Evidence that All States Are Unitarily Equivalent to X States of the Same Entanglement
Strong numerical evidence is presented suggesting that all two-qubit mixed states are equivalent to X states by a single entanglement-preserving unitary (EPU) transformation, so that the concurrence of such an X state equals that of the original general state. An X-state parameterization of a general two-qubit state is given, allowing all states to have their concurrence parametrically specified. A new kind of entanglement measure is proposed, relating a general state’s entanglement to that of a pure state in the same system. New states called “H States” are presented, having fully parametric concurrence and purity, with the intention of using them to construct entanglement-preserving depolarization channels, which may aid development of the new entanglement measure. A theory of “true-generalized” X states (TGX states) is proposed for the general case of -partite systems. While such states do not generally have the literal “X” shape, evidence is shown that they are the true generalizations of X states in larger systems, since they appear to always be EPU-equivalent to general states of all ranks, whereas literal X states generally are not. An example of this is given for , including the proposition of the maximally entangled mixed states (MEMS). If the claim that TGX states are universal is valid, then any entanglement measure may be computable in a simpler form by using the EPU-equivalence between general states and TGX states.
pacs:03.65.Ud, 03.67.Mn, 03.65.Aa
In the field of quantum information, much work has been done showing the benefits of preparing systems in states with special simple forms, such as X states (to be defined below), in which many of the density-matrix elements are zero. Experimentally, this simplicity can reduce equipment, complexity, time, and cost. From a theoretical viewpoint, such simple states can allow symbolic computation of entanglement, an advantage because entanglement of general states must usually be computed numerically, which can obscure the algebraic dependence of entanglement upon various parameters of interest. Therefore, it would be highly beneficial, both experimentally and theoretically, if we could somehow convert any given general state to a simple X state of the same entanglement.
The most important result of this paper is that it presents strong numerical evidence showing that any general state can indeed be transformed to an X state of the same entanglement. Section II of this paper provides this numerical evidence for two-qubit systems, the simplest case. Then, Sec. III generalizes the idea to -partite systems, giving numerical evidence for systems as an example. Since an explicit general multipartite entanglement measure has not yet been discovered beyond systems, Sec. II also suggests a new kind of entanglement measure that may enable further progress in the multipartite case, introducing a new two-qubit parameteric state family called “H states” which may be useful for this task if they can be generalized. Some important concepts for this paper are concurrence, X states, and maximally entangled mixed states (MEMS), all of which we now review before proceeding.
Hill and Wootters’ landmark papers Hill and Wootters (1997); Wootters (1998) presented the concurrence, a tool that allows the quantification of entanglement in any mixed two-qubit quantum state defined by the density matrix , where are probabilities such that and , and the are pure states. Originally developed to calculate the entanglement of formation Bennett et al. (1996), the concurrence is a measure of entanglement as well, and is defined as
where are the eigenvalues, in decreasing order, of the Hermitian matrix , where
where is a Pauli matrix. The are also the square-roots of the eigenvalues of the non-Hermitian matrix . Due to the reliance on eigenvalues of transformed quantities, the concurrence is generally only suited to numerical calculations, though it is still highly useful.
However, a useful special case was presented in Yu and Eberly’s work on the concurrence of X states Yu and Eberly (2007), defined as those states with the form
for which the concurrence was shown to be
which holds true for both pure and mixed X states. Since many well-known and useful families of states have X form, including the Bell states, Werner states Werner (1989), and isotropic states, (4) is a powerful tool for obtaining symbolic expressions for the concurrence of X states.
Furthermore, it was shown in Rafsanjani and Agarwal (2012) that the X-state part of any state, along with the anti-X matrix formed by looking at the remaining elements outside of the X form, can be used to find lower bounds on concurrence for non-X states. The same work also shows that literal X states in larger systems can be used to find lower bounds on generalizations of concurrence such as concurrence Rungta et al. (2001), though this is limited since such measures are merely sufficient to detect entanglement.
