Topology of Entanglement Evolution of Two Qubits
The dynamics of a two-qubit system is considered with the aim of a general categorization of the different ways in which entanglement can disappear in the course of the evolution, e.g., entanglement sudden death. The dynamics is described by the function , where is the -dimensional polarization vector. This representation is particularly useful because the components of are direct physical observables, there is a meaningful notion of orthogonality, and the concurrence can be computed for any point in the space. We analyze the topology of the space of separable states (those having ), and the often lower-dimensional linear dynamical subspace that is characteristic of a specific physical model. This allows us to give a rigorous characterization of the four possible kinds of entanglement evolution. Which evolution is realized depends on the dimensionality of and of , the position of the asymptotic point of the evolution, and whether or not the evolution is “distance-Markovian”, a notion we define. We give several examples to illustrate the general principles, and to give a method to compute critical points. We construct a model that shows all four behaviors.
Entanglement is one of the most intriguing aspects of quantum physics and is known to be a useful resource for quantum computation and communication (1); (2). However, its structure and evolution in time are not fully understood even for simple systems such as two qubits where a relatively computable entanglement measure, the Wootter’s concurrence , is available (3); ?; ?. The difficulty resides in the high dimensionality of state spaces (fifteen for two qubits) and the non-analyticity of the definition of the entanglement measure .
In the presence of external noise, pure states become mixed and entanglement degrades. These are distinct but related issues. The completely mixed state (density matrix proportional to the unit matrix) is separable: it has . Pure states, on the other hand, can have any value of between and inclusive. Purity (measured, for example, by the von Neumann entropy) tends to decrease monotonically and smoothly with time under Markovian evolution. The same is not true for entanglement. Apart from the expected smooth half-life (HL) decaying behavior, the sudden disappearance of entanglement has been theoretically predicted and experimentally observed (6); (7); ?; ?. Widely known as entanglement sudden death (ESD), this non-analytic behavior has been shown to be a generic feature of multipartite quantum systems regardless of whether the environment is modeled as quantum or classical (10); (11); (12); ?; ?; (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); ?; (26). While the monotonic decrease of is usually associated with Markovian evolution, non-Markovian evolution can also lead to entanglement sudden birth (ESB). It is believed to be related to the memory effect of the (non-Markovian) environment (27); ?; ?; (10); (30); ?; (24); ?; (26). Although most investigations have been focused on two-qubit systems and we will also focus on this case, ESD and ESB have been shown to exist in multi-qubit systems, and even in quantum systems with infinite dimensional Hilbert spaces, such as harmonic oscillators (32); (33); ?; ?; (36); (37); (38); ?; (40); (41).
Our aim in this paper is to formulate the problem of the evolution of entanglement in two-qubit systems in the polarization vector space, and to show that this formulation leads naturally to a categorization of entanglement evolutions into four distinct types, generalizing and making precise the concepts of HL, ESD, and ESB behaviors. It turns out that these categories are consequences of certain topological characteristics of a model. To show this, we proceed as follows. Sec. II characterizes in detail two manifolds in the polarization vector space: the manifold of admissible physical states and the manifold of separable states. Sec. III presents the concept of a dynamical subspace - a manifold that is associated with a physical model, and then gives several concrete examples of models of increasing complexity. In Sec. III we also show how to compute critical points: parameter values that separate one behavior from another. In Sec. IV, we prove that our categorization scheme is exhaustive. Sec. V presents the final results and discussion.
Ii Entanglement in Polarization Vector Space
The universality of the various entanglement behaviors suggests that they are derived from some structural property of entanglement in the physical state space, and that the system dynamics can be viewed as a probe of that property. To state this property precisely, we need to first characterize the space of all admissible density matrices , or equivalently, the space of all admissible polarization vectors.
For two qubits, the polarization vector is defined by the equation
where is the unit matrix and the are the generators of , satisfying
For our purposes the are most conveniently chosen as
where acts on the first qubit and acts on the second qubit. and sum over the unit matrix and the Pauli matrices , and . Thus, in Eqs. 1 and 2, is regarded as a composite index of and , but the term is singled out. This space has the usual Euclidean inner product (which corresponds to the Hilbert-Schmidt inner product on the the density matrices), and the inner product induces a metric and a topology in the usual fashion. The components of are physical observables and can be calculated by . For example, the average value of the z-component of the spin of the first qubit is . The six components , , , , and represent physical polarizations of spin qubits. The other nine components (, etc.) are inter-qubit correlation functions. The most common name for is “polarization vector”, but “coherence vector” and “generalized Bloch vector” are also in use. We note that different normalizations in Eqs. 1 and 2 are used in the literature (42); ?; (44); (45); (46); (47).
