Probing the structure of entanglement with entanglement moments
We introduce and define a set of functions on pure bipartite states called entanglement moments. Usual entanglement measures tell you if two systems are entangled, while entanglement moments tell you both if and how two systems are entangled. They are defined with respect to a measurement basis in one system (e.g., a measuring device), and output numbers describing how a system (e.g., a qubit) is entangled with that measurement basis. The moments utilize different distance measures on the Hilbert space of the measured system, and can be generalized to any -dimensional Hilbert space. As an application, they can distinguish between projective and non-projective measurements. As a particular example, we take the Rabi model’s eigenstates and calculate the entanglement moments as well as the full distribution of entanglement.
pacs:03.65.Ta, 03.65.Ud, 03.67.Mn
Quantifying entanglement has been of interest since Bell showed that this uniquely quantum feature was available for experimental verification Bell (1964); ?; ?. Since Bell, we have seen an explosion of potential applications in quantum information Ekert (1991) and computation Horodecki et al. (2009) as well as a whole body of theory to address the quantification of entanglement Plenio and Virmani (2007). There are many measures of entanglement for pure states and mixed state Plenio and Virmani (2007) which take as input a state and outputs a number telling you, very roughly speaking, how entangled a state is. However, these measures just tell one if a state is entangled, but not how it is entangled: two very different states can give the same number. To address this, we define a new set of functions called entanglement moments. (While we call these entanglement “moments”, they are not moments in the usual sense of distributions.) These quantities can tell us not only if and by how much a state is entangled but also how the distribution of entanglement looks by telling us how “clumpy” our distribution is.
For example, if we have a qubit entangled with another system and we make measurements on that other system, we will get a distribution of qubit states on the Bloch sphere. Two such examples are shown in Fig. 1. Note that in the first case, the distribution is centered around the north and south pole; while in the second case, the distribution is more evenly distributed about the sphere. We would like a measure that can distinguish these two instances – both of which have the same entanglement as given by the usual entanglement measures such as concurrence Wootters (1998); Rungta et al. (2001).
As an application, the property of entanglement moments to describe how the system is entangled allows them to characterize measurements from weak to strong/projective measurements Zurek (1981); ?. This uses the prescription for quantum measurement where the apparatus is treated quantum mechanically, becoming entangled with the system and mediating the collapse of the wave function Von Neumann (1955). This phenomenon has been exploited to understand the measurement process in the lab in terms of finite strength quantum measurement (see for instance Brune et al. (1996); ?; ?). In this situation, a measuring device is entangled with another system, and making measurements on the device indirectly probes the second system in what may be a non-projective way. Considering the apparatus and system, a projective measurement corresponds to a very “clumpy” distribution in the system’s Hilbert space while a non-projective measurement would be more evenly distributed. The entanglement moments can tell the difference between these two distributions and hence between projective and certain non-projective measurements.
In this letter, we first define entanglement moments for two systems and where is the measured system. Then, without specifying the system doing the measurement, we analyze the entanglement moments when our measured system is a qubit restricted to on the Bloch sphere and when is the entire Bloch sphere. The analysis generalizes to when is an arbitrary -dimensional Hilbert space in addition to the qubit case (see section 1 of the supplement sup ()). We then analyze the expressions with some informative examples.
Finally, to illustrate a physically relevant application of entanglement moments, we analyze the Rabi model Rabi (1936). This model shows up in many areas of physics including but not limited to circuit QED Niemczyk et al. (2010), cavity QED Schiró et al. (2012); Englund et al. (2007), photonics Crespi et al. (2012), and flux qubits Forn-Díaz et al. (2010). It comes into play when a qubit and a harmonic oscillator interact, and hence it finds its way into many of the approaches to quantum computation Pellizzari et al. (1995). While entanglement in the Rabi model has been studied before Fang and Zhou (1996), we show how the entanglement moments track the distribution of qubit states in particular eigenstates of the system – demonstrating how the moments discriminate between projective and non-projective measurements. The results are obtained numerically from the exact solution recently found by Braak Braak (2011). In addition, the distribution of qubit states shows one way in which the Rabi model can give qualitatively different results from the Jaynes-Cummings model such as the non-monotonic behavior of entanglement with respect to the interaction strength (seen in Fig. 5).
