Paradoxes of measures of quantum entanglement and Bell’s inequality violation in two-qubit systems
We review some counterintuitive properties of standard measures describing quantum entanglement and violation of Bell’s inequality (often referred to as “nonlocality”) in two-qubit systems. By comparing the nonlocality, negativity, concurrence, and relative entropy of entanglement, we show: (i) ambiguity in ordering states with the entanglement measures, (ii) ambiguity of robustness of entanglement in lossy systems and (iii) existence of two-qubit mixed states more entangled than pure states having the same negativity or nonlocality. To support our conclusions, we performed a Monte Carlo simulation of two-qubit states and calculated all the entanglement measures for them. Our demonstration of the relativity of entanglement measures implies also how desirable is to properly use an operationally-defined entanglement measure rather than to apply formally-defined standard measures. In fact, the problem of estimating the degree of entanglement of a bipartite system cannot be analyzed separately from the measurement process that changes the system and from the intended application of the generated entanglement.
Quantum entanglement Sch35 (); Ein35 (), being at heart of Bell’s theorem Bell (), is considered to be an essential resource for quantum engineering, quantum communication, quantum computation, and quantum information Nielsen (). There were proposed various entanglement measures and criteria to detect entanglement. Nevertheless, despite the impressive progress in understanding this phenomenon (see a recent comprehensive review by Horodecki et al. Horodecki-review () and references therein), a complete theory of quantum entanglement has not been developed yet.
It is a commonly accepted fact that the entropy of entanglement of two systems, which is defined to be the von Neumann entropy of one of the systems, is the unique entanglement measure for bipartite systems in a pure state popescu (). However, in the case of two systems in a mixed state, there is no unique entanglement measure. In order to describe properties of quantum entanglement of bipartite systems various measures have been proposed. Examples include Horodecki-review (): entanglement of formation, distillable entanglement, entanglement cost, PPT entanglement cost, the relative entropy of entanglement, or geometrical measures of entanglement.
It should be stressed that classification of entanglement measures of mixed states and effective methods of calculation of such measures are among the most important but still underdeveloped (with a few exceptions) problems of quantum information open-problems ().
Here, we shortly review counterintuitive properties of some entanglement measures in the simplest non-trivial case of entanglement of two qubits.
2 Measures of quantum entanglement
We will study quantum entanglement and closely related violation of Bell’s inequality for two qubits in mixed states according to some standard measures:
in terms of ’s, which are the square roots of the eigenvalues of , where is the Pauli spin matrix and asterisk stands for complex conjugation. The concurrence is related to the entanglement of formation, , as follows Wootters ():
and is binary entropy.
(ii) The PPT entanglement cost, which is the entanglement cost Horodecki-review () under operations preserving the positivity of the partial transposition (PPT), can be given as Audenaert03 (); Ishizaka04 ():
in terms of the negativity:
(iii) The relative entropy of entanglement (REE) Vedral97a (); Vedral98 () is a measure of entanglement corresponding to a “distance” of an entangled state from separable states. Precisely, the REE can be defined as the minimum of the relative quantum entropy
in the set of all separable states , i.e.,
where denotes the closest separable state (CSS) to . Numerical problems to calculate the REE are shortly discussed in Appendix A.
which is given in terms of the eigenvalues of , where is a real matrix with elements , is the transposition of and are Pauli’s spin matrices. For short, we refer to as “nonlocality” (measure).
For any two-qubit pure state , the nonlocality is equal to the entanglement measures and :
It is seen that for this case the measures , and correspond to the relative entropy of entanglement and von Neumann’s entropy:
where is given in Eq. (2).
In the following we describe somewhat surprising properties of the entanglement measures for two-qubits in mixed states. For brevity, by referring to the entanglement measures, we also mean the nonlocality .
3 Ambiguity in ordering states with entanglement measures
The problem can be posed as follows:
Two measures of entanglement, say and , imply the same ordering of states if the condition Eisert99 ()
is satisfied for arbitrary states and . The question is whether this condition is fulfilled for all “good” entanglement measures.
