Computable entanglement conversion witness that is better than the negativity

The primary goal of entanglement theory is to determine convertibility conditions for two quantum states. Up until now, this has always been done with the use of entanglement monotones. With the exception of the negativity, such quantities tend to be rather uncomputable. We instead promote the idea of conversion witnesses in this paper. A conversion witness is a function on pairs of states and whose value determines whether a state can be converted into another. We construct a conversion witness that can be efficiently computed for arbitrary states in systems of any size. This conversion witness is always better than the negativity at detecting when two entangled states are not interconvertible. Furthermore, when considering states of two-qubit systems, this new conversion witness is sometimes better than the entanglement of formation. This shows that the study of conversion witness is in fact useful, and may have applications in resource theories beyond that of entanglement.


I. INTRODUCTION
Entanglement is a necessary ingredient for many quantum information processing tasks, including the teleportation a quantum states [1], superdense coding [2], and numerous uses in quantum cryptography protocols [3]. Two principal features of entanglement are that it cannot be created among distant parties when there is none to begin with, and that it is depleted in the implementation of such protocols. Not all entanglement is created equal. Some entangled states may be more useful for certain applications than other entangled states. It is therefore of great interest to develop a detailed understanding of the properties of entanglement in terms of its nature as a resource [4].
In the paradigmatic setting for the study of entanglement, distant parties jointly share a state of a composite quantum system. Procedures that can be performed in such a setting are limited to those that can be implemented through local operations (LO) on the subsystems and exchange of classical communication (CC) between the parties. The primary goal of entanglement theory is to fully understand the structure of the entangled states that is induced by this restriction to LOCC operations. Given two resource states ρ and σ of a joint quantum system, the fundamental question that we want to answer is the following: Can we obtain σ from ρ using only LOCC? The possible transformations of resources establishes a partial order on the set of all possible states. Fully characterizing the structure of this partial order is of chief interest, since it will allow us to determine which states will be useful for a given task.
Entanglement is typically characterized via entanglement monotones. These are functions that quantify the resourcefulness of a state, and whose value may not increase through LOCC transformations. However, it is known to be a difficult problem to determine if a given state is entangled or not [5], so most monotones are unfor- * mwgirard (at) ucalgary.ca tunately difficult to compute in general [6]. Hence, finding entanglement monotones that can be efficiently computed, yet still yield useful information about the structure of entanglement, is an essential part of the study of entanglement theory.
The best (and perhaps the only) known such monotone is the negativity [7]. But individual monotones can only provide a limited amount of information about which state transformations are possible. When monotones fail to discern whether a particular transformation may be achieved, we must resort to other methods to help elucidate the partial order structure of entangled states.
The main motivation for this paper is to illustrate the importance of the concept of conversion witnesses (first introduced in [8]). These are functions that can detect whether or not a particular conversion is possible, without resorting to the computation of entanglement through monotones. Witnesses are not perfect, however, and do not in general detect every possible transformation. A single witness typically provides either a necessary or a sufficient condition for a conversion of resources to be possible, but not both. As with monotones, a witness is only useful in practice if it can be computed efficiently for any two states in consideration. In fact, entanglement monotones comprise a special case of a conversion witness, since a state ρ cannot be converted into σ if ρ is less entangled than σ with respect to any monotone.
To demonstrate that witnesses truly can be more useful than monotones, we construct a computable conversion witness for the theory of bipartite entanglement that is better than the negativity at detecting the existence of state transformations. To show that this witness does indeed improve upon the negativity, we present examples of pairs of states that have the same negativity, yet our witness detects that one cannot be converted into the other. The fact that they are interconvertible cannot be inferred from the negativity alone, since the states have the same level of resourcefulness with respect to the negativity. In particular, we show that an entangled pure state ρ of two qubits with negativity N (ρ) < 1 3 cannot be converted to any entanged Werner state σ with the same negativity as ρ.
The rest of this article is structured as follows. Section II begins with a review entanglement monotones to set the stage for the introduction of conversion witnesses, followed by a brief summary of positive operators and positivity under partial transposition. The notion of conversion witnesses is introduced and their rich structure is examined in section III. Our demonstrative example of a computable entanglement conversion witness is constructed in section IV. Construction of this witness, which is based on the negativity, is followed by a proof that this witness is indeed an improvement over the negativity. Analysis of this new witness is concluded with a few remarks about how further witness might be constructed. Section V concludes with a discussion on how conversion witnesses should be an important part in the study of all resource theories beyond that of entanglement.

II. PRELIMINARIES
The definition of an entanglement monotone is presented in such a way to introduce the idea of entanglement conversion witnesses, which will be defined in following section. The importance of studying operations that are positive under partial transposition (PPT) for entanglement theory is discussed, and a few important facts of positive operators are reviewed.

