossible bipartitions of the state ρ remains entangled.In particular, we show that ρ j is entangled with ρ ( j ) , ρ ij is entangled with ρ ( ij ) , ρ ijk is entangled with ρ ( ijk ) and so on, where ρ ( i ) = tr H i ρ which means that the i th subsystem has been traced out. To show this, notethat ρ ( j ) = tr H j ρ = P k λ k tr H j | Φ k ih Φ k | . However, ac-cording to Eq. (25), tr H j | Φ k ih Φ k | is a separable state.Then tr ( ρ ( j ) ) ≤ tr ( ρ i ) via Proposition 1 whichimplies that tr ( ρ ( j ) ) < tr ρ . By Proposition 2,this means that ρ ( j ) is entangled with ρ j . Similarly, ρ ( jk ) = P k ′ λ k ′ tr H / H j , H k | Φ k ′ ih Φ k ′ | . Again, separabilityof tr H / H j , H k | Φ k ih Φ k | implies that tr ( ρ ( jk ) ) ≤ tr ( ρ i ) < tr ρ , which means that the subsystem ρ ( jk ) is entangledwith ρ jk . We can do the same procedure to show that allthe rest of the possible bipartitions (3 and N −
3, 4 and N − N − ρ is genuine.Motivated by the above theorem, we define b N M,N ( d ) asthe class of states that belong to ˜ N M,N ( d ) as well as hav-ing the property that tr ( ρ i ) < tr ρ for all i . Thus, if wewant the class of entangled mixed states constructed inSection 2 via Theorems 1 and 2 to have genuine entangle-ment, then we restrict ourselves to the class of entangledmixed states belonging to b N M,N ( d ), i.e., the mixed state ρ ( ~λ M , ~ Φ) = P Mk =1 λ k | Φ k ih Φ k | such that tr ( ρ i ) < tr ρ for all i and | Φ k i ∈ N N ( d ). In other words, | Φ k i shouldsatisfy both Eqs. (14) and (25).As explicit examples, consider the mixture ρ A = 34 | GHZ +4 ih GHZ +4 | + 14 | GHZ − ih GHZ − | . (30)Here, tr ( ρ A ) = 0 .
625 and tr ( ρ Ai ) = 0 . i . Hence,tr ( ρ i ) < tr ρ for all i . Also, | GHZ ± i ∈ N N ( d ), hence, ρ A is entangled according to Theorem 2 and its entan-glement is genuine. On the other hand, the mixture ρ B = 34 | GHZ +4 ih GHZ +4 | + 14 | D ih D | (31)also has tr ( ρ B ) = 0 .
625 and tr ( ρ Bi ) = 0 . i . However, | D i ∈ M N ( d ) but | D i / ∈ N N ( d ), hence, ρ B is entangled according to Theorem 2 but its entan-glement may not be genuine. We emphasize that weare not claiming that the entanglement is not genuine.Rather, we are saying that the criterion is not enough tojudge whether the entanglement it detects for states notin b N N ( d ) are genuine or not.In general, Dicke states of the form | D NN/ i , likeEq. (27) for N = 4 belong to M N ( d ) / N N ( d ). To showthis, recall that for N -qubits with a parameter m between1 to n −
1, a Dicke state is generally written as | D Nm i = (cid:18) Nm (cid:19) − X |{ α }| = m | d { α } i (32) where the sum runs over all { α } ⊂ { , , . . . , N } whichare sets of m different integers between 1 and N and | d { α } i is the product state with | i in all subsystems andwhose numbers are contained in { α } and | i otherwise.Then tracing out all the subsystems except one gives usa state ρ i = 1Ω (cid:18)(cid:18) N − m (cid:19) | ih | + (cid:18) N − m − (cid:19) | ih | (cid:19) (33)where Ω = (cid:18) N − m (cid:19) + (cid:18) N − m − (cid:19) . For Eq. (33) tobe maximally mixed, we want (cid:18) N − m (cid:19) = (cid:18) N − m − (cid:19) .This implies that m = N . Now, for Dicke states ρ = | D NN/ ih D NN/ | , tracing out all but two of the sub-systems leaves a state ρ ij which has a concurrence of C ( ρ ij ) = N − . Hence, Dicke states satisfy the propertyin Eq. (14) but not Eq. (25), in other words, | D NN/ i ∈M N ( d ) but | D NN/ i / ∈ N N ( d ).To summarize this section, if we are only interested inconstructing entangled states without regard to whetheror not the entanglement is genuine, then we can dropthe property in Eq. (25), giving us a larger set of purestates that we can mix from. However, if we want toensure that the constructed mixed states contain genuineentanglement, then we do so at the expense of making theset of pure states we can mix from smaller, and this isdone by adding the property in Eq. (25). V. CONSISTENCY WITH CONCURRENCE
In this section, we will discuss the consistency of ourtheorems in the previous section with the measure forquantifying bipartite entanglement known as the concur-rence. In particular, we will construct states that areentangled according to our theorems and see if it is entan-gled under concurrence as well. We limit our comparisonsto the case of bipartite qubits ( d = 2 and N = 2) for tworeasons. First, for the two-qubit case, the concurrenceprovides a computable formula for the entanglement offormation, a measure based on convex roof. Second, westill don’t have a consensus as to what is the correct en-tanglement measure for states with arbitrary d and N .In fact, many multipartite entanglement measure existsthat define what it means to be maximally entangled indifferent ways [15–23].Theorem 1 is trivially satisfied by d = 2 since it wouldrequire us to mix M = 1 states out of the Bell states.In general, a mixed state comprised of the Bell states isgiven by ρ = α | ψ + ih ψ + | + β | ψ − ih ψ − | + γ | φ + ih φ + | + (1 − α − β − γ ) | φ − ih φ − | (34)where α + β + γ <