Much work has also been done in exploring the relationship of concurrence as a function of purity such as in Ishizaka and Hiroshima (2000); Horst et al. (2013); Ziman and Bužek (2005), which discuss the idea of maximally entangled mixed states (MEMS), the latter of which investigates the action of local channels on X states. MEMS are states with the maximum entanglement possible for a given purity, and are defined in Ziman and Bužek (2005) for two qubits as all states local-unitarily equivalent to
where is a Bell state. The concurrence-purity (CP) plot of gives the maximum entanglement possible for all mixed two-qubit states. Note that the shown in (5) are X states.
In this paper, we will see strong evidence that for two qubits, X states can access all possible concurrence and purity (CP) combinations, and that for every general state , there is an X state to which it can be transformed using a single entanglement-preserving unitary (EPU) matrix. Such EPU transformations are not necessarily a tensor product of unitary matrices, and a more general form for them will be given in Sec. II and App. A.
It will then be shown that if the above transformation is always possible, then general density matrices can be parameterized entirely in terms of an X state transformed by a single EPU matrix, so that Yu and Eberly’s formula in (4) allows us to parametrically specify the concurrence of any general mixed state through its X-state core.
Then, a new kind of entanglement measure will be briefly proposed, based on relating the concurrence of any state to that of a pure state in the same system. To assist in the development of this entanglement measure, this paper will also present a new type of state that has fully parametric concurrence and purity.
Finally, to generalize these ideas, in Sec. III we will define and examine methods to find “true-generalized X states” (TGX states), for all discrete quantum systems, which yield TGX states even in systems for which literal X states are not the same as TGX states, as in . The hypothesis that TGX states are universally equivalent to all quantum states up to an EPU transformation is further supported by examples in , and is conjectured to hold true for all multipartite systems.
Ii Universality of X States for Two-Qubit Systems
Here, we focus only on two-qubit systems (having dimensions), with the goals of showing that all states are EPU-equivalent to X states and of finding an X-state parameterization for all two-qubit states. A new kind of entanglement measure is also proposed, and new states called H states are presented, having the feature of parametric concurrence and purity.
For most physicists, the new term “entanglement-preserving unitary” (EPU) probably calls to mind the local-unitary operations, which are tensor products of unitary operators in each subsystem, such as , where is a unitary operator in subsystem . Indeed, local-unitary matrices do qualify as EPU. However, the special form of X states actually enables a wider class of unitary matrices to be entanglement-preserving, as is proved in App. A. Therefore, local-unitarity is merely sufficient for ensuring entanglement preservation, and thus we need a more general term for more general cases. The most transparent term for these is “entanglement-preserving unitary” (EPU) transformations. In general, labeling these EPU transformations as , they can be formally defined, for a given input state , by
where is any valid entanglement measure and is any state. Thus, (6) means that is unitary, and that given input state , the transformed state has the same entanglement as .
First, note that the particular entanglement measure used in (6) is not important because all that matters is that its value is unchanged by the application of . Secondly, notice that if the set of states is restricted to a particular type of state, such as the X states, then that can affect the definition of which unitary matrices qualify as EPU, due to the dependence of (6) on .
As proven in App. A, the set of EPU matrices acting only on X states includes nonlocal unitary matrices, so that these EPU matrices will generally not have product-form. See Sec. II.4.1 for a compact summary of the form of these EPU matrices, or see App. A for more details.
As a preview of the most important results of this section, the main numerical evidence that shows that there always exists an EPU matrix that transforms a general two-qubit mixed state into an X state of the same concurrence is given in Fig. 2 compared to Fig. 1. These show that it is always possible to find an X state with the same concurrence-purity-rank (CPR) combination as any general state. Then, given that numerically-supported fact, (17) shows how to find the EPU matrix that causes the desired transformation, while Fig. 3 shows that it works on a large number of consecutive arbitrary mixed states.
ii.1 X States Contain Maximal Concurrence-Purity Combinations for All States
The premise of this paper is that if X states contain all of the same concurrence-purity (CP) combinations available to general states for each rank, then by virtue of the fact that entanglement-preserving unitary (EPU) matrices preserve purity, concurrence, and rank of all states, then we should be able to transform any general state to an X state using a single EPU operation.
Therefore, first we must investigate all possible CP combinations available to general states, and compare them to the MEMS of (5) to verify that they indeed represent a maximum for all states, and not just X states.
Although it is popular to use the participation ratio , we use the purity here since it is more ubiquitous in quantum information, and powers of it are directly proportional to , rather than inversely proportional.
