Spin squeezing and entanglement for arbitrary spin
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin- particles to entanglement conditions for spin- particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in [A. Sørensen et al., Nature 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.
pacs:03.67.Mn 03.65.Ud 05.50.+q 42.50.Dv
One of the most rapidly developing areas in quantum physics is creating larger and larger entangled quantum systems with photons, trapped ions, and cold neutral atoms review (); KS07 (); WK09 (); CG12 (); LZ07 (); HH05 (); MS11 (); JK01 (); revatom (); MG03 (); IA12 (); MB12 (). Entangled states can be used for metrology in order to obtain a sensitivity higher than the shot-noise limit SD01 (); PS09 (); fisher_kentanglement () and can also be used as a resource for certain quantum information processing tasks RB03 (); G96 (); G97 (); crypto (). Moreover, experiments realizing macroscopic quantum effects might give answers to fundamental questions in quantum physics D89 (); FD12 ().
Spin squeezing is one of the most successful approaches for creating large-scale quantum entanglement K93 (); W94 (); HS99 (); VR01 (); SD01 (); SM99 (); WS01 (); KB98 (); HM04 (); HP06 (); EM05 (); EM08 (); F08 (); Achip (); BEC (); BEC2 (); BEC3 (). It is used in systems of very many particles in which only collective quantities can be measured. For an ensemble of particles with a spin the most relevant collective quantities are the collective spin operators defined as
for where are the components of the angular momentum operator for the spin.
Spin-squeezed states are typically almost fully polarized states for which the angular momentum variance is small in a direction orthogonal to the mean spin K93 (). They can be used to achieve a high accuracy in certain very general metrological tasks PS09 (); fisher_kentanglement (). On the other hand, in spin- systems spin squeezing is closely connected to multipartite entanglement. A ubiquitous criterion for detecting the entanglement of spin-squeezed states is SD01 ()
Any fully separable state of qubits, that is, a state that can be written as W89 ()
Apart from the original inequality Eq. (2), several other generalized spin-squeezing entanglement conditions have been presented RevNori (); GT04 (); WVB05 (); KC05 (); KC06 (); GT06 (); KK09 (); CP11 (); SM01 (); HP11 (); DC02 (); MZ02 (); MK08 (); Relquadspin (); DuanPRL (); MGD12 () and even the complete set of such criteria for multi-qubit systems has been found in Ref. TK07 (). While most of the conditions are for a fixed particle number, conditions for the case of nonzero particle number variance have also been derived HP10 (); HP12 ().
So far most of the attention has been focused on ensembles of spin- particles. The literature on systems of particles with has been limited to a small number of conditions, specialized to certain sets of quantum states or particles with a low spin SM01 (); GT04 (); WVB05 (); MK08 (); HP11 (); DC02 (); MZ02 (). The reason is that known methods for detecting entanglement for spin- particles by spin-squeezing cannot straightforwardly be generalized to higher spins. For example, for Eq. (2) can also be violated without entanglement between the spin- particles, as we will discuss later F08 ().
In spite of the difficulties in deriving entanglement conditions for particles with a higher spin, they are very much needed in quantum experiments nowadays. As most of such experiments are done with atoms with such conditions can make the complexity of experiments much smaller: The artificially created spin- subsystems must be manipulated by lasers, while the physical spin- particles can directly be manipulated by magnetic fields. Moreover higher spin systems could make it possible to perform quantum information processing tasks different from the ones possible with spin- particles or to create different kind of entangled states TM10 (); UH12 (); KTqutrit (); LS11 (); HG12 (); RP01 (); SU12 ().
In this paper, we will start from the complete set presented for spin- particles in Ref. TK07 (). All spin-squeezing entanglement criteria of this set are based on the first and second moments of collective angular momentum coordinates. It has been possible to obtain a full set of tight inequalities by analytical means only due to certain advantageous properties of the spin- case. For the case of particles with the inequalities presented in the literature are either based on numerical optimization SM01 () or are analytical but not tight MK08 (). The reason for this is that for the second moments of the collective observables are not only connected to the two-body correlations, as in the spin- case, but also to the local second moments.