In early fundamental works on quantum information, it is often claimed that good entanglement measures should fulfill this condition. For example, in Ref. Vedral97a () it was stated that: “For consistency, it is only important that if is more entangled then for one measure than it also must be for all other measures.”
For qubits in pure states, condition (10) is always fulfilled, since all good measures are equivalent. However, standard measures can imply different ordering of mixed states even for only two qubits. This was first shown numerically by Eisert and Plenio Eisert99 () by analyzing their results of Monte Carlo simulations of two-qubit states. The problem was then analyzed by others Zyczkowski (); Virmani (); Wei (); H4 (); H5 (); H6 (); H7 (); Ziman (); Kinoshita07 (); H8 ().
To our knowledge, the first analytical examples of two-qubit states violating condition (10) were given in Refs. H4 (); H5 (). In Ref. H6 (), to find analytical examples of extreme violation of Eq. (10), we applied the results of Verstraete et al. Verstraete01a () concerning allowed values of the negativity for a given value of the concurrence .
Note that the violation of condition (10) cannot be observed for pure states of two-qubit systems. By contrast, for three-level systems (the so-called qutrits), analytical examples of violation of the condition are known even for pure states Zyczkowski (); Virmani (); Wei ().
The property that ordering of states depends on the applied entanglement measure sounds counterintuitive. Nevertheless, it is physically sound, since states, which are differently ordered according to two measures, cannot be transformed into each other with 100% efficiency by applying local quantum operations and classical communication (LOCC) only. Virmani and Plenio Virmani () proved in general terms that all good asymptotic entanglement measures are either identical or have to imply a different ordering on some quantum states.
In Ref. H7 (), the three measures (the negativity, concurrence, and the REE) were compared and found analytical examples of states (say and ) for which one measure implies state ordering opposite to that implied by the other two measures:
There can be found other analytical examples of states exhibiting even more peculiar ordering of states according to these three measures. Examples include pairs of states for which a degree of entanglement is preserved according to one or two measures but it is different according to the other measures, e.g.:
The comparative analyses presented in Refs. H4 (); H5 (); H6 (); H7 () are not only related to a mathematical problem of classification of states according to various entanglement measures. They could also enable a deeper understanding of some physical aspects of entanglement.
3.1 Nonequivalent states with the same entanglement according to , and
Find analytical examples of *nonequivalent* two-qubit states and exhibiting the same entanglement of formation , the same PPT entanglement cost , and the same relative entropy of entanglement ?
As a first attempt to find such an example, let us compare two different pure states:
fulfilling the condition
which guarantees the same degree of entanglement according to the measures , and . However, states and can be transformed into each other by local operations. Namely, by applying local rotations, can be converted into ()
for which the negativity and concurrence are equal to . The same value is obtained also for , but this state can be transformed into by applying the NOT gate to each of the qubits. This shows that pure states are not a good example of states satisfying the conditions specified in Problem 2.
As a second attempt, let us compare two Bell diagonal states described by and with the same maximum eigenvalue . These states have the same entanglement according to the measures , and . However, as shown in Ref. H7 (), states and exhibit different nonlocality, i.e., violate Bell’s inequality to different degree. Specifically, the nonlocality for a Bell diagonal state is given by H7 ():
where subscripts correspond to cyclic permutations of . It is seen that violation of Bell’s inequality depends on all values of , while the entanglement measures , , and depend only on the largest value . Thus, states and , fulfilling the conditions and , have the same entanglement measures: , and , but the states are not equivalent as they exhibit different nonlocality, .
4 Ambiguity of robustness of entanglement
4.1 Maximally entangled pure states in lossy cavities
Let us analyze the following problem:
Which maximally entangled pure states are the most fragile or robust to decoherence of two qubits in lossy cavities?
where . State can be obtained from by applying Hadamard’s gate to the second qubit.