For CP relations, a more useful parameterization for is in terms of purity . In fact, evaluating this reveals a third case is necessary to ensure that a MEMS exists for every possible purity value, which yields
where , and , , , and . This then reveals the MEMS CP relation as
As the top plot of Fig. 1 shows, the CP plot of does indeed appear to match the extreme upper bound of all general two-qubit states. Of course, this is not a proof, but rather strong evidence, since the main sample is an even distribution over all ranks of random general states. The bottom plots show that the CP values accessible to general states of rank are bounded only below at a purity-wall of , with the “separable ball” having maximum purity , Życzkowski et al. (1998).
The need to look at rank-specific CP values is because a single unitary matrix cannot change the rank of a state. Thus, to show EPU-equivalence, we need to show that for each rank, the X states access all the same CP values available to general states for that same rank.
The proof that contain the maximum concurrence values for each possible purity value was given in Ishizaka and Hiroshima (2000) for states up to rank , and checked numerically for rank . Therefore, since MEMS are also X states, it is a fact that X states contain the upper bounds of all CP values up to rank 3, and we shall tentatively accept the numerical results of the rank- case of Fig. 1 as evidence in favor of the hypothesis that X states contain the upper bound of CP values for all general states and ranks.
ii.2 Evidence that X States Contain All CP Combinations
Now that we have given evidence that X states contain the highest extreme CP values available to all states, we need to show that they contain all lower values, as well. To accomplish this, we now define several parametric state families that will be useful in what follows.
Bell States and States
As a starting point, recall the Bell states,
which may generalized by two-parameter pure states,
where and , , and . These states are separable at , and maximally entangled at , where they are equal to Bell states when . Intermediate values cause varying degrees of entanglement, and hence we may call these states.
Some noteworthy features of states are that they are all X states, and they contain all possible maximally entangled pure X states (though keep in mind that maximally entangled non-X pure states exist as well), as well as the four separable standard basis states , , , and . We can even define real-valued states as , , , and .
The purpose of defining these states is to parameterize all possible X states, which we will explore next.
Generalized Two-Qubit X States
Here, we wish to find a general form for all possible X states, so that we can search the full range of CP values accessible to X states. To obtain a general X form, note that all X states have at most, two unique non-zero off-diagonal elements, and . These are generally formed by convex sums of complex numbers.
Since any complex number can be represented as a vector on a complex plane, and since off-diagonal elements of pure states are the geometric mean of the corresponding diagonal elements, the minimum number of pure states required to decompose either of the off-diagonal X elements alone is two. Therefore, since there are two off-diagonal X elements, we need a minimum of four states to decompose a general mixed X-state.
Thus, we define the most general mixed state as an -parameter mixed state,
where the probabilities have hyperspherical form
with parameters on and . All possible X states can be obtained with suitable choice of these parameters, making the most general X state parameterization.
We can remove unnecessary parameters by considering
where the probabilities are still given by (12), but the number of parameters is only nine. The reason for only setting the first and third phase angles to zero is that each consecutive pair of pure states in (13) share off-diagonal elements, and thus for their sum to roam all complex phase values, only one of them has to be complex.
While (13) is the essential parameterization of the most general X states, we will find it convenient to study the following real, rank-specific X states,
where we will take as our canonical minimal X state parameterization, since it can be linked to all other X states of rank by diagonal unitary phase transformations. These states each have only parameters. For each state in (14), it is assumed that , so the parameterizations of the probabilities of the first three are different than those of (12). Note that in (14), it is implied that each of the probabilities shown must be nonzero to ensure the specified rank, and for the same reason, the must be chosen to ensure that and . Now we are ready to investigate the CP qualities of X states.
Evidence of All CP Combinations in X States
Here, we take advantage of the fact that contains all possible X states to see if the set of X states has access to the same CP region accessible to general states. Since symbolic evaluation of the concurrence of as a function of purity is difficult even with the help of (4), we limit ourselves to a random sample of states by randomly choosing angles for , as shown below in Fig. 2.
As Fig. 2 shows, the X states appear to have access to all of the same CP values available to general states, as seen by comparing Fig. 2 to Fig. 1. More importantly, however, the rank-specific plots show that the X states match the CP values of general states by rank, as well. Due to this correspondence, a single EPU matrix can convert any to an X state, which is one of the central claims of this paper.