In order to address this question of “how” two systems can be entangled, we lift one of the requirements of entanglement measures: that the measure must be invariant under local unitary transformations Plenio and Virmani (2007). To understand why we need to lift this requirement in order to get at the nature of the entanglement, consider a qubit coupled to any system with . Given any basis for , , we can always perform a unitary operation, , on the Hilbert space of such that and . This unitary operation has destroyed the information describing how the basis is entangled with our qubit. While it is true that only two states at any given time are entangled with a qubit, rarely can an experiment know what those two states are a priori. On the other hand, we must take into account the converse of this: We can rotate and into any two vectors (such that ). Thus, the choosing of a basis must have some physical relevance and hence we call it the measurement basis.
To construct these entanglement moments, consider systems and and let be the measurement basis in system . We can write a general state vector as
where are unnormalized vectors in the Hilbert space of . Defining the weight of the vector by , we write the expression for th entanglement moment as
where for the normalized ,
and the quantity normalizes the maximal value of to unity – we will specify its value for specific cases later. The quantity is a distance function on the Hilbert space of system . If , we get the Hilbert-Schmidt distance measure (which leads to the Fubini-Study metric), and if we let , we get the trivial distance measure. The presence of the distance measure is to quantify how system changes as we measure system ; if system changes upon measurement of system , we know they are entangled. Therefore, a state is separable if and only if each entanglement moment is zero.
For the specific value , we actually reproduce I-Concurrence Rungta et al. (2001) in general (or just concurrence in the case of a qubit):
where is the reduced density matrix of system and is the order-2 Rényi entropy Bengtsson and Zyczkowski (2006). Thus, the quantity is invariant under all local unitary transformation. Not only is this quantity I-Concurrence, but it begins with a clearer, geometric, and intuitive definition (Eq. (2)).
To illustrate what happens when , we first assume that not only is system on the Bloch sphere but that our states are constrained to be on the great circle defined by the -axis. (This example is illustrative: we consider the entire Hilbert space in Eq. (10).) We can define the distribution of the entanglement such that is the probability distribution of states in the Hilbert space of given measurements in with a particular basis (each corresponds to a particular state in Hilbert space). This allows us to rewrite the entanglement moments as
(In the case of a countable number of ’s, is a sum of delta functions.) In this representation, we know the exact form of the distance measure . With simple trigonometric identities, it can be shown that
Substituting this into Eq. (5) and using the fact that we have normalized such that , we obtain
We can read off the normalization
These entanglement moments are picking up the features of this distribution in terms of its Fourier components – the distance functions are diagonal in this basis.
This expression gives us insight into our original expression for entanglement moments. First, the th moment decreases from unity for each Fourier coefficient up to the th that is non-zero. Corollary to this, since is invariant under local unitary transformations, the norm of the first Fourier component will remain the same no matter what measurement basis is chosen (or in the general case, the occupation in the first harmonic will remain the same; see section 1 of the supplement sup ()). By the uncertainty principle, highly localized (separable) states have a large distribution in frequency space, so such states will have large numbers of suppressed entanglement moments.
Consider the equation for entanglement moments Eq. (2). As we increase , the distance measures interpolate between points on the circle to the trivial distance measure (every point is a distance away from every other point). If we now consider a state localized near the north and south poles, the points near the north pole are roughly a distance from all points near the south pole for all distance measures . Thus, each decreases solely because the normalization decreases. In this way, a decreasing of the moments is indicative of there being more than one “clump” in the entanglement distribution. Hence, if measuring system corresponds to a projective measurement on , one should see the entanglement moments all decrease.
