Evaluation of convex roof entanglement measures
We show a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples.
Quantum entanglement plays a central role in quantum information science and quantum optics GT09 (). There are now efficient methods to detect entanglement, that have even been used in many experiments expdet (). These mostly answer the yes or no question: ”Is the quantum state entangled?” or ”Is the quantum state genuine multipartite entangled?” After verifying the presence of entanglement, the next step is quantifying it. Calculating measures is becoming increasingly important in experiments in quantum information science GR08 (); EB07 (); WP09 () and it also plays a crucial role in investigations in quantum statistical physics, e.g., in studying phase transitions OA02 ().
Most entanglement measures are based on the convex roof of a quantity on pure states such as the entropy of the reduced state WK02 (); WG03 (); CK00 (). Measures of this type can also be used to classify states according to their membership in some convex sets, for example, based on their Schmidt rank AB01 (); SB01 (). They play a central role in quantum information theory, however, in most of the cases they are not computable as there are no efficient ways to calculate convex roofs. Most importantly, the simplest multipartite entanglement measure, the three-tangle for three-qubits, cannot be computed for a general state.
Thus, for obtaining entanglement measures in theory and experiments, it would be crucial to find methods to calculate convex roof constructions efficiently, at least for not too large systems. This seems to be a very difficult task since straightforward numerical search means an optimization over an infinite number of convex decompositions of the density matrix. Such an approach will lead to an upper bound on the measure, since a multivariable numerical optimization is not guaranteed to find the global optimum RL09 (). Upper bounds, however, are often not very useful as the amount of entanglement can be much lower or even zero even if the procedure signals considerable entanglement.
In this paper, we present a method that produces a series of very good lower bounds on important entanglement measures. Our method has the following characteristics: (i) It is based on semidefinite programming. The series of bounds obtained converge in a controllable way to the true value. Even the first lower bound in the series is non-trivial. (ii) We have a clear physical picture for what states our method yields a nonzero value for the measures. (iii) The set of separable states is used in the optimization procedure. This way we connect calculating convex roofs to the separability problem, which might help to find applications of the separability problem in other areas of physics. We will demonstrate the use of our method with the example of computing bipartite entanglement measures for bound entangled states, computing the convex roof of the tangle for various three-qubit states, and even quantities outside of quantum information science. Our method can also be used to compute a lower bound from incomplete data of the quantum state or in device independent scenarios VMMW02 (); LV11 (); MB14 (); pusey ().
Convex roof of linear entropy. For pure states, the linear entropy of entanglement is given as
where we used the definition of the linear entropy Hence, the linear entropy of entanglement for pure states equals also where is the concurrence WK02 (), and it is also equal to the -tangle Itangle (). The definition (1) can be extended to mixed states by a convex roof construction as
where is a decompositon to pure states
It can be shown that does not increase under local operations and classical communication (LOCC) on average, hence it is an entanglement monotone elinmonotone (). Consequently, has also been used to characterize entanglement even in the multipartite setting ElinEnt ().
Next, we will show a method to compute Eq. (2). For this aim, first we write the liner entropy of entanglement as an expectation value of an operator acting on two copies of a bipartite pure state as H03 ()
Here, and denote the parties of the first copy while and denote the parties of the second copy. Moreover, the projector to the antisymmetric space is defined as is the flip operator, and we explicitly wrote out for clarity MK05 ().
Next, we will consider mixed states. Let us assume that is the decomposition attaining the convex roof. Then, for a state with such a decomposition we obtain
where the state on the two-copy space is defined as
The density matrix has three important properties. It is a mixture of product states, i.e., a separable state W89 (). Moreover, all the pure product components are symmetric. Thus, is supported on the symmetric subspace. In fact, any symmetric separable states can be written in the form (6) SymSep (). Finally,
Hence, we arrive at our first main result.