Again, while this randomly generated sample is no proof that this is true, it provides strong evidence in favor of the hypothesis that X states access the same set of CP combinations that general states do. Furthermore, since the rank-specific plots use the real-valued X states of (14), Fig. 2 shows that real-valued X states access all CP combinations as well.
ii.3 Convertibility of All States to X States
Finding the EPU X-Conversion Matrix If It Exists
Suppose, as Fig. 2 suggests, that X states access all CP values available to general states by rank. Then, if that is true, here we prove the existence and form of a matrix that converts any general state into an X state.
First, if a general state and a particular X state have the same purity and rank, then they can be related by a single unitary matrix. Furthermore, if also has the same concurrence as , then the unitary matrix that relates them also preserves the entanglement, which can mean, but does not necessarily mean it has a product form , though in general, it does not. Also, since unitary matrices cannot change eigenvalues, and must have the same eigenvalues.
Therefore, expressing the states as and , where and are eigenvector matrices whose columns are the eigenvectors of their respective states and is the diagonal matrix of eigenvalues, where we use the same for each in accordance with the above argument, then we can eliminate as
from which we obtain the X-conversion transformation,
so the EPU matrix that transforms to is
and is entanglement-preserving since it cannot change the concurrence of , which is true by the above definition that this particular has the same concurrence as . Therefore, (17) requires additional constraints to ensure entanglement preservation, which we will discuss soon.
Thus, we have shown that if X states access all possible CP values for all ranks, then every general state is EPU-equivalent to an X state of the same concurrence, where the EPU matrix relating them is given by (17), provided that . See Sec. II.4.1 and App. A for details about the form of such EPU matrices.
Demonstration that Arbitrary General States Can Be Transformed to X States
Here we perform a simple test of the EPU-equivalence of general states with X states. The test is as follows.
First, generate a random general state . Then, measure its concurrence, purity, and rank. Next, search a large number of real-valued rank-specific X states from (14) until finding one that yields with equal to that of within a certain tolerance, for the defined in (17). Then, is an X state with the same as general state . As a measure to quantify how close this new state is to X form, measure the sum of square-magnitudes of its anti-X unique off-diagonals by using , where
Since exactly only when all eight complex parts of the anti-X elements are zero, this can only happen for X states. In the case when a state is the most “non-X” it can be, the sum of all anti-X-element magnitudes is , in which case . Thus, is a good measure for how close a state is to an X state.
As Fig. 3 shows, the general states are all unitarily transformed to X states of the same concurrence to close approximation, and we may hypothesize that this can be done to any desired accuracy. Note that the successes of this test were produced consecutively using a while-loop for each input, meaning that no failures were encountered in this sample of states, only the first of which are shown in Fig. 3. Therefore, the above test provides compelling evidence to support the hypothesis that every general two-qubit state can be transformed to an X state with a single EPU transformation. The fact that such a transformation is unitary is because is a product of unitary eigenvector matrices, and its entanglement-preservation is because also did not change the entanglement. Thus, we have demonstrated strong evidence in favor of the main hypothesis of this paper.
As a more tangible example, given the random state,
the above procedure produces the transformed state
where the anti-X elements in (20) are zeros to at least decimal places, and within .
The prime importance of being able to find these kinds of transformations is that X states are always easier to work with, and can even enable symbolic computation of entanglement. Therefore, the ability to transform any state to X form while preserving the original entanglement opens a door to symbolic computation of entanglement for any input state.
Simple Symbolic Example of EPU X Conversion
In general, symbolic proof for EPU X conversions is very difficult, despite the ease with which numerical examples can be generated as in Fig. 3. However, to illustrate the process with a simple case, we now look at an example simple enough to permit symbolic proof.
First, choose a non-X state from the restricted set,
where , and we shall abbreviate quantities of this “general” input state as , , and . Then, from purity and normalization, we know that and , which yields
using descending-order convention. Now, consider an X state of parametric concurrence and purity , defined as
which exactly match those of in (22). Then, solving for the descending-order eigenvector matrix of gives
where and , and the column vectors of were verified to satisfy the eigenvalue equations for , where the eigenvectors are column vectors of such that . Thus, we obtain the following eigenvalue relation between the two states,
Thus, the X transformation we seek is
Thus, we have proven that, for the subset of general states defined in (21), it is always possible to unitarily transform into while preserving exactly.