In order to solve this problem, we define modified second moments and the corresponding variances as follows
where The modified quantities do not contain anymore the local second moments. We will show that by using the first moments and the modified second moments of the collective operators, it is possible to write down tight entanglement conditions analytically also for the case Note_tilde (). We will also discuss that the local second moments are related to single-particle spin squeezing (see Sec. VI.1).
The main results of our paper are as follows.
(i) We will find the complete set of conditions for the case, which we will call optimal spin-squeezing inequalities for spin- particles. They are a complete set since, for large they detect all entangled states that can be detected knowing only the first moments and the modified second moments. For instance, they can be used to verify the entanglement of singlet states, symmetric Dicke states and planar squeezed states HP11 ().
(ii) We also present a generalization of the original spin squeezing parameter defined in Eq. (2) that can be used for entanglement detection even for particles with
(iii) Finally, we will show that, in the large particle number limit, the entanglement condition based on the following entanglement parameter
is strictly stronger than the condition based on Note that is defined only for In this way will always be non-negative. In Eq. (6), the subscript “” refers to the optimal spin-squeezing inequalities since we obtain , essentially, by dividing the left-hand side of one of the inequalities by the right-hand side. For clarity, we give Eq. (6) explicitly for the case
If then the state is entangled. The parameter (5) is appropriate only for spin-squeezed states with a large total spin depicted in Fig. 1(a), while the parameter (6) detects also states that have zero total spin, as shown in Fig. 1(b). Moreover, we will also show that for large particle numbers, if then we also have
Thus, is a better indicator of entanglement than
The paper is organized as follows. In Ref. PRL (), we have already presented a generalization of the complete set of spin-squeezing inequalities valid for systems of spin- particles with In this paper, we extend the results of Ref. PRL () in several directions. In Sec. II, we present the optimal spin-squeezing inequalities for spin- particles and discuss some of their fundamental properties. In Sec. III, we study states that violate the inequalities maximally. In Sec. IV, we show a method for mapping existing entanglement conditions for spin- particles to analogous conditions for spin- particles with Using the mapping, we derive the spin-squeezing parameter . In Sec. V, we present the spin-squeezing parameter and examine its properties. In Sec. VI, we consider various issues concerning the efficient application of our spin-squeezing inequalities.
Ii Complete set of spin-squeezing inequalities for spin- particles.
In this section, we present our spin-squeezing inequalities for particles with an arbitrary spin and we also examine the connection of these inequalities to the entanglement of the reduced two-particle state, and to the criterion based on the positivity of the partial transpose.
ii.1 The optimal spin-squeezing inequalities for qudits
Observation 1.—The following generalized spin-squeezing inequalities are valid for separable states given by Eq. (3) for an ensemble of spin- particles even with
Here may take all the possible permutations of If a quantum state violates one of the inequalities (9), then it is entangled.
Proof. We will prove that for separable states the following inequality holds
where is a subset of indices including the two extremal cases and . We consider first pure product states of the form For such states, the modified variances and the modified second moments can be obtained as
Substituting Eq. (LABEL:subs) into the left-hand side of Eq. (10), we obtain
The two inequalites in Eq. (LABEL:opt_gen) follow from the inequality TK07 ()
and from the well-known bound for an angular momentum component Hence we proved that Eq. (10) is valid for pure product states. Due to the left-hand side of Eq. (10) being concave in the state, it is also valid for separable states.
which is a consequence of the identity angular ()
Hence, we proved that Eq. (9) is valid for separable states.