Observation 1.—The convex roof of the linear entropy can be written as
Observation 1 connects the separability problem of symmetric bipartite states, i.e., answering the question ”Is the state entangled?” mentioned in the introduction, to entanglement quantification. In principle, to obtain a lower bound on any necessary condition for separability could be used. We will consider the method based on the positivity of partial transpose (PPT) PPT () and obtain a lower bound as
Horodecki state.—We test our method to calculate entanglement measures for the one-parameter family of the bound entangled state introduced by P. Horodecki H97 (). We mix the state with white noise according to and calculate the entanglement as a function of and The results can be seen in Fig. 1. The critical noise for which agrees with the calculations of Ref. MK05 () and Ref. CM12 (). We note that we made the computer program calculating with all other programs used for this publication, publicly available mathworks (). Other methods for calculating entanglement measures are in Refs. CA05 (); G03 ().
It is a surprise that, while the bound relies on the PPT criterion, the method is still able to detect PPT entangled states. In order to obtain more information on what kind of states are detected, we need to know the separability criterion based on symmetric extensions PPTsymext (). A given bipartite state is said to have a symmetric extension if it can be written as the reduced state of a multipartite state which is symmetric under and for all If we also require that the state is PPT for all bipartitions, then it is a PPT symmetric extension. Separable states have such extensions for arbitrarily large and while the lack of such an extension signals the presence of entanglement.
Observation 2.—For all non-PPT states and for all states that do not have a symmetric extension we have Moreover, for all states having a PPT symmetric extension holds. The proof can be found in the Supplement SUPP ().
Before we continue let us point out that we can also obtain a lower bound on if we choose any other entanglement condition, such as the method based on local uncertainty relations lur (), the covariance matrix criterion GH07 (), or the computable cross norm or realignment criterion (CCNR) ccnr (). However, for symmetric states these are all equivalent to the PPT condition TG09 ().
Therefore, to strengthen the bound a stronger criterion must be employed. Here again the method of PPT symmetric extensions can be used PPTsymext (). Rather than approximating by PPT states, we demand that has an PPT symmetric extension symext (). In this way we obtain a sequence of lower bounds with increasing accuracies. The corresponding optimization can similarly be carried out by semidefinite programming. Note that the PPT symmetric extensions converge to the set of separable states in a controlled way NO09 (). Finally, note also that semidefinite programs not only detect entanglement, but through solving the dual problem, it is possible to find entanglement witnesses PPTsymext (). In our case, these witnesses can even bound entanglement measures, as explained in the Supplement SUPP ().
Generalization and further examples.—The previous ideas can straightforwardly be generalized to compute the convex roof of any quantity that can be written as a polynomial of expectation values for pure states as
where are operators and are constants (see e.g., OS12 (); G05 ()). It is possible to define an operator whose expectation value on several copies reproduces Eq. (9). Then, the convex roof of Eq. (9) can be obtained as an optimization over -copy symmetric fully separable states SymSep ()
Three-tangle.—Our next example is the calculation of the three-tangle, a three-qubit entanglement monotone DV00 (). For pure states, it has been defined by Coffmann, Kundu and Wootters CK00 (). Remarkably, it can be written as a fourth-order polynomial in expectation values OS12 (). Hence, for mixed states, the tangle can be defined through a convex roof extension, which we can now map to the optimization problem
The optimization can be carried out for symmetric multiqubit states that are PPT with respect to all bipartions rather than symmetric separable states, leading to the lower bound The results are shown in Fig. 2 for states of the form
where and Note that a lower bound for the convex roof of the tangle for general states, which is exact for states with certain symmetries, has been developed ES12 ().
As a practical comment, we add that the numerical computation is challenging, but can be computed on a standard laptop with standard free packages for semidefinite programming SDP (), if the state has some symmetries, or has a rank up to six. Calculations for general three-qubit states of rank eight are realistic with computer clusters and professional packages.
Schmidt rank.—Let us consider the quantities that are nonzero for states with a Schmidt rank larger than For example, where are the eigenvalues of the reduced state. The quantities are proven to be entanglement monotones SZ02 (). We can calculate the convex roof of with our method. Convex roofs for such quantities allow us to bound the dimensionality of the entanglement from below. A powerful bound can be obtained by carrying out the optimisation for -qudit symmetric states that are PPT with respect to all bipartitions. An alternative is computing the negativity VW02 (); ES13 (). In particular, signals that the Schmidt number is larger than We show that our method outperforms the negativity as a dimension witness in Fig. 3(a) for the family of states
with and colored noise with .