A. Entanglement monotones and conversion witnesses
One of the main goals of entanglement theory is to understand the structure that is induced by the restriction to LOCC operations. Essentially, given any two states ρ and σ of a bipartite system, we want to be able to answer the question: can ρ be converted to σ via LOCC operations? If such a transition is possible, this is denoted as ρ → σ. Arbitrary compositions of LOCC operations are again LOCC operations. That is, if ρ → σ and σ → τ , then also ρ → τ . Moreover, ρ → ρ for any state ρ by simply doing nothing. This induces a partial order on the set of states.
Quantifying the entanglement in states is the standard method for characterizing this partial order structure in entanglement theory. If it is possible to convert a state ρ into another state σ, then ρ is at least as useful for any task that requires σ. Hence ρ must be at least as entangled as σ under any measure of entanglement. Finding useful entanglement measures, or entanglement monotones, is important for describing which state transformations may or may not be possible under LOCC.
Definition 1 (Entanglement monotones). An entanglement monotone is a real-valued function f on quantum states of bipartite systems that does not increase under LOCC operations. That is, the function f is a monotone if f (E(ρ)) ≤ f (ρ) for all states ρ and LOCC operations E ∈ C.
Equivalently, a monotone is a function such that ρ → σ implies f (ρ) ≥ f (σ). However, a single monotone does not typically supply enough information to determine if any pair of quantum states is convertible. Indeed, even if ρ is more entangled than σ under some monotone, it may be the case that ρ cannot be converted into σ.
A family of monotones f i indexed by i ∈ I is said to be complete if ρ → σ if and only if f i (ρ) ≥ f i (σ) for all i ∈ I. A complete family of monotones can always be trivially defined. For each state τ , define the function Although its value cannot be straightforwardly computed, f τ is indeed a monotone, and the family (f τ ) τ naturally comprises a complete family of monotones. Even though a complete family of monotones exists, this family may not necessarily be useful. Given states ρ and σ, one would need to be able to actually compute the values f (ρ) and f (σ) for the monotone f to be practical. When considering bipartite entanglement of only pure states, such a complete family of computable monotones is well known [9,10]. However, a complete family of finitely many computable monotones does not exist for determining convertibility of arbitrary mixed states [11]. Hence, other methods of easily determining whether a particular state transformation is possible are desired.

B. LOCC and PPT operations
Although LOCC emerges as the natural class of operations for most tasks in quantum information, its mathematical structure is highly complex and difficult to characterize [12] (for a precise definition of the LOCC class see [13,sec. XI]). Equally troublesome is the task of characterizing the separable states. In fact, the separability problem in arbitrary dimensions of the subsystems is known to be NP-Hard [5]. It is therefore desirable to consider other classes of operations that still provide interesting insights regarding entanglement.
Perhaps one of the most elegant results in the early days of quantum information was the characterization of entangled states through partial transposition [14,15]. If a state ρ of a bipartite system has ρ Γ ≥ 0, then ρ must be entangled, where Γ indicates the partial transpose of ρ with respect to one of the subsystems. The set of states that are positive under partial transposition (PPT) include the separable ones. Moreover, all PPT entangled states are bound entangled. That is, they are 'useless' for generating entanglement [16] within the framework of LOCC. Hence, the set of PPT states is not only easy to characterize but also very useful in the study of entanglement.
In addition to the class of LOCC operations, we also consider the class C Γ of operations that are positive under partial transposition (PPT) [17,18]. This is the set of all completely positive trace-preserving maps E such that the partially transposed map E Γ , defined by is also completely positive. It is well known that all LOCC operations (and all separable operations [19]) form a subset of the PPT operations [17,18,20]. Hence, any function of states that is a no-go conversion witness for convertibility under PPT operations is also a no-go entanglement conversion witness. With this fact, in section IV we construct a computable no-go conversion witness based on the structure of PPT maps.

C. Useful properties of positive operators
We recall a few useful facts about positive operators. Denote by H n the set of n × n hermitian matrices. Let H n,+ denote the cone of positive semi-definite hermitian matrices, and furthermore let H n,+,1 denote the subset of those with unit trace. Hence H n,+,1 is equivalent to the set of states of a n-dimensional quantum system. Every operator A ∈ H n has a unique decomposition into its positive and negative parts, where A + , A − ∈ H n,+ . Furthermore, for any two positive semi-definite operators A, B ∈ H n,+ , we have the following operator inequalities: (see Appendix A for proof). We also recall that the 1norm of an hermitian operator A 1 = Tr |A| is the sum of the absolute values of its eigenvalues, where the oper- This useful characterization will be exploited in the construction of our conversion witnesses.

III. CONVERSION WITNESSES
In this section, we introduce the concept of entanglement conversion witnesses, a generalization of monotones. Such computable witnesses can be more effective than monotones at determining the convertibility of quantum states. The rich structure of these conversion witnesses is also explored.
In the previous section, it was noted that the extent of the usefulness of entanglement monotones in comparing quantum states is inherently limited, and thus other methods must be found. The most general technique for characterizing the convertibility of states is through conversion witnesses 1 (see [8, sec. II.A.]).