Now that we have seen a simple example, we are ready to talk about the conditions for constructing the most general EPU X transformation.
ii.4 Expression of General States using X States
Parameterizing General States with X States
If the claim that all states are EPU-equivalent to X states is true, as is supported by the test in Sec. II.3.2, then we should be able to parameterize all states in terms of an X state transformed by an EPU matrix. Therefore, using a real-valued X state , the most general two-qubit state is
where from (14) and is the most general EPU operation on X states, characterized by
where is any valid entanglement measure, and is a single-qubit unitary matrix with only one superposition angle and one relative phase angle, where the parenthetical subscripts indicate the subspace upon which acts by referencing its only nonzero off-diagonal element, and is a diagonal unitary matrix.
The entanglement of each successive single-qubit-transformed state, such as , is generally not equivalent to the final entanglement. Specifically, each successive single-qubit transformation typically has a different concurrence which can be higher or lower than that of , even though the final state in (29) does have the same concurrence as .
In fact, using single-qubit factorization of the EPU matrices of the test in Fig. 3, it has been verified that EPU matrices do generally have nontrivial unitary rotations on all single-qubit subspaces, of the form in (30). Thus, in general, as proved in App. A, is nonlocal.
Furthermore, as seen in Fig. 2, we only need real X states to reach all CP values, and in general, has degrees of freedom (DOF), as seen in (14). Then, to see whether could have enough DOF to upgrade to a fully general state, note that discarding global phase, intrinsically has DOF, of which belong to all the collectively, while belong to with its global phase discarded. However, the only DOF that matter are those that persist in the transformed state. Since is adjacent to the X state, the zeros of the X state reduce the DOF of to only DOF, so has DOF. That leaves the variables of to contribute the necessary DOF still needed to produce the DOF of a general state.
While it is true that a local unitary matrix contains up to DOF, the more general unitary form of (30) offers a greater range of possibilities for supplying the DOF, without limiting us to local transformations. Since (30) describes the most general way in which one could make any unitary matrix, and since numerical tests confirm that all single-qubit rotations are generally present in EPU matrices, then (30) is the most general form for EPU matrices (though admittedly an explicit method to determine the angular parameters is still unknown). See App. A for more details.
The strong numerical evidence provided by Fig. 3 suggests that every general state can be unitarily related to an X state of the same concurrence as . Therefore, since is the most general such unitary matrix, then (29) is the most general way in which a general state can be parameterized with an X state.
Parametric Concurrence for General States
Given that the most general two-qubit state can be parameterized as (29), then due to the fact that
where is given in (30), the concurrence of any general state is then conveniently given by Yu and Eberly’s explicit concurrence of its corresponding X state using (4). Therefore, to obtain a general two-qubit state with a given concurrence, purity, and rank (CPR), first find an X state with that CPR, and then the set of all general states of that same CPR combination is parametrically accessible by putting that X state into (29) and choosing a . In that way, the concurrence of a general state can be determined by that of an X state.
ii.5 Entanglement-Preserving Depolarization as a Universal Measure of Entanglement
Recalling the conceptual definition of MEMS as states that maximize the entanglement (however it is measured) for any given purity, consider the following observation.
Domain of Constant Entanglement Theorem: For every general state with entanglement , there exists a MEMS and a pure state that have the same entanglement so that . Thus, the domain of constant entanglement is a family of states whose members of minimum purity are MEMS and whose members of maximum purity are pure, all of which share the same entanglement.
Note that the above observation is true regardless of the propositions we have made about X states. Figure 4 illustrates the essence of the Domain of Constant Entanglement Theorem (DCET) for CP values.
As Fig. 4 shows, since all general states are bounded by MEMS and pure states, then for every general state , a horizontal line of constant connects it to both a MEMS and a pure state of equal .
Thus, the DCET suggests a new kind of entanglement measure. Suppose there exists a transformation that preserves entanglement while simultaneously depolarizing the input state as much as possible. Since any depolarizing operation cannot increase the purity, that means is always towards the left of , unless is pure and maximally entangled.