The generators satisfy
where is totally anti-symmetric and is totally symmetric. These structure constants can be found in Appendix A.
Eq. 1 holds for a -level system. It has an obvious generalization to -level systems; the just become the generators of . For , is the usual Bloch vector in a real -dimensional vector space. It is important to stress that the correspondence between and is one-to-one; they give completely equivalent descriptions of the physical system. Certain physical concepts have geometric interpretations when stated in terms of ; as we shall see below. This is not so true of . In our opinion, is the more convenient quantity for most purposes. has been the traditional language in which to describe mixed states, but some experimental groups now favor (48); (49).
We shall refer to the set of all admissible as , the state space. What shape does have? Eq. 1 guarantees that is Hermitian and has unit trace. To guarantee that is positive (all its eigenvalues are non-negative), we also need the condition that all coefficients of the characteristic polynomial are non-negative (50). Note by definition.
For two-qubit systems there are four of them, which are
Note that is trivially satisfied for all density matrices.
For one qubit , is the usual Bloch vector, and only the constraint applies. Thus the positivity requirement is that and is the familiar -dimensional spherical volume. For the -qubit case that we are concerned with, there are cubic and quadratic inequalities to be satisfied, so the surface that bounds is not so simple. The main point, however, is that is convex: the line joining any two points in is also in . This follows from the convexity argument for : if and are positive, then so is for all . This argument clearly also holds for .
All of the positivity requirements can be written in terms of but the higher-order ones are fairly complicated. The requirement Tr is of particular interest, since it has a simple expression in terms of
Hence the vectors in lie within a sphere of radius . Technically, is a -dimensional manifold with boundary. We will follow physics usage and also employ the term “space” for , though of course it is not closed under vector addition. Note that pure states satisfy Tr , so the pure states are a subset of the -sphere in with . To be more specific, the two-qubit pure states are of measure zero on that sphere since they can be parametrized by real parameters
An overall phase has been dropped in writing since it does not appear in .
Further insight into the shape of can be gained by noting that must be invariant under local unitary transformations (rotations of one spin at a time), which means that has cylindrical symmetry around the single-qubit axes. This is verified by making some -dimensional sections of with exactly two components of non-zero. In contrast to Ref. (51) where a different basis was used (generalized Gell-Mann matrices), we find only two types of shapes, as shown in Fig. 1 and tabulated in Table 1. Using the structure constants and , it can be shown that the square and disc sections are the only possibilities along the plane. When commutes with the section is a square and anticommutes with the section is a circular disc. Details can be found in Appendix A.
The discs correspond to the local rotations between single-qubit-type axes, such as the section. If, on the other hand we rotate from a definite polarization state of qubit 1 to a definite polarization state of qubit 2, we find a square cross-section; examples are the or sections. Rotations of that mix single–qubit-type and correlation-type directions can be of either shape; the section is square, while the section is a disc. Finally, rotations between correlation-type directions can have either shape. The section corresponds to a local rotation of qubit 2; hence it is a disc. Rotations involving both qubits, such as that which generates the section, generally give square sections.
We may conclude that is a highly dimpled ball, perhaps most similar in shape to a golf ball. Its minimum radius is and its maximum radius is .
Since our aim is to quantify entanglement in , we need an entanglement measure. We will employ , the concurrence of Wootters (4). The concurrence varies from for separable states to for maximally entangled state, i.e., the Bell-like states. It is defined as , and
where are the square roots of the eigenvalues of the matrix arranged in decreasing order and
is a spin-flipped density matrix. is the complex conjugate of the density matrix . It is not possible to write the function in a simple explicit form unless further restrictions on apply (52), but it is clear from the form of the continuous function and the presence of the function that is a continuous but not an analytic function of .
We next consider , the manifold of separable states, which we define as those for which the concurrence vanishes: . is a subset of and is the set of entangled states. includes the origin since and . Since is continuous, actually includes a ball of finite radius about the origin: it can be shown that if , then (53). Thus the manifold of separable states has finite volume in : is also -dimensional. We will also refer below to the interiors and boundaries of and and denote these by Int, , Int, and . Since the various sets we encounter in this paper are not linear subspaces, we need the general topological definitions of “boundary”, “interior” and “dimension”. These may be found, for example, in Ref. (54).
is also a convex set. What else can we say about the shape of ? It is easily seen that the surface of , like that of , is rather non-spherical. Indeed along any of the basis vector directions. Coupled with the fact that is convex, we see that must contain a large hyperpolygon with vertices at , , etc. is invariant under local rotations, so it has the same hyper-cylindrical symmetry as . Again we may consider -dimensional sections in order to understand the shape of the surface. Two examples are shown in Fig. 1.