For further illustration, consider two distributions on :
where is an arbitrary parameter, and . These distributions are chosen such that they normalize to unity, and for every , they give the same entanglement . While on rather than the entire Bloch sphere, they are qualitatively like the two distributions in Fig. 1. The first is a projective measurement, interpolating between the unentangled state at and equal probable projection as . The second is a state localized around , delocalizing as increases to eventually cover the whole circle (it is in fact the Green’s function of the heat equation). Plotting the higher entanglement moments in Fig. 2, we see a stark contrast. In the first all moments get smaller as we increase , and in the second they get larger. While we have explained this in terms of Eq. (2), consider now the mode decomposition Eq. (7): the higher modes in are exponentially suppressed; while in , half of the higher modes stay constant.
If we know our system is highly entangled, we can use the moments to tell if measuring system will correspond to a projective measurement on system . To illustrate this, we step from to the entire Bloch sphere . The results are
(The generalization to an -dimensional Hilbert space is found in section 1 the supplement sup ().) We see that instead of Fourier coefficients, we now deal with harmonics: . To showcase this analysis on the sphere, Fig. 3 shows the entanglement moments for three distributions that all have the same, maximal entanglement as measured by concurrence (i.e. ): evenly distributed about the sphere, evenly localized along the equator, and localized to the north and south poles. Note how in each case, the higher entanglement moments indicate how localized our state is. The highly localized state at the north and south pole asymptote down to , indicating it is in fact localized to two points. For the distribution localized to the equator, it dips down, showing that it is localized, but rises back up to asymptote to 1. The reason it rises is that no state in its distribution is localized to a set of measure zero – if we go to our original expression for entanglement and let , then if and are not the same and it is 1 if they are. So if we have such an even distribution of states, just by the normalization of our states: .
For a more in depth example, we consider a qubit and a harmonic oscillator described by the Rabi Hamiltonian
where () is the annihilation (creation) operator for the harmonic oscillator and are the Pauli matrices for the qubit.
If we specify our measurement basis as the eigenbasis for the operator , we can write a vector in this Hilbert space as
where is a vector on the Bloch sphere. This set of vectors can be mapped onto a distribution on the Bloch sphere.
Now consider explicitly the eigenstates of Eq. (12). They can be labeled by an integer and as shown in the exact solution given by Braak Braak (2011). These states, , only live on a circle of the Bloch sphere due to the exclusion of from the Hamiltonian. As such, we use Eq. (5) for the entanglement moments (for details, see section 2 of the supplement sup ()). The entanglement moments and full distributions for are plotted for the ground state in Fig. 4 and for the sixth excited state in Fig. 5.
Notice in these figures how the distribution changes with for a given eigenstate. For large , the higher moments asymptote to a value less than 1 while . This represents the localization described in Fig. 3 as well as how a measurement in corresponds to a projective measurement in of the qubit. On the other hand, the figures show other instances where along with other entanglement moments, and in those cases the states are more evenly distributed about the circle. In fact when we see moments rise for higher values of , we know the state is becoming more evenly distributed just as in the case of Fig. 2. The cross-over from the moments getting larger as increases to the point where they start decreasing with represents the cross-over from non-projective to projective-like measurements.
Entanglement moments could also be used in various dynamical questions and in many other systems (such as the system considered in Wilson et al. (2012)). Additionally, generalization to density matrices remains an open question at this time. Such a theory would need to generalize I-concurrence and separate out the classical probabilities from the quantum.
In this letter, we have defined the new concept of entanglement moments. These moments contain and surpass traditional entanglement measures, describing not only if a system is entangled, but also how. Taken all together, they can qualitatively and quantitatively describe how projective a measurement is. As seen in Figs. 2 and 3, if higher moments increase, the distribution will be quite distributed and the measurement less projective. As an example, we calculated these moments for eigenstates of the Rabi model, showing complex behavior for higher excited states.
This research was supported by DOE-BES-DESC0001911 (V.G. and J.M.) and the JQI-PFC (J.W.)