Definition 2 (Entanglement conversion witnesses).
Let W be a real-valued function on pairs of quantum states. If W (ρ, σ) ≥ 0 implies that ρ → σ under LOCC, then W is said to be a go witness. If W (ρ, σ) < 0 implies that ρ → σ, then W is said to be a no-go witness. Finally, W is said to be a complete witness if it is both a go and a no-go witness.
Given a monotone f , we can define a no-go witness by , and thus ρ → σ by the monotonicity of f . So entanglement monotones can be considered as a special case of entanglement conversion witnesses.
The set of all no-go witnesses is endowed with the structure of a partially ordered set. Indeed, given two no-go witnesses W 1 and W 2 , we say that That is, W 1 W 2 means that the witness W 1 tells us more information about the convertibility of states than W 2 does. If W 2 detects the inconvertibility ρ → σ for some states ρ and σ, this same information can already be obtained by W 1 . But W 1 might be able to detect the inconvertibility of other pairs of states that W 2 cannot.
The partial order structure of no-go witnesses is illuminated in the following example: Given a family of no-go witnesses (W i ) i∈I , we can construct a new no-go witness W I by minimizing over all witnesses in the family This is indeed a witness, since W I (ρ, σ) < 0 implies that W i (ρ, σ) < 0 for at least one i ∈ I and thus ρ → σ. Hence W I W i and the resulting witness W I is an improvement over each of the sub-witnesses W i . Similarly, given a family (f i ) i∈I of monotones, one can define a witness If the family (f i ) i∈I is complete, then the resulting W I is a complete witness. Furthermore, if W is a complete witness, then W W for any no-go witness W .
An example hierarchy of no-go conversion witnesses is depicted in Fig. 1. Note that two no-go witnesses W 1 and W 2 may be incomparable in general. That is, it may be that both W 1 W 2 and W 2 W 1 .
An analogous partial order exists for go witnesses. If W 1 and W 2 are two go witnesses and W 1 (ρ, σ) ≥ 0 implies that W 2 (ρ, σ) ≥ 0 for all states ρ and σ, then we say that W 1 W 2 . Given a family of go witnesses (W i ) i∈I , a new go witness can be constructed such that W I W i for each subwitness. Additionally, we have W W for any complete witness W and any go witness W .

IV. NEW ENTANGLEMENT CONVERSION WITNESSES
In this section, we construct a computable example of a no-go conversion witness for entangled states that is based on the construction of the negativity. We first recall a few useful properties of positive operators. The proof that the negativity is a monotone under PPT operations is then reproduced, since it is needed to construct our new conversion witness. After establishing our new no-go witness, we subsequently show that this witness is, in fact, better than the negativity at detecting convertibility under LOCC. Although the conversion witness is not computable as it is initially introduced, we consider simplified versions of the witness that are computable and still supply valuable information about conversion that is not available from the negativity alone. Finally, we compare these witnesses with the negativity using the partial order structure introduced in the previous section.

A. Negativity is a PPT monotone
We recall that the negativity [7,22,23] of a bipartite quantum state is defined as which is the sum of the negative eigenvalues of the partial transpose of a normalized state. The negativity is known to be a monotone under LOCC operations and hence comprises an entanglement measure that can be computed effectively. A more convenient definition for the negativity that we employ here is which does not depend on the normalization of ρ, and where ρ Γ− = (ρ Γ ) − represents the negative part of the partial transpose of the operator ρ.
Following the proof in [4], we now show that the negativity defined in (3) is a monotone under PPT operations. Let E ∈ C Γ be a PPT operation and ρ be a state of a bi- The inequality in the last line follows from (1), since E Γ is a positive map and the operators ρ Γ+ = (ρ Γ ) + and ρ Γ− are both positive. Hence, we have the operator inequality for all states ρ and PPT operations E. Since E and E Γ are trace-preserving, taking the trace of the right side of (4) yields , which is just the negativity of the state ρ. Taking the trace of both sides of (4) yields the inequality i.e. the negativity is in fact a monotone under PPT operations. Recall that the LOCC operations are a subset of PPT ones, so the negativity is a monotone under LOCC as well.
Since the negativity is a monotone under PPT operations, the convertibility condition ρ → σ implies N (ρ) ≥ N (σ) for any states ρ and σ of a bipartite system.