A simple-sampling Monte Carlo study shows that there are more entangled states than separable states in . The details can be found in Appendix B.
Iii dynamical evolution in
The dynamical evolution or trajectory of a quantum system is a function with and . The initial point is and, in the cases of interest here, the trajectory approaches a limiting point as and we can define . The entanglement evolution is the associated function . . For studies of decoherence the main interest is in entanglement evolutions such that and , i.e., the system starts in an entangled state and ends in a separable state. Four distinct categories of entanglement evolution of this type have been seen in model studies (56). They are shown in Fig. 2. These four categories are topologically distinct, as may be seen by considering the set . In category , is the null set; in category , is a single infinite interval; in category , is a set of discrete points; in category , is a union of disjoint intervals.
These categories also reflect how the trajectory traverses and . Entanglement evolutions in category approach the boundary of separable and entangled regions asymptotically from the entangled side. The trajectories never hit while the decrease in entanglement may or may not be monotonic, as seen in Ref. (57). Entanglement evolutions in category bounce off the surface of at finite times but never enter . Overall, entanglement diminishes nonmonotonically. Entanglement evolutions in category enter at finite time and entanglement stays zero afterwards. This is the typical ESD behavior. Entanglement evolutions in category give ESB: after ESD, entanglement suddenly appears after some dark period.
We shall focus on models with associated linear maps , i.e., . More general non-linear models may be contemplated, but they seem to have unphysical features (58). It is known that is completely positive (CP) if and only if there exists a set of operators such that (2)
We require to be trace preserving so that it maps density matrix to density matrix. This condition is equivalent to the completeness condition . In terms of the polarization vector, the dynamics is described by an affine map acting on the initial polarization vector , i.e.
where is a real matrix and is a real vector (47). is zero for all time only when is unital, i.e., it maps to (in terms of , the unital property means that maps identity matrix to identity matrix). and .
Coherent dynamics is described by unitary transformations on the density matrix (single Kraus operator). The dynamical map is then linear which translates to orthogonal transformations acting on the polarization vector . Decoherent dynamics (multiple Kraus operators) is characterized by the nonorthogonality of the transfer matrix . Markovian dynamics is conventionally defined by possessing the semigroup property , which translates to
We shall adopt a slightly different definition of Markovianity for the present paper. An evolution will be said to be “distance Markovian” if is a monotonically decreasing function. Note “distance Markovian” is a weaker condition than Markovian, though the two are usually equivalent. Given the semigroup property Eq. 12 and Eq. 13, we have
since all eigenvalues of have their norms in the range to , i.e., cannot increase the purity of the quantum state.
Any model of an open quantum system defines a set of possible dynamical evolutions. This is done by specifying the equations of motion, which give , and the initial conditions, which give . We define the dynamical subspace of a model as the set of all trajectories allowed by the set of initial conditions and the equations of motions. Eq. 11 shows that, as long as the set of all initial conditions is a linear space (the usual case), then is a linear space intersected with : we first choose a basis that spans the set of all possible , then evolve this basis according to Eq. 11, giving a linear subspace in the space of all . A precise and general definition of is given in Appendix C. The set of admissible is then given by intersecting this linear subspace with . is a manifold of any dimension from to in the two-qubit case. We note could be smaller than . This happens when both and are expandable by identity and a true subalgebra of . is then equal to the number of independent elements in . For example, if is a two-qubit “X-state” and the dynamics can be described by the action of Kraus operators in the X-form, the dynamical subspace will be -dimensional (59).
It is the nature of the intersection of with and the position of relative to that determines the categories of entanglement evolution of a model.
The aim of the remainder of this paper is to show how to determine the topological structure of and the position for various illustrative models of increasing complexity, and then to deduce the possible entanglement evolutions from this information. We note that in general can be determined without fully solving the dynamics. Thus it is possible to gain qualitative information of the entanglement evolution of the model with simple checks.
Our first model consists of two qubits (A and B) with a Heisenberg interaction and classical dephasing noise on one of the qubits. The Hamiltonian is
where is a random function. This is a classical noise model. To compute we need to average over a probability functional for , which we will specify more precisely below. Note that the manifold spanned by is decoupled from the manifold spanned by under the influence of this Hamiltonian. Thus if the initial density matrix lies in one of the two subspaces, the four-level problem decouples into two two-level problems and we can use the Bloch ball representation to visualize the state space and entanglement evolution.