We add that we checked several random edge states to test the conjecture of Sanpera, Bruß and Lewenstein claiming that all bound entangled states in such systems have a Schmidt rank and did not find a counter example SB01 ().
Evaluation of entanglement measures based on incomplete information.—Experimentally it is very important that entanglement measures can be evaluated based on incomplete knowledge on the quantum state. There are efficient methods to bound entanglement measures based on an operator expectation value from below GR08 (); EB07 (); WP09 (). The current method can be adapted straightforwardly to the partial information case by replacing the condition with the set of linear constraints where are the measured observables and are the corresponding expectation values. As an example, see Fig. 3(b), where the entanglement is bounded from below based on complete information and based on and measurements for the state
where is a two-qubit Bell state embedded in the system and acts on this two-qubit system.
Device independent scenario.—The amount of entanglement can be bounded exclusively from the observed data but independent of the quantum description. Depending whether only one or both sides are untrusted one distinguishes between a steering-type or a Bell-type scenario. The necessary steps to lift the method using only partial information to such device independent scenarios employs the translation idea highlighted in Ref. MB14 () and is explained in more detail in the supplement SUPP (). As an example, in Fig. 3(c) we plot a lower bound on the linear entropy of entanglement given as a function of the violation of the CHSH Bell inequality GT09 ().
Concave roof.—Besides convex roofs, concave roofs can also be computed. For example, if in Eq. (2) a concave roof is used instead of a convex roof, then we compute the linear entanglement of assistance DF99 (), which is the maximal entanglement available if the mixed state is given as a purification to us, and a third party which holds the ancilla needed for the purifucation is assisting us. In this case, in our method minimum must be replaced by maximum. In this way, we obtain a converging series of upper bounds on the entanglement of assistance. The results are shown in Fig. 3(d) for the family of states
where and . As a reference, the linear entropy of entanglement is also shown for the same state.
Conclusions.—We have shown a general framework for calculating convex roof-based entanglement measures. We demonstrated its use in calculating the entanglement for bipartite systems, as well as, the three-tangle for three-qubits. We discussed several other quantities for which it can be applied. In the future, we would like to explore further possibilities of using our algorithm to compute convex roofs, in calculating the linear Holevo capacity BW97 (); OV06 (), the quantum Fisher information based on incomplete information qfi (), or the convex or concave roofs of sums of several variances, as outlined in the Supplement SUPP ().
We thank C. Eltschka, M. Kús, J. Siewert, and M. Tiersch for stimulating discussions. We thank the EU (ERC Starting Grant GEDENTQOPT, CHIST-ERA QUASAR, Marie Curie CIG 293993/ENFOQI), the MINECO (Project No. FIS2012-36673-C03-03), the Basque Government (Project No. IT4720-10), the OTKA (Contract No. K83858), the UPV/EHU program UFI 11/55, the FQXi Fund (Silicon Valley Community Foundation), and the DFG.
.1 Supplemental Material
In this supplemental material, we give some further details of our derivations.
Proof of Observation 2.
Let us assume that the state has Then, from Eq. (8) it follows that there is a symmetric PPT state such that and Hence, for this state and The symmetry of means that Hence, also holds. We can write
Since is unitary, implies Finally, we obtain
Hence, implies This proves the first part of our observation.
To prove the second part, note that based on the discussion above is a 2:2 symmetric extension of It is not necessarily a PPT symmetric extension since for the partition it can also be non-PPT.
Finally, the third part can be proved as follows. Let us assume that has a PPT symmetric extension denoted by Hence, and Moreover, is a PPT state. Hence,
Note that Theorem 2 can be generalized to states that have involving PPT symmetric extensions and symmetric extensions to several parties.
Quantitative entanglement witnesses
In this section, we describe how our method can be used to construct quantitative entanglement witnesses. As an example, we present a condition for entanglement witnesses, such that the expectation value of all witnesses satisfying the condition gives a lower bound on defined in Eq. (8). We also prove that for every state there is a witness of this type that gives not only a lower bound, but gives the value of exactly.