B. A no-go witness for PPT conversion
We are now ready to construct no-go entanglement conversion witnesses that are based on the construction of the negativity discussed in the previous subsection.
Let ρ and σ be arbitrary quantum states. Suppose that we want to determine if ρ → σ under PPT operations. If σ can be obtained from ρ via PPT operations, then σ = E(ρ), and thus for some PPT operation E ∈ C Γ . On the other hand, if σ violates this operator inequality for all PPT operations, then we have the condition that and so σ = E(ρ) for all E ∈ C Γ . It follows that ρ → σ under PPT operations.
Taking the trace of both sides of (7), this 'witness' simply reduces to the statement that N (σ) ≤ N (ρ) implies ρ → σ, which already follows from the fact that the negativity is a PPT monotone. However, the operator inequalities of (7) contain much more information that is 'lost' by taking the trace and reducing it to the one-dimensional inequality N (σ) ≤ N (ρ). The idea that more useful information may be extracted from this operator inequality in (7) implies that a family of no-go witnesses might be constructed from it.
Before introducing such no-go witnesses explicitly, we present a few more useful concepts. The support function of a subset C ⊂ H n is defined as [24] h C (ρ) := sup γ∈C Tr[γρ] (8) for any ρ ∈ H n . If C is also compact, then sup in (8) may be replaced with max. We now define some sets of states that we use in the following analysis. For every c ≥ 0, define the set which is the set of normalized states with negativity at most c. For c = 0, the set N 0 is just the set of PPT states. Given a class of CPTP operations C, we can consider the 'orbit' of states that are reachable from a given state ρ via operations from C. This is denoted as Consider the orbit of a state ρ under PPT operations. All states in the orbit C Γ (ρ) must have negativity at most N (ρ), since the negativity is a monotone under PPT operations C Γ . This implies the containments for all normalized states ρ. Hence, the respective support functions of these sets obey the inequality whenever ρ is a normalized state.
We now return to the task of constructing no-go witnesses for PPT conversion. For some states ρ and σ, suppose that ρ → σ. Then σ = E(ρ) and hence the operator inequality in (6) holds for some E ∈ C Γ . Using the property of positive operators from (2), the fact that the operator inequality σ Γ− ≤ E Γ (ρ Γ− ) holds for some E ∈ C Γ is equivalent to the statement that the family of inequalities holds for all τ ∈ H n,+,1 and some PPT operation E ∈ C Γ . Note that the partial transpose operator E Γ of a PPT operation is again another PPT operation, hence for all τ ∈ H n,+,1 , where ρ Γ− is a non-normalized state. Therefore, if the conversion ρ → σ is possible under PPT operations, the inequality must hold for all τ . It follows that, for each τ ∈ H n,+,1 , the function is a valid no-go conversion witness for PPT operations. That is, if W τ (ρ, σ) is negative for some τ , then it must be the case that ρ → σ.
If ρ itself is a PPT state, then ρ Γ ≥ 0 and thus ρ Γ− = 0 so the negativity of ρ vanishes. Hence σ can be obtained from ρ only if σ is also a PPT state with vanishing negativity. For any interesting applications of these conversion witnesses, we may assume that the initial state ρ is entangled with non-vanishing negativity and thus ρ Γ− = 0. For each non-PPT state ρ we can renormalize the operator ρ Γ− to define a normalized statẽ where we use the fact that N (ρ) = Tr[ρ Γ− ]. The inequality in (11) is then equivalent to Employing of the support function for the orbit ofρ under PPT operations, we have Tr[τ E Γ (ρ)] ≤ h C Γ (ρ) (τ ) and thus for each τ ∈ H n,+,1 if the conversion ρ → σ is possible. This yields a no-go witness for each τ , since Note that if we chose τ = 1 n I to be the maximally mixed state, evaluating the support function simplifies to since each E ∈ C Γ is trace preserving and Trρ = 1. So the witness (14) simplifies to which is just the difference of negativities of the two states. Thus, the family of witnesses W τ yields at least as much information regarding the convertibility as the negativity does. In fact, minimizing W τ over all possible τ ∈ H n,+,1 yields the witness In the syntax of the partial order of no-go witnesses, we have that W W τ for each τ , and W W N , where W N is the witness formed from the negativity. This means that the witness W may be able to yield more information about the convertibility of arbitrary states under PPT operations than the best-known monotone, the negativity. However, the support function h C Γ (ρ) (τ ) is very difficult to calculate in general, since it involves an optimization over all PPT operations. So this witness is not very computable in practice. In the following, we construct a more useful generalization of this witness.
Replacing the orbit C Γ (ρ) in (13) with the set N N (ρ) (i.e. the set of states whose negativity is bounded by N (ρ)) yields the inequality This inequality must hold for all τ if ρ → σ. Hence, we obtain a family of witnesses defined by for each τ ∈ H n,+,1 . If W τ (ρ, σ) < 0 for any τ , then ρ → σ. Note that W τ (ρ, σ) ≤ W τ (ρ, σ), so the witnesses W τ supply less information about convertibility than the witnesses W τ do. However, choosing τ = 1 n I, we see that so the new witnesses W τ still yields at least as much information as the difference of negativities. As before, minimizing over all τ ∈ H n,+,1 yields the witness such that W (ρ, σ) ≤ N (ρ)−N (σ). In particular, we have W W τ for each τ . Furthermore, the hierarchy of no-go witnesses W W W N holds. This witness is still not very useful in practice, since h Nc (τ ) is difficult to compute for arbitrary τ . In the following section, we show how to compute h Nc (τ ) for certain highly symmetric states in order to construct computable versions of the witnesses W τ and W .