Take the initial state to be in span for example. The dynamical subspace is a -dimensional ball, as shown in Fig. 3. This makes it relatively easy to visualize the state and entanglement evolution. However, note that the center of is not the state . In fact, all the points on the -axis belong to because every neighborhood of any of these points contains points for which .
The square roots of the non-zero eigenvalues of are , where and are the spherical coordinates of the ball, and is the spin-flipped density matrix, as in Eq. 9. The concurrence is given by
The maximally entangled states are on the equator and the separable states are on the -axis, as seen in Fig. 3. The concurrence has azimuthal symmetry and is linear in the radial distance from the -axis. The separable states in form the -dimensional line that connects the north and south poles of .
The key point is that has a lower dimension than itself. Now consider the possible trajectories with on the -axis. No function with continuous first derivative can have a finite time interval with . The trajectories either hit the -axis at discrete time instants which puts them in category , or approach the -axis asymptotically which puts them in category .
Let us specify in more detail to demonstrate how those two qualitatively different behaviors are related to Markovianity. Qubit A sees a static field while qubit B sees a fluctuating field . All fields are in the -direction. We will take the noise to be random telegraph noise (RTN): assumes value and switches between these two values at an average rate . RTN is widely observed in solid state systems (60); (61); (62); (63); ?.
For this dephasing noise model, the above-mentioned decoupling into two 2-dimensional subspaces occurs. In the block labelled by we find
wherer is the Pauli matrix in the subspace.
This Hamiltonian can be solved exactly using a quasi-Hamiltonian method (46). The time-dependent decoherence problem can be mapped exactly to a time-independent problem where the two-value fluctuating field is described by a spin half particle. The quasi-Hamiltonian is given by
where are the Pauli matrices of the noise “particle”. is the generator in the space.
The transfer matrix is given by
is the dephasing function due to RTN and it describes the phase coherence in the x-y plane (26). since the dynamics is unital. Note has qualitatively different behaviors in the and regions, as the trigonometric functions become hyperbolic functions (46).
Taking as initial state, the effective Bloch vector is
The state trajectory is fully in the equatorial plane, as seen in Fig. 4. The dephasing function modulates the radial variation and the static field provides precession.
and the dynamics is distance Markovian if ( being monotonic). In this parameter region, can be approximated by and the dynamics is approximately Markovian as well. Thus we do not need to distinguish Markovian and distance Markovian in this model. In the Markovian case, the monotonicity of gives rise to spiral while in the non-Markovian case the state trajectory periodically spirals outwards with frequency . In both cases, the limiting state is the origin of the ball.
The concurrence evolution is given by
Thus Markovian noise gives rise to entanglement evolutions in category while non-Markovian noise gives rise to that in category . Entanglement evolutions in the other two categories can not occur due to fact that .
In the previous section we saw that dynamical subspaces spanned by two computational basis states does not possess the property . A natural question is whether simply increasing the number of basis states helps. This can be done by choosing a Hamiltonian that connects only the triplet states in the original Hilbert space. One example would be
as in Ref. (65), where the Heisenberg coupling has time dependence and is modeled as a classical random process.
Note this Hamiltonian conserves the total angular momentum of the two qubits. As a result, the triplet space spanned by is decoupled from the singlet space. Thus the dynamical subspace has its basis elements in if we choose the initial state to be in the triplet subspace.
Using Gell-Mann matrices as the elements of algebra, the state space is a subset of a ball in (42).
The are linearly related to the previously defined .
The square roots of the eivenvalues of are
Thus the set of separable states is composed of two geometric objects: which is in and
which is in . In addition to a concurrence-zero hyperline, we have a hyperplane in . Hence and this model only displays entanglement evolutions in categories and .
Introducing extra dimensions thus helps to form non-zero volume of in but a Hilbert space spanned by three of the four computational basis states is still not enough. and both avoid the region near the fully mixed state , where most separable states reside (66); ?. This region is included in the dynamical subspaces in the next two sections.
Note that for and , the symmetry of the Hamiltonian and the specification of the initial conditions allow us to fully describe the dynamical subspace without explicitly solving the dynamics. This feature can be seen in the more complicated models in the following sections as well: entanglement evolution categories, as a qualitative property of the system dynamics, can be determined from symmetry considerations of the model (dynamics plus initial condition), position of in and the memory effect of the environment.
Yu and Eberly considered a disentanglement process due to spontaneous emission for two two-level atoms in two cavities. In this case the decoherence clearly acts independently on the two qubits. Nevertheless they found that ESD occurs for specific choices of initially entangled states (11).
The decoherence process is formulated using the Kraus operators
and and .
It is possible to choose initial states such that the density matrices have the following form for all
where . Here the parameter determines the initial condition.