For the linear entropy of entanglement we needed to minimize the expectation value of the operator over all symmetric separable states with a fixed reduced marginal . Consider now an operator that acts on the original bipartite Hilbert space. We require that is satisfies
where . Here is an operator acting only on the symmetric subspace of the two copies , while acts on the full tensor product but we only used the projected symmetric part of the partial transpose. For such a decomposition, it can be shown that its expectation value for is
The projectors onto the symmetric subspace can be dropped in the third line since is supported only on it. In the last line we used , while nonnegativity holds because all occuring operators are positive semidefinite. Hence, Eq. (S4) can be rewritten as
where we have further simplified the right-hand side using that has a fixed reduced density matrix. Since Eq. (S5) holds for any valid state , it holds in particular for the one yielding the linear entropy of entanglement, thus we arrive at
Hence the expectation value of our witness provides a lower bound on the linear entropy of entanglement.
Next, we will show that for a given quantum state if we optimize over all such witness operators, it is always possible to find one that saturates the inequality (S6).
Observation 3.—For the linear entropy of entanglement we obtain
with being the set of all operators of Eq. (S3).
Proof. The proof is given by applying the dual form of a semidefinite program SDP (), which has been employed in a variety of different quantum information problems. In particular we refer to Ref. PPTsymext () which explains such a procedure very nicely for the separability criterion based on symmetric extensions. We have structured the proof in two parts: In the first part, we show an equivalent formulation on the two-copy level. Afterwards we further simplify this dual problem to interpret it as an operator acting on a single density operator using techniques that were introduced in Ref. PPTsymext ().
In the first part, we parse the original problem as given in Observation into the form of a semidefinite program and invoke its dual, which provides the same solution. In order to achieve this one should note that the two conditions, just supported on the symmetric subspace and the linear equations can be satisfied automatically with an appropriate ansatz . Here is the fixed part of two-copy density operator such that the marginals equal to (its precise form being discussed later), while the remaining part is the yet to be determined part on the symmetric subspace, i.e., the set of operators is a Hermitian operator basis for the symmetric subspace . With this the primal problem reads
where one should note that acts on the symmetric subspace, while acts on the full tensor .
Taking this into account, it is straightforward to invoke the dual and to derive an equivalent optimization problem. That this dual program provides the same solution is certified for instance via the Slater regularity condition SDP (), which holds since this problem has an inner point, i.e., if has full rank; otherwise one should constrain anyway to the range of . Since such reformulations have been carried out quite frequently, we refer here only to the literature, and continue with its solution, which is given by
where similarly acts on and on the full tensor product space. This finishes the first part.
In the remaining part we show how the objective of Eq. (S9) can be interpreted as an operator on the single copy. For that we need some structure of the fixed part that is given by the reduced state . The idea follows closely the ideas of Ref. PPTsymext (), though we need to do it here for the symmetric subspace.
To start, note that any given density operator can be written as with being an operator basis for the traceless Hermitian operators. Next let us define . The expectation values of all these operators are completely determined by the reduced state , and since all these state coefficients are independent this means that the set is linearly independent. This implies a positive definite Gram matrix , a unique inverse , and the existence of the operators . These operators are the corresponding orthogonal operators , so that the fixed part becomes
Note that also the desired dimensionality of matches, since are excatly independent linear equations. To transfer this to the single copy level we write this solution in terms of a map applied to ,
This map has the adjoint map, i.e., the map satisfying for all matrices ,
Via this we can finally make the connection to the single copy level by
where we defined the single copy witness in the last equation, parametrized in terms of the coefficients of . However since we want to have the witness as the open parameter we need to parametrize in terms of the witness . Setting
achieves Via this we can finally replace all occurrances of in Eq. (S9) by and we obtain the stated result of the observation.
Note that one can obtain other quantitative entanglement witnesses if one replaces the decomposable structure, as given in Eq. (S3), by a different entanglement witness condition. It is easy to see that if the operator is non-negative on separable states then gives a lower bound. Compared to other possibilities, the advantage of the witness (S3) is that the optimization (S7) to get the lower bound can be carried out with semidefinite programming.