1. Also, without loss of generality,we can assume that α > β > γ . Note that the state has d < M and so it does not fall under the condition requiredby Theorem 1. Are states not falling under Theorem1 necessarily separable? We can readily calculate theconcurrence of Eq. (34) giving us C ( ρ ) = 1. Thus, we seethat there are entangled states which are not part of theclass of entangled states given by Theorem 1 (i.e., thosethat have d > M ) and these entangled states which arenot detected by Theorem 1 automatically belong to thatof type II. Note that if we let α = β = γ = , then wefind that C ( ρ ) = 0.Now, let us investigate Theorem 2 for bipartite qubits.Here, Theorem 2 requires that d = M = 2 and so ingeneral, the mixed state will be given by ρ = λ | φ ih φ | + (1 − λ ) | φ ih φ | (35)where | φ i and | φ i could be any of the distinct Bellstates. Calculation of the concurrence of Eq. (35) yields C ( ρ ) = | λ − | which is equal to zero only for λ = . Inother words, mixing d number of entangled d -dimensionalpure states that are not equally weighted automaticallyyields an entangled mixed state and this state belongs tothat of type I. We conjecture that this holds for a general N -partite qudit systems although we can only check it forthe bipartite qubit case since a generalized measure foran N -partite qudit mixed state (one that does not involveoptimization) is not yet available. VI. CONCLUSION
We have considered mixtures of entangled pure stateswith maximally mixed one-qudit reduced density matri- ces and studied the conditions were they remain entan-gled using purity as our separability criterion. We havefound that in order for the resulting mixed state to re-main entangled, then the number of pure states withmaximally mixed one-qudit reduced density matrices tobe mixed must be less than or equal to its dimension.For the d < M case, we found that there are entangledstates which are not detected although these entangledstates belong to type II which are “undesirable” in termsof their utility for the reduction of communication com-plexity or quantum cryptography. For the d = M case,we found that the resulting mixed states are entangledprovided that the entangled pure states with maximallymixed one-qudit reduced density matrices to be mixedare not equally weighted. We have shown that it is con-sistent with what is predicted by the concurrence for thecase of bipartite qubits. However, such comparison can’tbe made for the general case of mixed N -partite quditsdue to a lack of computable measure similar to concur-rence. Also, we’ve shown that we can obtain genuinelyentangled mixed states at the expense of restricting theset of entangled pure states that we can mix from.However, there still exist open problems for future re-search that will be considered elsewhere like the physi-cal significance of the relationship between d and M aswell as an understanding of the d > M case where thecriterion fails. Also, it will be fruitful to look at much“stronger” separability criteria where the entropic-basedones fail. Another interesting case to consider would bethe mixing of pure states of different dimensions and itseffect on the entanglement of the resulting mixed state. [1] O. Guhne, G. Toth, Phys. Rep. , 1 (2009).[2] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, Rev. Mod. Phys. , 865 (2007).[3] C. Eltschka, J. Siewert, J. Phys. A: Math. Theor. ,424005 (2014).[4] A. Uhlmann, Entropy , 1799 (2010).[5] S. Popescu, Phys. Rev. Lett. , 6 (1994).[6] M. Horodecki, J. Oppenheim, and A. Winter, Nature, , 673 (2005).[7] A. Higuchi, A. Sudbery, Phys. Lett. A , 213 (2000).[8] C. Sabin, G. Garcia-Alcaine, Eur. Phys. J. D, , 435(2008).[9] A. Peres, Phys. Rev. Lett. , 8 (1996).[10] P. Horodecki, R. Horodecki, Quant. Inf. and Comp. , 1(2001).[11] P. Horodecki, M. Horodecki, and R. Horodecki, Phys.Rev. Lett. , 5 (1999).[12] R. Horododecki, P. Horodecki, and M. Horodecki, Phys.Lett. A, , 377 (1996).[13] R. Horodecki, M. Horodecki, Phys. Rev. A, , 3 (1996).[14] W. Wootters, Phys. Rev. Lett. , 2245, eprint quant-ph/97099029, (1998).[15] V. Vedral, Nature , 07124 (2008).[16] T. Wei, K. Nemoto, P. M. Goldbart, P. G. Kwiat, W.J. Munro and F. Verstraete, Phys. Rev. A, , 022110(2003).[17] W. Dr, G. Vidal and J. I. Cirac, Phys. Rev. A , 062314(2000).[18] P. Love, A. Brink, A. Smirnov, M. Amin, M. Grajcar, E.Ilichev, A. Izmalkov and A. Zagoskin, Quant. Inf. Pro-cess., , 187 (2007).[19] N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A, , 1 (1998).[20] I. D. K. Brown, S. Stepney, A. Sudbery and S. L. Braun-stein, J. Phys. A: Math. Gen. , 1119 (2005).[21] A. Osterloh and J. Siewert, New J. Phys. , 075025(2010).[22] G. Gour and N. R. Wallach, J. Math. Phys. , 112201(2010).[23] W. Helwig, W. Cui, J. I. Latorre, A. Riera and H. Lo,Phys. Rev. A86