C. A computable no-go witness
Although the support function h Nc (τ ) cannot be determined in general for arbitrary τ , it can be evaluated explicitly for certain operators τ that exhibit high degrees of symmetry, such as the Werner states and isotropic states [23,25,26]. Rather than performing the minimization in (17) over all states τ , we can instead minimize over classes of states for which h Nc (τ ) is computable.
Recall from (15) that h Nc (τ ) is computable for τ = 1 n I, and that W 1 Thus, restricting the minimization in (17) to only be over states where h Nc is computable will still yield a witness that is at least as good as the difference of negativities. In this section, we perform such a minimzation over a small class of states to construct a computable example of a no-go witness.
We restrict to states of a bipartite d × d-system, where H n,+,1 comprises the states of the system and n = d 2 .
The Werner states are those that are invariant under all unitaries of the form U ⊗ U , where U is any unitary on the d-dimensional subsystems. Furthermore, the Werner states are invariant under application of the twirling operation of the form where dU denotes the standard Haar measure on the group of all d × d unitary matrices. Not only do we have T U ⊗U (ρ) = ρ for any Werner state ρ, but applying T U ⊗U to any state always yields an Werner state. The Werner states on a d × d-system form a one-dimensional family that may be conveniently parametrized by where F is the so-called 'flip' operator on a d × d-system such that F |ψ ⊗ |φ = |φ ⊗ |ψ for all product vectors. Explicitly, the flip operator is defined by |ij ji|.
Note that this can also be given by F = d|Φ Φ| Γ , where |Φ is the maximally entangled state of a d × d-system, Another family of states with a high degree of symmetry that is typically studied in bipartite entanglement consists of the isotropic states. Similar to the Werner states, the isotropic states are invariant under all unitaries of the form U ⊗Ū , where theŪ denotes the complex conjugate of U . The isotropic states are invariant under the application of a twirling operation of the form such that T U ⊗Ū (ρ) = ρ for any isotropic state ρ. Applying T U ⊗Ū to any state always yields an isotropic state. We can conveniently parameterize the isotropic states of a d × d-system as Making use of this high degree of symmetry, the support function h Nc (τ ) can be explicitly evaluated for these families of states. Indeed, when the state τ = ω d p is Werner, it suffices to maximize only over the Werner states, rather than maximizing over all states, with negativity at most c. Analogously, when τ = η d p is isotropic, it suffices to maximize only over the isotropic states. Explicit calculations to evaluate the support function h Nc on Werner and isotropic states are given Appendix B.
Using these calculations, computable witnesses can be defined by optimizing over the isotropic and Werner states. These are Explicit calculations can again be found in Appendix B 1, but closed-form results for this witnesses can be given as and where W iso and W wer are sub-witness given by and Here F − is the negative part of the flip operator. From the expressions for the witnesses in (20) and (21), it is clear that W iso (ρ, σ) ≤ W N (ρ, σ) for all ρ and σ, and thus W iso W N . Similarly, we have W wer W N . So both new witnesses do in fact supply at least as much information about convertibility of states as the negativity does.
However, in the next subsection, we show that all of the witnesses W N , W wer and W iso are incomparable with respect to the partial order on witnesses. Minimizing over all three of these witnesses yields a computable witness that is certainly an improvement over the negativity.
Interestingly, the witnesses in (22) and (23) do not depend on the explicit form of ρ, but only on the negativities N (ρ) and N (ρ). Furthermore, note that Φ| σ Γ− |Φ ≤ Tr[σ Γ− ] = N (σ), and thus Therefore, this new witness W iso can only supply new information about whether or not so the new witness W wer can only supply new information about whether or not ρ → σ if N (ρ) < 1 d .

D. Incomparability
Our new no-go conversion witnesses W wer and W iso in (20) and (21) give us at least as much information as the negativity, but it is not obvious that they are actually an improvement. In this section, we present examples of pairs of states (ρ and σ) to illustrate that W N W iso and W N W wer . In addition, we also show that neither of the new witnesses W wer and W iso are better than the other. Proposition 1. The witnesses W N , W wer and W iso are all incomparable with respect to the partial order. That is, Hence the no-go witness W Γ obtained in (24) from minimizing over all three of the above witnesses truly does give more information about the conversion of states under PPT than the negativity. This proposition is proved by finding pairs of states ρ and σ where one witness detects the inconvertibility ρ → σ while the other two do not, Hence none of these three witnesses is greater than any other with respect to the partial order.

Proof that W iso is not less than WN or W wer
We first show that W N W iso and W wer W iso . That is, there are states ρ and σ such that W iso (ρ, σ) < 0, but W N (ρ, σ) ≥ 0 and W wer (ρ, σ) ≥ 0. In a bipartite d × dsystem, such an example can only exist if d ≥ 3.
Let d ≥ 3 and consider σ to be an entangled Werner state of a d × d-system as given in (18). The negativity of the Werner states is given by Note that the Werner state ω d p is entangled if and only if N (ω d p ) > 0. In this case, we have (ω d p ) Γ− = N (ω d p )|Φ Φ| and note that the most entangled Werner state has negativity 1 d with p = 1 1−d . Let ρ = |x x| be a pure entangled state of two qubits, which we may consider to be in Schmidt form where λ 0 ≥ λ 1 are the Schmidt coefficients such that λ 2 0 + λ 2 1 = 1. This state has negativity N (|x x|) = λ 0 λ 1 . Since |x is entangled, we must have λ 0 λ 1 = 0. To find ρ, note that |x For ρ = |x x|, this yieldsρ = 1 N (ρ) ρ Γ− = |ψ − 01 ψ − 01 | and thus N (ρ) = 1 2 . Hence, with states ρ = |x x| and σ = ω d p , the witness in (23) evaluates to In particular, to show that this witness really does provide more information than the negativity, we can choose such that N (|x x|) = 1 d . Furthermore, we choose σ = ω d p such that N (σ) = 1 d . Thus, ρ and σ have the same negativity, i.e W N (ρ, σ) ≥ 0, so we cannot determine from the negativity alone if ρ → σ. However, the no-go witness W iso for these states evaluates to and we conclude that ρ → σ and W N W iso . Furthermore, note that Tr[|Φ Φ|F − ] = 0 and thus W wer (ρ, σ) ≥ 0 for the above choice of ρ and σ. This shows that W wer W iso .