In order to evaluate Eq. (9), six operator expectation values are needed. These are the vector of the expectation values of the three collective angular momentum components
and the vector of the modified second moments
For the spin- case, the modified second moments can be obtained from the true second moments since For spin- particles with the elements of typically cannot be measured directly. Instead, we measure the true second moments
and the sum of the squares of the local second moments
Then, can be obtained as the difference between the true second moments and the sum of local second moments as
In Sec. VI.3, we discuss how to measure based on the measurement of and
For any value of the mean spin Eq. (9) defines a polytope in the -space. The polytope is depicted in Figs. 2(a) and 2(b) for different values for It is completely characterized by its extremal points. Direct calculation shows that the coordinates of the extreme points in the -space are
where The points and can be obtained in an analogous way. Note that the coordinates of the points and depend nonlinearly on
Let us see briefly the connection between the inequalities and the facets of the polytope. The inequality with three second moments, Eq. (9a), corresponds to the facet in Fig. 2(a). The inequality with three variances, Eq. (9b), corresponds to the facet The inequality with one variance, Eq. (9c) corresponds to the facets and The inequality with two variances, Eq. (9d), corresponds to the facets and
ii.2 Completeness of Eq. (9)
In this section, we will show that, in the large limit, all points inside the polytope correspond to separable states. This implies that the criteria of Observation 1 are complete, that is, if the inequalities are not violated then it is not possible to prove the presence of entanglement based only on the first and the modified second moments. In other words, it is not possible to find criteria detecting more entangled states based on these moments. To prove this, first we can observe that if some quantum states satisfy Eq. (9) then their mixture also satisfies it. Thus, it is enough to investigate the states corresponding to the extremal points of the polytope. We will give a straightforward generalization of the proof for the spin- case presented in Ref. TK07 ().
Observation 2.—(i) For any value of there are
separable states corresponding to for
(ii) Let us define
If is an integer then
there exists also a separable state corresponding to Similar
statements hold for and Note that this condition is
always fulfilled, if and is even.
(iii) There are always separable states corresponding to points such that their distance from is smaller than In the limit for a fixed normalized angular momentum the points and the cannot be distinguished by measurement, for that a precision or better would be needed when measuring which is unrealistic. Hence in the macroscopic limit the characterization is complete.
Proof. A separable state corresponding to is
Here are the single-particle states with .
If is an integer, we can also define the state corresponding to the point as
Since there is a separable state for each extreme point of the polytope, for any internal point a corresponding separable state can be obtained by mixing the states corresponding to the extreme points.
If is not an integer, we can approximate by taking as the largest integer smaller than defining the state
It has the same coordinates as except for the value of where the difference is
The extremal states that correspond to the vertices of the polytope defined by the optimal spin-squeezing inequalities are, in a certain sense, generalizations of the coherent spin states defined as RevNori (); Bcoh ()
where is a state with maximal All states of the form (26) saturate all the inequalities, as can be seen by direct substitution into Eq. (9). Further extremal states can be obtained as tensor products or mixtures of coherent spin states. Note that they exist for all the possible values of the mean spin while spin coherent states Eq. (26) were fully polarized.
ii.3 Relation of Eq. (9) to two-particle entanglement
Since the optimal spin-squeezing inequalities (9) contain only first moments and modified second moments of the angular momentum components, they can be reformulated with the average two-body correlations. For that, we define the average two-particle density matrix as
where is the two-particle reduced density matrix for the and particles.
Next, we formulate our entanglement conditions with the density matrix
Observation 3.—The optimal spin-squeezing inequalities Eq. (9) for arbitrary spin can be given in terms of the average two-body density matrix as
where we have defined the expression as the sum of all the two-particle correlations of the local spin operators
Proof. Equation (10) can be transformed into
Next, let us see how Eq. (30) behaves for symmetric states. We know from angular momentum theory that Eq. (9a) of the optimal spin-squeezing inequalities is saturated only when the state is symmetric. For the case, all symmetric states saturate Eq. (9a), while for only some of the symmetric states saturate it. Based on these and Eq. (14), we know that, for spin- particles in a symmetric state the right-hand side of Eq. (30) is zero. On the other hand, for spin- particles with in a symmetric state, the right-hand side can also be negative.