We also add that if one only has measured a few observables then to get a lower bound one merely has to add the constraint which means that the witness is a linear combination of the measured observables with coefficients Then, we have to optimize where are the corresponding expectation values
Finally, if one also wants quantitative entanglement witness for the other tasks one can proceed similarly. For instance, if one likes to bound the tangle one demands that is a non-negative on all fully separable states, thus it is an entanglement witness to test against full separability.
Other quantities that can be calculated by our approach
Convex roof of the Meyer-Wallach measure.—The Meyer-Wallach measure is an entanglement measure for pure states defined as MW03 ()
where is the reduced state of the qubits. This measure can be generalized to include the reduced states of multi-qubit groups S04 (). Our method can calculate the convex roof of the measure (S15) and the generalized measures as well.
It is a capacity measure for a channel For qubit channels, explicit formula is given in Ref. OV06 ().
Convex and concave roofs in entanglement conditions with the quantum Fisher information and the variance.— First, let us see simple entanglement conditions with the quantum Fisher information and the variance. We start from the fact that for pure -qubit states
holds. Next, we need the fundamental properties of the quantum Fisher information in our criteria qfi (): (i) For pure states equals four times the variance (ii) For mixed states, it is a convex function of the state. Hence, for separable states follows T12 ()
Due to the concavity of the variance, we can obtain a similar entanglement condition with variances as TK07 ()
Any state that violates these is entangled. Numerical evidence shows that Eq. (S20) is stronger than Eq. (S18). Moreover, numerical evidence shows also that Eq. (S21) is stronger than Eq. (S19). These ideas can be extended to improve other entanglement conditions based on variances G04 ().
We note that Ref. LP13 () shows that covariance matrices can always be decomposed as the
where has the decomposition as in Eq. (3). Hence, we know that the bound on the sum of two variances cannot be improved this way. However, Ref. VP13 () demonstrates that such a decomposition is not always possible for covariance matrices. This is connected to the fact that the bound for separable states for the sum of three variances can be improved.
Quantum Fisher information based on incomplete data.— The quantum Fisher information can be bounded from below from partially known data. That is, we know the expectation value of some operators, and want to find a lower bound for the quantum Fisher information. The problem can be mapped to a semidefinite optimization in the two-copy space. A very good lower bound can be obtained if we optimize over PPT states.
For that we can use that the quantum Fisher information is, apart from a constant factor, the convex roof of the variance TP13 ()
The variance of a pure state can be expressed on two copies as
Hence, a lower bound on the quantum Fisher information can be obtained as
where the constraints are given with the expectation values The optimization (S25) can straightforwardly be carried out with semidefinite programming.
In Fig. S1(a), we present a simple example where a lower bound on the quantum Fisher information is shown based on measurements of the fidelity with respect to the GHZ state. Below a fidelity of the bound for is zero. This is due to the fact that the product state reaches this fidelity value, while is zero for this state. If the fidelity is we obtain and as expected T12 ().
In Fig. S1(b), we present a bound on the quantum Fisher information based on collective measurements, relevant to spin squeezing. Note that for well polarized ensembles, increasing leads to decreasing On the other hand, for small increasing leads to increasing Some of the curves have points only in certain ranges of as there are no physical states corresponding to measurement results outside of these ranges, assuming a given value for and
Similar methods can be used for bounding the variance of an observable from above based on the expectation value of other observables. We can use that the variance is the concave roof of itself TP13 ()
The difference between the two cases is that for the quantum Fisher information we have to look for the minimum, while for the variance we have to look for the maximum.
Genuine multipartite entanglement.—It is possible to define quantities that detect true multipartite entanglement and can be evaluated with our method. Let us define
where is the linear entropy for the bipartition of the qudits. To be more precise, is the linear entropy of the reduced state of the qudits in one of the two partitions for the bipartion. If then the state is biseparable, otherwise it is genuine multipartite entangled.