Proof that Wwer is not less than WN or Wiso
Let ρ = |y y| be a pure state of two qutrits, where |y may be given in Schmidt form as

10
and λ 1 = λ 2 = √ 5 10 . The negativity of the state ρ = |y y| is It is also straightforward to calculate the negativity of the normalized stateρ = 1 N (ρ) ρ Γ− , Let d = 2 and define the constant b as Note that, in a 2 × 2-dimensional system, the negative part of the flip operator is just F − = |ψ − 01 ψ − 01 |. Choose σ = η p to be an isotropic state of two qubits with negativity b < N (σ) ≤ N (ρ), where b is defined in (25), so that the parameter is in the range 4b+1 and thus W N (ρ, σ) ≥ 0, so we cannot determine from the negativity alone if ρ can be converted to σ. However, and thus ρ → σ. So we conclude that W N W wer . Furthermore, note that Tr[|Φ Φ|F − ] = 0 and thus W iso (ρ, σ) ≥ 0 for the above choice of ρ and σ. This shows that W iso W wer .

E. Further PPT conversion witnesses
In this section, we show how further conversion witnesses might be constructed that are better than those presented in the preceding section.
While the witnesses constructed above are in fact computable and are indeed an improvement over the negativity, these quantities have the unfortunate property that they are not invariant under local transformations. That is, for local unitaries U and V , we don't necessarily have that In principle, since the application of local unitaries is an invertible operation in the class of LOCC (and PPT) operations, we should have for any unitaries U and V . That is, if ρ can be converted to σ, then it can also be converted into any state that is equivalent to σ up to local unitaries. Hence, if W is a no-go witness and there exist some U and V such that Since our witnesses constructed in the previous sections depend on a particular choice of bases relative to the two systems, this suggests that we might be able to build an improved witness by minimizing over all local unitaries (LU): where W is either W iso or W wer . This extension of these witnesses makes them no longer directly computable. Yet, in contrast to the computation of the support function h N (ρ) for arbitrary ρ, the minimiziation in (26) can be performed numerically, so perhaps not all is lost. In particular, the witness defined by might be useful for determining convertibility of quantum states within the context of entanglement theory. The witnesses W wer and W iso that we have constructed here are computable because the support function h Nc (τ ) is computable when τ is a Werner or isotropic state. This computability is due to the high degree of symmetry present in these families of states which comes from the fact that they are invariant under a particular group action. However, there are other group actions that one might be able to consider, through which other families of highly symmetric states could be constructed where the support function h Nc is also computable. Indeed, if G is a group with an action on states given by U (g)ρU † (g) for g ∈ G, we can make use of a twirling operation T G (ρ) = G U (g)ρU † (g) dg, where dg is the Haar measure of G. If this twirling operation is a PPT operation and τ is invariant under the action of G, then sup γ∈Nc Tr[γτ ] = Tr[γ τ ] will be maximized by state γ that is invariant under G. Hence, finding a group G with a suitable group action could lead to the construction of other computable conversion witnesses.
For example, the Werner states are invariant under all bipartite unitaries of the form U ⊗U , where U is any d×d unitary matrix. But the Werner states can be considered as a special class of states of the form (27) which is the family of states that are invariant under application of bipartite unitaries of the form U ⊗U where U is diagonal in the {|i } basis with We shall call the states in (27) the generalized Werner states. This family of states has been studied before in relation to the entanglement distillation problem [27].
Making use of the symmetry in these states, it is also possible to compute h Nc (τ ) whenever τ is a generalized Werner state. However, minimizing W τ (ρ, σ) over all of the generalized Werner states turns out to be no less then W wer , which is obtained from minimizing over just the Werner states, so this particular family of symmetric states does not yield any improved witness (see Appendix D for calculations). This indicates that the witness W wer that we have constructed above is in fact quite good, since its generalization is not any better. Finally, we show how one can construct refinements of our no-go witnesses that are not necessarily computable. Although they are not computable, they may yield insight into the problem of constructing other useful witnesses for PPT conversion. Note that key idea in constructing the PPT conversion witness in the preceeding sections was that a state σ cannot be obtained from ρ if σ ∈ C Γ (ρ), i.e. if σ is outside of the PPT-orbit of ρ. But each state that can be obtained from ρ must have negativity at most N (ρ). The essential ingredient in the construction of our computable no-go conversion witness was thus the approximation of the PPT-orbit by the set of states N c with negativity at most c = N (ρ), where we note that C Γ (ρ) ⊂ N N (ρ) . This observation, however, only yields the information that N (σ) > N (ρ) (and thus σ ∈ C Γ (ρ)) implies that σ can't be obtained from ρ.
Our witness makes a stronger statement than this. In particular, we have that is a witness for each τ . But the value of the support function h C Γ (ρ Γ − ) cannot be found in general, since minimizing over the PPT-orbit of ρ Γ− is difficult. Instead, we approximated C Γ (ρ Γ− ) with the larger set whose support function h N1(ρ) (τ ) is in fact computable for certain τ , in particular for Werner and isotropic states. Note that the set N 1 (ρ) is precisely the unnormalized version of the states in the set N N (ρ) in (9). Yet even this approximation can be further refined. For sets S that are refinements of N 1 (ρ), i.e. such that C Γ (ρ Γ− ) ⊂ S ⊂ N 1 (ρ), we can construct a witness for each state τ . The derivation of the refinements S are left to Appendix C, since the details are rather cumbersome and may not lead to an improved witness which is computable.