Let us now turn to the reformulation of Eq. (30) in terms of the two-body reduced density matrix. The modified second moments and variances can be expressed with the average two-particle density matrix as
Not that, as in the spin- case, there are states detected as entangled that have a separable two-particle density matrix TK07 (). Such states are, for example, permutationally invariant states with certain symmetries for which the reduced single-particle density matrix is completely mixed. For large due to permutational invariance and the symmetries mentioned above, the two-particle density matrices are very close to the a completely mixed matrix as well and hence they are separable. Still, some of such states can be detected as entangled by the optimal spin-squeezing inequalities. Examples of such states are the permutationally invariant singlet states discussed later in Sec. III.2.
ii.4 Relation of Eq. (9) to the criterion based on the positivity of the partial transpose
Our inequalities are entanglement conditions. Thus, it is important to compare them to the most useful entanglement condition known so far, the condition based on the positivity of the partial transpose (PPT) PPT ().
In Ref. TK07 (), it has been shown for the spin- case that the optimal spin-squeezing inequalities can detect the thermal states of some spin models that have a positive partial transpose for all bi-partitions of the system. Such states are extreme forms of bound entangled states: they are non-distillable even if the qubits of the two partitions are allowed to unite with each other. We found that for the case, the inequality (9b) also detects such bound entangled states in the thermal states of spin models. An example of such a state for and is
The state (32) is detected by our criterion below the temperature bound while it is detected by the PPT criterion below the bound
Finally, we will consider the special case of symmetric states. In this case, the PPT condition applied to the reduced two-body density matrix detects all states detected by the spin-squeezing inequalities.
Observation 4.—The PPT criterion for the average two-particle density matrix defined in Eq. (27) detects all symmetric entangled states that the optimal spin-squeezing inequalities detect for The two conditions are equivalent for symmetric states of particles with
Proof. We will connect the violation of Eq. (28) to the violation of the PPT criterion by the reduced two-particle density matrix If a quantum state is symmetric, its reduced state is also symmetric. For such states, the PPT condition is equivalent to TG09 ()
holding for all Hermitian operators Based on Observation 3, it can be seen by straightforward comparison of Eqs. (28) and (33) that, for Eq. (28) holds for all possible choices of and for all possible choices of coordinate axes, i.e., all possible if and only if Eq. (33) holds for all Hermitian operators For there is no equivalence between the two statements. Only from the latter follows the former.
Iii States that violate the optimal spin-squeezing inequalities for spin
In this section we will study, what kind of states violate maximally our spin-squeezing inequalities. We will also examine, how much noise can be mixed with these states such that they are still detected as entangled by our inequalities.
iii.1 The inequality with three second moments, Eq. (9a)
The first two equations of Eqs. (9) are invariant under the exchange of coordinate axes and . As a consequence of basic angular momentum theory, Eq. (9a), the inequality with three second moments is valid for all quantum states, thus it cannot be violated. As discussed in the proof of Observation 3, for the case, all symmetric states saturate Eq. (9a), while for only some of the symmetric states saturate it. In both cases, states of the form (26) are a subset of the saturating states.
iii.2 The inequality with three variances, Eq. (9b)
The states maximally violating Eq. (9b) are the many-body singlet states. The characteristic values of the collective operators for many-body singlets are shown in Table I. States violating Eq. (9b) have a small variance for all the components of the angular momentum as shown in Fig. 1(c).
Let us see now some examples of many-body singlets states. For a pure singlet state can be constructed, for example, as a tensor product of two-particle singlets of the form
Any permutation of such a state is a singlet as well. The mixture of all such permutations is a permutationally invariant singlet defined as
For even and the state is the only permutationally invariant singlet state. For all singlets are outside of the symmetric subspace.
In the case of spin- particles, the following two-particle symmetric state
is also a singlet. It is very important from the point of view of experimental realizations with Bose-Einstein condensates that for there are singlet states in the symmetric subspace.