Similar idea can work such that only a sum of entropies must be computed by defining
where is linear entropy for the bipartition. If then the state is biseparable, otherwise it is genuine multipartite entangled. If, instead of , we calculate given in Eq. (8) then Eq. (S28) can be obtained via a semidefinite program. The advantage of Eq. (S28) is that only two copies of the original state are needed to calculate the value with our approach, while for the formula (S27) we need much more copies. The formalism of Eq. (S28) is in the spirit of the PPT mixer detecting genuine multipartite entanglement PPTmixer ().
Note that a three-qubit state mixed from states that are PPT with respect to some partitions have been found that is genuine multipartite entangled Tobi (). Thus, detecting genuine multipartite entanglement is a non-trivial task.
.2 Device independent programs
In this section, we explain the methods to obtain lower bounds on the linear entropy of entanglement for the device independent scenarios; either in the steering case where only the apparatus of one side is uncharacterized, or in Bell-type scenarios where both sides are unknown.
We will use the tool presented in Ref. MB14 (), resting on ideas from Ref. NPA1 (); doherty08 (), which transforms the problem of estimating entanglement in a device independent scenario into the more common problem to lower bound the entanglement of a given fixed finite-dimensional system having only partial information. The method uses instead of the quantum state of unkown dimension, a finite dimensional object which captures most of the properties of the state.
To set the stage, let us assume that on a given side, say system , one only knows the number of settings and respective outcomes . This measurement scheme is described by a collection of POVM elements , which act on a Hilbert space of unknown dimension. To this measurement scenario one now associates a specific completely positive local map: with Kraus operators . Here and are respective basis states of the input and output Hilbert spaces, while are operators out of a chosen set on which we comment shortly. However, via this structure, first observe that this map transforms a given input state to
hence an output with matrix elements given by certain expectation values. At this stage the specific operator set becomes important, since so far we know nothing about because we neither know nor . The only knowledge that we have are certain generic properties of the POVM elements , more precisely we have positivity , normalization and that each operator is a projector. Here note that by Naimark’s extension any measurement can be written as a projector onto a larger dimensional space. Since for most device independent tasks this extension does not change the underlying tasks this property can be assumed without loss of generality. . In addition note that the expectation values of each measurement operator is observable, .
Via these four properties one can thus choose specific operator sets such that one has at least some partial information on . For instance, if one chooses to consist of the measurement operators one knows for instance
while other entries like with are still unknown. Nevertheless via this one gets some partial knowledge and some structure of , which can be captured by an explicit parametrization as
using appropriate operators and . Here the first part represents the known part of , while the second one is the restricted open unknown part.
Such a structure can be inferred for any choice of . For instance, one could remove some linear dependencies of the just given example set if one adds the identity and erases the last outcome for each measurement setting . In addition note that one could also enlarge this set by including also products up to POVM elements , so for instance , already removing trivial parts. In this way one gets further relations like
if . The advantage of including products is that one gets a tighter, more constrained, description. This set of operators is precisely the one which has been mostly used MB14 (), since it is very straightforward to “decode” all the known structure. Still there are other possibilities, like . Here it might be harder to deduce all the structure but it has for instance the advantage that the associated map is then even trace-preserving, thus can be completely interpreted as an output quantum state; something which is not directly possible if one uses .
Now let us come to the concrete cases. At first let us discuss the fully device independent case where both sides are completely uncharacterized. If we locally apply the just described trace-preserving physical map (using for instance the choice ) we transform any state into another bipartite state . Since an entanglement monotone does not increase under local operators and classical communication, we get and thus we obtain a valid lower bound by estimating the entanglement of the output state. Hence if we want to bound the linear entropy of entanglement by seeing a certain value of a Bell inequality we use
Now let us turn to the steering case, where we assume that Alice’s side is uncharacterized while Bob obtains complete tomography. Then the data are given by the collection of unnormalized density operators for Bob with . In principle we can use the same method as for the fully device independent case by employ the trace-preserving local map only on one side and then bounding the linear entropy of entanglement of the output state.
However in this case we can do slightly better, since it is possible to bound the linear entropy of entanglement more directly on the original state . This is in similar spirit as the negativity of Ref. MB14 () and Ref. pusey (). Suppose we apply the same local, not necessarily trace-preserving, local map to the two copies