To address Problem 3, let us analyze two entangled qubits in a superposition of vacuum and single-photon states (so-called photon-number qubits) in a lossy cavity (or, equivalently, in two cavities). Then, one can apply the standard master-equation approach to describe the effect of radiative decay of cavities (i.e., zero-temperature reservoirs) on entanglement of two qubits according to the concurrence , negativity , and nonlocality H4 (). In Fig. 1, it is assumed that the qubits are initially in the MES for and the cavity damping rate is . By analyzing Fig. 1, one can conclude that entanglement decays in this model fulfill the inequalities:
It is worth noting that due to the Markov approximation assumed in the derivation of the master equation, our conclusions are valid for evolution times short in comparison to reservoir decay time , and much longer than correlation time of reservoir(s), i.e., , where is the initial evolution time. Thus, in this specific dissipation model, the most fragile to dissipation is according to the negativity , according to the concurrence , and according to the nonlocality . The results seem to be contradicting, but it should be remembered that measures , and describe different aspects of mixed states even if for pure states they coincide . Results of Refs. H4 (); H5 () clearly confirm the relativity of state ordering by , and . This example of Ref. H4 () was probably the first demonstration of this property in a real physical process.
4.2 Maximally entangled mixed states in lossy cavities
Here, we analyze decay of Werner’s states, which can be defined for as Werner89 ():
which is a mixture of the singlet state, , and maximally mixed state, given by , where is identity operator. Original Werner’s state can be generalized for mixtures of other Bell states with . Thus, one can define Werner-type state as follows ():
Werner’s states can be considered as maximally entangled mixed states (MEMS) of two qubits since the amount of entanglement of these states cannot be increased by any unitary transformation ishizaka00 () and they are maximally entangled (according to the concurrence) for a given value of linear entropy munro01 ().
Let us ask more specific question related to Problem 3:
Which MEMS are the most robust to dissipation in the discussed model of lossy cavities?
Even for such formulated question there is no simple answer. To show this we analyze the same model of decaying photon-number qubits in a lossy cavity (or cavities) as studied in Sect. 4.A, but for qubits initially in Werner’s states for and . Let us compare the decays of the negativity as shown in Fig. 2 and also described in detail in Table I in Ref. H4 (). It is seen that a given Werner state can be more robust to decay than another Werner’s state at short evolution times but, in turn, less robust at longer times. The differences between the negativity values for various states shown in Fig. 2 are not very large but still distinct.
5 Mixed states more entangled than pure states
Can two-qubit *mixed* states be more entangled than *pure* states according to some entanglement measure at a fixed value of another entanglement measure assuming for any state ?
It can be shown analytically that pure states are the upper bound for the negativity for a given value of the concurrence Verstraete01a (), as shown in Fig. 3(a), and the upper bound for the REE as a function of the concurrence Vedral98 (), as presented in Fig. 3(b). Similar conclusions can be drawn for, e.g., the nonlocality for a given value of the concurrence [see Fig. 3(c)], and the nonlocality as a function of the negativity.
Thus, it is reasonable to conjecture that pure states are the upper bound also for the REE, e.g., for a given value of the negativity. But it was shown in Refs. H7 (); H8 () that this conjecture is wrong [see Fig. 3(e)]. This property can be demonstrated analytically on the example of, e.g., the Horodecki state Horodecki-review () defined as a mixture of the maximally entangled state [e.g., the singlet state ] and a separable state orthogonal to it (e.g., ):
where . The negativity and REE for the Horodecki state are equal to
where and , which corresponds to point in Fig. 4. These inequalities were shown analytically by expending and in power series of () for values close to 0 (1). Moreover, mixed states corresponding to blue region in Fig. 4, for which the inequality in Eq. (26) holds, can be obtained by mixing the Horodecki state with a separable state closest to H7 ():
where , and . The closest separable state is given by ():
where and . With this choice of , parameter is just the negativity of . States corresponding to blue region in Fig. 4 can be obtained as special cases of state for in the range and proper values of . Thus, it is seen that there are mixed states for which the REE is greater than that for pure states at least in the range . Later, in Ref. H8 (), it was shown that the generalized Horodecki states exhibit this property in slightly larger range as shown by yellow region in Fig. 4. There is some evidence H8 () that the upper bound of the REE as a function of the negativity is likely to be given by these states.