Then, since maximally depolarizes along the line of constant , it will hit a purity wall as shown by the vertical gray line in Fig. 4, which is imposed by the MEMS for that . For separable states, the purity wall is , where the dimension for two qubits.
The method for finding entanglement of an arbitrary input state is then to first find , and then search a wide set of pure states with a uniform distribution of entanglement values and find . Finally,
which says that if both and have the same minimal purity in a channel of entanglement-preserving depolarization , then the entanglement of the general state can be computed directly as the entanglement of the pure state .
Since finding entanglement of pure states is always possible, and since states allow us to parameterize this entanglement smoothly, then we can test an arbitrarily fine resolution of entanglement values for candidate pure states until finding the one whose minimal purity in matches that of the input state to any desired tolerance.
Thus, if can be found in all multipartite systems, this offers a means of devising a universal entanglement measure. Furthermore, note that this does not require that we know anything about MEMS or even that the pure states have X form.
The difficulty is finding the entanglement-preserving depolarization channel . One candidate is the generalization of the local-unitary rotation, the so-called doubly-stochastic local channel, defined as
where and . Unfortunately, such channels do not generally preserve entanglement.
Thus, at present, there is no known form of . Yet, it is likely that the DCET holds true in all larger systems, as well. The conceptual existence of is an intriguing thought that prompts us to try to develop this new kind of constant-entanglement purification, where in this context we would not enlarge the system as with conventional purification, but rather merely find the pure state in the same system that has equal entanglement to an input state. For now, we leave this as an open problem.
States of Constant Entanglement
As a possible aid to finding the entanglement-preserving depolarization channels , here we investigate a parametric family of states that allows us to precisely specify both concurrence and purity .
First, consider the prototype for such states, given by
where , and we shall call these H states due to the fact that changes in only and cause the CP plot to be horizontal, as desired. Notice that the first term of the second state is a state with , and in this form, the two parts overlap.
The full CP parameterization of the H states is
where , and to which the first case has been added to describe the region of purity below the separable cutoff. These states are defined on intervals,
The H states have the potential to be incredibly useful. Since they span the full range of physical CP values, then any state EPU-equivalent to an H state can be linked directly to a value. The procedure would be as follows. First, from an input state of unknown , measure . Then, holding constant, search an even distribution of different values of for a number of H states. The one that that has the correct will have the same eigenvalues if the input state has the same rank as the H state. Unfortunately, the rank requirement makes this method only useful in some cases.
At present, H states of rank have not yet been derived. Alternatively, if there were a rank-changing transformation that preserved , one could use that to adapt an input state to the rank of the H states.
If such a transformation could be found, and if H states could be found in all larger systems, then this would provide a method of obtaining a general entanglement measure. Then, would be a parameter directly related to the superposition angle of the pure H states, and we could call such a measure the angle of entanglement.
At present we leave this as an open challenge. Now, we move to the last part of our discussion, which is the search for a generalized X form in larger systems that is also EPU-equivalent to all states.
Iii True-Generalized X States for Multipartite Systems
Here, we generalize the idea of X states in all possible multipartite discrete systems by observing qualitative physical facts about the two-qubit case and applying those observations to larger systems.
The two main observations are firstly, that the anti-X elements (the zeros in (3)), are related to the partial trace operation, and secondly, that the X elements are identifiable by collectively observing a basis of maximally entangled states of a particularly simple form.
We shall then show that these qualitative observations are equivalent and generalizable to all systems, working out several examples along the way, which will reveal that such states do not have a literal X shape. Finally, we will examine the idea of the universality of such states by comparing these new true-generalized X states (TGX states) with literal X states.
iii.1 TGX Definition by Relation to Reductions
Relation of X Form to Partial Trace For Two Qubits
Recall that the operation of partial tracing is tracing over only subsystems not to be retained while operating on the desired subsystems with the identity. Thus, for two qubits, in terms of the full , the reductions are
where superscripts refer to subsystem labels and are in parentheses to distinguish from matrix power notation.
Then, in complement to (3), define the anti-X matrix,
where the dots represent zeros. Now, notice that the anti-X elements