V. CONCLUSION
Although the concept of conversion witnesses in quantum resource theories was first introduced elsewhere, we have demonstrated the importance of studying conversion witnesses within the framework of entanglement theory. The central problem in entanglement theory is to determine conditions for when one state may be converted into another. Such conditions have, up until now, only been studied in terms of monotones. We have shown that conversion witnesses, which may be considered as a generalization of monotones, can be used to determine convertibility of quantum states when the best-known monotones fail to do so. Moreover, and more importantly, we constructed a computable no-go conversion witness that is an improvement over the negativity, the previously best-known computable entanglement monotone.
The main goal of this paper was to illustrate the importance of conversion witnesses in the context of entanglement theory, but entanglement is not the only concept in quantum information that can be considered as a resource. Other "resource theories" that are determined by sets of restricted operations other than LOCC can be considered. The task of determining what is possible given certain these allowable quantum operations is exactly the study of quantum resource theories (for a recent review, see [28]). Although our treatment of conversion witnesses was limited to entanglement, the concept of a computable resource conversion witness could (and should) be studied in all resource theories. Furthermore, beyond the quantum framework, resource theories can be studied as mathematical entities in their own right [29,30], and conversion witnesses may have applications in such abstract resource theories as well.
While we have only studied examples of conversion witnesses in the resource theory of entanglement, the search for conversion witnesses in other resource theories that yield improvements to the best-known monotones may be fruitful.

Appendix A: Proof of operator inequalities
We now prove the operator inequalities in (1). For positive semi-definite operators A, B ∈ H n,+ , we have Indeed, note that (A − B) + = P † (A − B)P where P is a projector onto the subspace spanned by the eigenvectors of A − B with corresponding positive eigenvalue. For any vector |ψ ∈ C n , denote |ψ = P |ψ . Since A and B are both positive operators, we have since P † AP ≤ A for any projection operator P and positive operator A. Thus (A − B) + ≤ A as desired. The second inequality in (1) follows from the first, since

Appendix B: Computation of support function hN c (τ ) and the witnesses Wτ for Werner and isotropic τ
In this section, we introduce the Werner states and isotropic states of a d × d-system, then calculate h Nc (τ ) when τ is Werner or isotropic. This is used to determine the form of the witnesses W τ when τ is Werner or isotropic, and new witnesses are defined by minimizing W τ over the Werner and isotropic states.

Werner states
Recall that a Werner state is a state of a d × d-system that is invariant under the action of all unitaries of the form U ⊗ U . That is, a state ρ is Werner if for all unitaries U acting on a d-dimensional Hilbert space. The Werner states of a d × d-system may be parametrized by a one-dimensional parameter in the following manner where F is the 'flip' operator defined in (B7) and I is the It is important to determine the negative part of the partial transpose of the Werner states. First note that we can write The negativity of the Werner states is given by and the negative part of the partial transpose of ω d p is Note that the maximum negativity of all d-dimensional Werner states is 1 d . We now make use of the high degree of symmetry of the Werner states to calculate the support function h Nc for Werner states. Recall that the Werner states are invariant under the twirling operator For all Werner states τ = ω d p we have T U ⊗U (τ ) = τ . Furthermore, the twirling operator in (B2) is self-adjoint with respect to the Hilbert-Schmidt inner product, i.e.

Tr [T U ⊗U (A)B] = Tr [A T U ⊗U (B)] for all hermitian operators A and B.
To calculate h Nc , suppose that the operator γ ∈ N c is optimal such that Tr[γ τ ] = h Nc (τ ) = max γ∈Nc Tr[γτ ]. Then for τ = ω d p we have Note that T U ⊗U (γ ) is also a Werner state and T U ⊗U is a PPT operation, so the negativity of T U ⊗U (γ ) must not be greater than c. Hence, to calculate h Nc (τ ) whenever τ = ω d p is a Werner state, it suffices to maximize over the Werner states with negativity bounded by c, and thus We are now ready to minimize the witnesses W τ over all Werner states τ = ω d p to define the new witness The value of the witness W ω d p is piecewise linear in p, so it suffices to check only the endpoints of the segments in which W ω d p is linear, i.e. p = − 1 d−1 , p = 0, and p = 1 d+1 . For any p ≥ 0, note that for any states ρ and σ. Performing the minimization in (B3), this reduces to where W Wer is the sub-witness defined by for the witness.