Recently, we also analytically demonstrated Horst () that the entanglement REE for a given nonlocality for mixed states exceeds that for pure states [see Fig. 3(f)]. Moreover, this effect occurs in the larger range of abscissa values in comparison to the dependence of the REE on the negativity, as seen by comparing Figs. 3(e) and 3(f).
In this short review, we presented a few intriguing properties of some standard entanglement measures for two qubits. Our examples include a comparison of the negativity corresponding to the Peres-Horodecki criterion Peres (); Horodecki96 (), the Wootters concurrence Wootters (), and the relative entropy of entanglement of Vedral et al. Vedral97a (). Moreover, the predictions of these measures were also compared with the Horodecki measure Horodecki95 () of the violation of Bell’s inequality, referred here to as “nonlocality”.
We discussed the following three counterintuitive properties of entanglement measures: (i) entangled states cannot be ordered uniquely with the entanglement measures, which also implies that (ii) fragility or robustness of entanglement of dissipative systems cannot be uniquely classified by entanglement measures, and (iii) there are two-qubit mixed states, which are more entangled (according to the REE) than pure states for a given negativity or nonlocality.
It is well known that there is no unique entanglement measure for mixed states. But the relativity of entanglement measures and its implications are more counterintuitive. Our demonstration might indicate that operational approaches to the quantum entanglement problem are more meaningful rather than standard approaches based formally-defined measures. We find the problem of defining operational entanglement measures analogous to operational approaches to the quantum phase problem 111Noh et al. in Ref. Noh92 () wrote: “There has been a good deal of discussion in the past of the most appropriate dynamical variable to represent the phase of a quantum field, and many candidates have been studied. Our analysis suggests that this question may not have a general answer with respect to the measured phase operators, because different measurement schemes lead to different operators. As in many other quantum-mechanical problems, it seems that questions about the value of a dynamical variable cannot be divorced from the measurement process that generates the ensemble.” posed by Noh et al. Noh91 (); Noh92 (). The idea is to define entanglement (or phase) measures in terms of what actually is, or can be, measured.
We hope that the discussed problem of non-unique ordering of states according to formally-defined entanglement measures can stimulate investigations of operationally-defined measures oriented for some specific experiments.
Acknowledgements The work was supported by the Polish Ministry of Science and Higher Education under Grant No. N N202 261938.
Appendix A Notes on calculation of the REE
The concurrence, negativity and nonlocality can be calculated easily. By contrast, there has not yet been proposed an efficient method to calculate the REE for arbitrary mixed states even in the case of two qubits Eisert (). Analytical formulas for the REE are known only for some special sets of states with high symmetry (see Horodecki-review (); H9 () and references therein). Thus, usually, numerical methods for calculating the REE have to be applied Vedral98 (); rehacek (); doherty ().
It is a long-standing problem, posed by Eisert Eisert (), of obtaining an analytical compact formula for the REE for two qubits. The problem is equivalent to finding the closest separable state for a given entangled state . In Ref. H9 (), a few arguments were given indicating that this problem, probably, cannot be solved analytically for arbitrary states. Nevertheless, there exists a solution to the inverse problem of finding an analytical formula for for a given closest separable state as derived by Ishizaka et al. Ishizaka03 (); H9 ().
The complexity of the problem can be explained (see, e.g., Vedral98 ()) by virtue of Caratheodory’s theorem, which implies that any separable two-qubit state can be decomposed as
where the th () qubit pure states can be parametrized, e.g., as follows
and with . Thus, the minimalization of the quantum relative entropy , given by Eq. (5), with described by Eq. (31), should be performed over real parameters. Usually (see, e.g., Refs. Vedral98 (); rehacek ()), gradient-type algorithms are applied to perform the minimalization. Řeháček and Hradil rehacek () proposed a method resembling a state reconstruction based on the maximum likelihood principle. Doherty et al. doherty () designed a hierarchy of more and more complex operational separability criteria for which convex optimization methods (known as semidefinite programs) can be applied efficiently. One can also use an iterative method based on Ishizaka formula Ishizaka03 (); H9 () for the closest entangled state for a given separable state in order to find the closest separable state for a given entangled state. Our algorithms for calculating the REE are based either on the latter method or on a simplex search method without using numerical or analytic gradients.
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