The two-qubit entanglement is
In the polarization vector representation, the dynamics defined by Eq. 22 can be given explicitly by the transfer matrix and the translation vector
The non-zero components are
and the limiting state is
where the coordinates are .
This shows that the dynamical subspace is a 3-dimensional section of where the non-zero components are , and but also and , such that it can also be visualized in three dimensions. Interestingly, the limiting state due to spontaneous emission is on the boundary of set of separable states, and the purity of the state increases with time.
Although we have so far fully solved the system dynamics, for the purpose of describing , it is enough to know that are the basis of and that the Kraus operators preserve the equalities , from the initial conditions. is then determined from the positivity condition of the density matrix.
The positivity condition for the density matrix is given by
where and . At , . The positivity constraint gives rise to the range of the possible initial conditions, parametrized by .
The square roots of the eigenvalues of are
The ordering of the ’s can change during the course of an evolution. When is the largest one finds which is helpful in dtermining .
is a tetrahedron with vertices at , , , and , as seen in Fig. 5. is a hexahedron that shares some external areas with . The -dimensional section of with is shown in Fig. 6. On the other hand, if the section is done with , we get a upside down triangle made of separable states, as shown in Fig. 7.
Yu and Eberly showed that a sudden transition of the entanglement evolution from category to category is possible as one tunes the physical parameter . This phenomenon can be easily understood in our formalism, as seen in Fig. 8. The curvature of the state trajectories vary as the initial state changes. Thus there is a continuous range of initial states parametrized by whose trajectories enter within finite amount of time and also a continuous range of initial states whose trajectories never enter Int. Note Fig. 8 is a schematic drawing since the true state trajectories are truly three dimensional.
To be more quantitative, the transition between the category and category behaviors in the YE model could be determined by examining the angle between and the tangent vector of the state trajectory in the long time limit. We denote the limiting tangent vector by and it is given by
The relevant in the YE model is a plane passing defined by the following three points: , and . It is parametrized by
where is the unit normal of the plane pointing into the separable region . Note is proportional to and its sign tells us whether the state trajectory approaches from the separable region or the entangled region. Since , falls in the range and both the ESD and HL behaviors are possible.
i.e., , gives rise to the critical trajectory which approaches along . When , i.e., , the state trajectory approaches from the entangled region and we get entanglement evolutions in category . These trajectories are represented by the brown curves in Fig. 8. On the other hand, when , i.e., , the state trajectory approaches from the separable region and we get entanglement evolutions in category . These trajectories are represented by the black curves in Fig. 8.
The key point about the model is that the limiting state : it is on the boundary of the entangled and separable regions. That is why entanglement evolutions in both category and category are possible.
Here we present a physically motivated dynamical subspace where the dynamics satisfies the following conditions: 1) the two qubits are not interacting; 2) the noises on the qubits are not correlated; 3) the effect of dephasing and relaxation can be separated. This model shows all four categories of entanglement evolution.
For this model the two-qubit dynamics can be decomposed into single-qubit dynamics (26). The extended two-qubit transfer matrix is where
is the extended transfer matrix of individual qubits. The top left entry describes the dynamics of and is there only for notational convenience. Here describes dephasing process, is the longitudinal relaxation rate and is static field in the direction that causes Larmor precession. and if dephasing occurs. Note this dynamical description of decoherence is completely general as long as one can separate dephasing and relaxation channels.
The dynamical subspace in this model is a specially parametrized -dimensional section of the full two-qubit state space . Only the components , , , and are non-zero and we further have constraints and . We thus use as independent parameters and the state space can be visualized in three dimensions. This dynamical subspace has been previously considered in Ref. (56) and we will call it .
The fact that relies on judicious choice of the initial states . In Ref. (26), more general initial states are considered such that is expanded into a dynamical subspace with elements in the algebra, i.e., .
Note is conserved in . The positivity of the density matrix requires
The concurrence is given by
Separable states form a spindle shape on top and entangled states form a torus-like shape on bottom with triangular cross sections. A section along is shown in Fig. 6.
We have thus fully described the entanglement topology of and now we construct entanglement evolutions that induce . A model similar to that of Eq. 14 that satisfies the three conditions is
Note the RTN has both dephasing () and relaxation () effects in this case. Situations with at intermediate qubit working point (with the presence of both dephasing and relaxation noise) have been considered in Ref. (56). Here we choose the initial state to be the generalized Werner state (55)
is a Bell state.
The state trajectory is then given by
Similarly, if the Werner state derived from the Bell state
is used as initial state, the state trajectory is
In both cases, the evolution of the concurrence is