Isotropic states
Recall that the isotropic states [25] of a d×d-system are those that are invariant under the action of all unitaries of the form U ⊗Ū , whereŪ denotes the complex conjugate of U . Analogous to the Werner states in the preceding section, the isotropic states may be parametrized by a one-dimensional parameter in the following manner: where − 1 d 2 −1 ≤ p ≤ 1 and |Φ is the maximally entangled state of a d × d-system.
It is useful to determine the negative part of the partial transpose of isotropic states. The flip operator can be written as where |ψ ± ij = 1 √ 2 (|ij ± |ji ), and we can split F into its positive and negative components and that the negativity N (η d p )) = Tr[(η d p ) Γ− ] of the isotropic states can be given by As before, we make use of the high degree of symmetry to calculate the value of the support function h Nc for isotropic states. Recall that the isotropic states are invariant under the twirling operator For isotropic states τ = η d p we have T U ⊗Ū (τ ) = τ . Furthermore, the twirling operator in (B8) is also self-adjoint with respect to the Hilbert-Schmidt inner product.
Using the same arguments as for Werner states, it suffices to optimize only over the isotropic states when calculating the support the function h Nc for isotropic states. So the support function reduces to The isotropic states of a d × d-system have negativity of at most d−1 2 , so this calculation must be considered in two cases. If c ≥ d−1 2 , then the maximum is taken over , and the maximum is taken over ]. Since the function Tr[η d q η d p ] is linear in q, it suffices to check only the endpoints of the range of q. The support function for isotropic states evaluates to The calculations for minimizing the witnesses W τ over all isotropic states τ = η d p to define the new witness are analogous to those for the Werner states. Performing the minimization in (B9), this reduces to where W iso is the sub-witness defined by for the witness.
The key idea in this paper is that we can construct a no-go witness by determining if a state σ is outside of the PPT-orbit of the initial state ρ. However, the PPTorbit is difficult to characterize, so we resort to other more easily computable techniques. Namely, instead of considering the orbits C Γ (ρ), we consider the larger sets N N (ρ) of states that have negativity less N (ρ). In this appendix, we construct further sets of states that contain the orbit C Γ (ρ) but are smaller than N N (ρ) .

Appendix D: Generalized Werner states
In this section, we consider the generalized Werner states, first analyzed in the context of the distillability problem [27]. This is a symmetric class of states that contains the Werner states. The support function h Nc (τ ) can be computed when τ is a generalized Werner state, which we show here, so a conversion witness W Gwer can be constructed by minimizing over all W τ (ρ, σ) where τ is a generalized Werner state. However, in the following computation, we show that W wer W Gwer , so this new witness is not any better than W wer .
The generalized Werner states of a bipartite d × dsystem are defined in the following manner. Consider the subgroup of d-dimensional unitary matrices that are diagonal with respect to some fixed basis {|i }. These are matrices U such that j| U |i = δ ij e iθi . A state ρ is a generalized Werner state if it is invariant under any bipartite unitary U ⊗ U , where U is diagonal. These states may be parametrized by two real parameters. For a, b ∈ [0, 1] with a + b ≤ 1, the generalized Werner states can be given by For brevity, we can instead write these states as where P , P + , and P − are orthogonal projection matrices with P + P + + P − = I.
To determine the negativity of the ρ ab states, we need to compute their partial transpose. This can be given by Note that |Φ Φ| is another projection operator with |Φ Φ| < P , and thus P = P − |Φ Φ| is also a projection operator. Hence Note that only the coefficients in front of the P and |Φ Φ| can be negative. In particular, the negativity of ρ ab is given by A schematic of the ρ ab states can be seen in Figure 2. For a fixed state ρ ab and a value 0 ≤ c, we want to determine the value of the support funciton h Nc (ρ ab ). Due to the symmetry of the generalized Werner states, this is reduced to determining max {Tr[ρ ab ρ a b ] | N (ρ a ,b ) ≤ c}.
For different values of c, the sets of states ρ a b with negativity not greater than c are displayed in Figure 3.
We now determine Tr[ρ ab ρ a b ] for a fixed ρ ab : Note that this is a linear function of a and b . Let For ρ a,b with A = d+1 d−1 a + b − 1 and B = d+1 d−1 b + a − 1, the trace overlap is given by To maximize this value for a given state ρ ab , we split our analysis into the following cases: b + 1−cd d 2 .
• 0 < B ≤ A: The state that maximizes h Nc (ρ ab ) should have 0 < b ≤ a . Hence the optimal (a , b ) is ( 1+cd 2 , 1−cd 2 ), and thus h Nc (ρ ab ) = 1+cd We can now examine the value of the witnesses W ρ ab . Since the value of the witness W ρ ab is linear in a and b within each of the regions in Fig. 4, it suffices to check only the extremal points of each of these regions, which we have labelled in Fig. 5. The values of the witness evaluated at each of these points are listed in Table I. Unfortunately, the only case when any of these witnesses is an improvement over the negativity is when ρ a,b is the most entangled Werner state, which is just the value of the witness W wer analyzed in the main body of this paper. Hence, the witness W Gwer generated by minimizing W τ over all generalized Werner states is no better than the witness W wer generated by minimizing over the Werner states.