Appendix A Supplementary material: Probing the structure of entanglement with entanglement moments
In this supplement, we discuss entanglement moments when the measured system is an dimensional Hilbert space, and we detail the Rabi model calculations presented in the main text. We described how to apply entanglement moments when the measured system is and the Bloch sphere, but the analysis is far more general. By utilizing the mathematics of , we can apply entanglement moments in dimensions. For completeness, we also discuss our calculations in the Rabi model in depth. A wealth of information in the Rabi model is obtainable with exact calculations and simple numerics.
Appendix B Extending analysis to any finite dimensional Hilbert space
We assume a bipartite system where the Hilbert space is the direct product of two other Hilbert spaces . We can write states in terms of an orthonormal basis of , , and (unnormalized) vectors ,
The vector while in general.
Let system be an arbitrary -dimensional Hilbert space. The space of normalized vectors is , but there is a U(1) gauge freedom in the distance measures given by
so the space is actually . This is in fact a Hopf fibration from to over the U() fiber. As with the other cases considered in the main text, we map onto the function .
The entanglement moments are then given by
As in the main text, the distance functions are known:
The distance function can be considered as a function of the on the angle set, so that we write
Since Eq. (18) is an th ordered polynomial in , we can expand it into Jacobi polynomials, .
Now we need to use an addition formula for complex projective space as derived by Koornwinder (1972a); ?; Shatalov (2001). To develop the formula, we should write the space of functions, , as a direct sum of orthogonal subspaces in the following way. Dividing into the spaces of spherical harmonics, we have , where is the finite-dimensional vector space of harmonic polynomials homogeneous of degree of real variables that are restricted to . These should be further restricted to those that are just U() invariant since . With this restriction, we follow the notation of Grinberg (1983) and write
where are just the U() invariant parts of .
Given this, we now state the addition theorem as written in Shatalov (2001). Let and be an orthonormal basis in the space . Then the Jacobi polynomials become
Note that we can also calculate from formulae given in Shatalov (2001). It is
Just as before, we can expand our the distance function in terms of , then expand that by the addition theorem and obtain
is the norm of the function in the finite subspace – i.e., the norm in the th harmonic. So the th entanglement moment captures the information about the 1st through th harmonic of the distribution.
Proper normalization of our distribution gives us , since is the space of constant functions. We can read off the normalization as
This entire analysis reduces to the case of a Bloch sphere for , and we reproduce the Bloch sphere formula from the main text exactly.
Appendix C Entanglement calculations in the Rabi model
For the Rabi model, is a two level system and is a harmonic oscillator, and we have
where () is the annihilation (creation) operater, and are the and Pauli matrices respectively, and , , and are constants (frequency of the oscillator, coupling, and Zeeman splitting, respectively).
The Rabi model’s eigenstates have a particular form since the operator , where is the parity operator on the harmonic oscillator, commutes with the Hamiltonian. The states are
where and are the eigenstates of with eigenvalues , and are unknown vectors in the Hilbert space of the harmonic oscillator.
The concurrence Rungta et al. (2001) for this system can be easily calculated, and happens to be
so the entanglement of these states just depends on the expectation value of the parity operator with harmonic oscillator wave functions.
We can exactly calculate things if we let . In that case the eigenstates are just
where is the th state of the harmonic oscillator centered at .
The resulting concurrence is then
where are Lagueere polynomials. As the coupling increases, we see oscillations in the entanglement due to the Laguerre polynomials.
Braak Braak (2011) solved for the eigenstates when . The eigenvalues can be calculated from and
where the coefficients satisfy
The unnormalized eigenstates, written in Bargmann space Bargmann (1961), are
Taking normalization into account, we can obtain
The higher moments, , do not admit a closed form expression when we specify the measurement basis as the eigenstates of . However, it is straightforward to numerically calculate them. The eigenvalues can be calculated from Eq. 30 using a relatively simple rootfinding algorithm. Then with the eigenfunctions from 34, the probability distribution on the Bloch sphere is given from
The entanglement moments, , can be plotted by extracting the Fourier components of the surface and adding them together appropriately.
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