Next, we mix the spin- singlet state with white noise and examine up to how much noise it is still violating Eq. (9b). The noisy singlet state is the following
where is a singlet state maximally violating Eq. (9b), and is the amount of noise and we defined the completely mixed state as
where the dimension of the qudit is The vectors of the collective quantities are shown in Table I for the completely mixed state. Based on these, simple calculations show that the state (38) is detected as entangled by Eq. (9b) if
Hence, the white-noise tolerance decreases with .
|Singlet state discussed in Sec. III.2|
|Completely mixed state defined in Eq. (39)|
|Symmetric Dicke state, discussed in Sec. III.3|
Finally note that for any the modified second moments of the collective angular momentum components are zero for the completely mixed state, i.e.,
Thus, the completely mixed state belongs to a point at the origin of the coordinate system of the modified second moments for In contrast, in the space of true second moments the singlet state is at the origin, since for the singlet we have for
Eq. (9b) has been proposed to detect entanglement in optical lattices of cold atoms GT04 (). A related inequality was presented for entanglement detection in condensed matter systems by susceptibility measurements WVB05 (). Experimentally, it has been used for entanglement detection in photonic systems IA12 () and in fermionic cold atoms MB12 (). An ensemble of -state fermions naturally fills up the energy levels of a harmonic oscillator such that all levels have fermions in a multipartite singlet state. Such a state is also a singlet, maximally violating the optimal spin-squeezing inequality with three variances, Eq. (9b). Singlets can also be obtained through spin squeezing in cold atomic ensembles TM10 (); UH12 (). Finally, the ground state of the system Hamiltonian for certain spinor Bose-Einstein condensates is a singlet state HG12 ().
iii.3 The inequality with only one variance, Eq. (9c)
Next, we will consider the optimal spin-squeezing inequality with one variance Eq. (9c). This entanglement criterion is very useful to detect the almost fully polarized spin-squeezed states shown in Fig. 1(a). It can also be used to detect symmetric Dicke states with a maximal and States close to such symmetric Dicke states have a small variance for one component of the angular momentum while they have a large variance in two orthogonal directions as shown in Fig. 1(b).
Dicke states are quantum states obeying the eigenequations
where is a label used to distinguish the different eigenstates corresponding to the same eigenvalues and In particular, we will show that Eq. (9c) is very useful to detect entanglement close to the symmetric Dicke state
where must be even for half integer ’s. In this case, the label is not needed, as the two eigenvalues determine the state uniquely. The state state (43) for has already been known to have intriguing entanglement properties GT06 () and it is optimal for certain very general quantum metrological tasks fisher_kentanglement ().
The state (43) maximally violates Eq. (9c) for since it maximizes all terms with a positive coefficient and minimizes all terms with a negative one on the left-hand side of Eq. (44). This statement is almost true also for the case except for the term with the local second moments which has a value
The proof of Eq. (45) is given in the appendix. Based on these, our symmetric Dicke state is detected as entangled for any
In a practical situation, it is also important to know how much additional noise is tolerated such that the noisy state is still detected as entangled. Next, we look at the noise tolerance of the inequality (44) for our case. We mix the symmetric Dicke state (43) with white noise as
The expectation values and the relevant moments of the collective angular momentum components for the Dicke state (43) are given in Table I. Based on these, a noisy Dicke state is detected as entangled if
For large the bound on the noise is
Entangled states close to Dicke states have been observed in photonic experiments with a condition similar to the optimal spin-squeezing inequality with one variance, Eq. (9c) KS07 (); WK09 (); CG12 (). Symmetric Dicke states can be created dynamically in Bose-Einstein condensate LS11 (); HG12 (). Cold trapped ions also seem to be ideal to create symmetric Dicke states, thus the use of our inequalities is expected even in these systems HH05 (); KC06 (); UF03 ().
iii.4 The inequality with two variances, Eq. (9d)
As the last case let us consider the optimal spin-squeezing inequality (9d). Typical states strongly violating Eq. (9d) have a small variance for two components of the angular momentum while having a large variance in the orthogonal direction, see Fig. 1(d). As we will see, for certain values for singlet states [Fig. 1(c)] also violate Eq. (9d).
Now it is hard to compute the maximally violating state, because an independent optimization for the different terms does not seem to lead to a state maximizing the whole expression even for Thus, we will consider examples of important states violating the inequality and compare it to other similar conditions.
Let us consider the multi-particle spin singlet states. Based on and given in Table I, we find that the optimal spin-squeezing inequality (9d) is violated whenever
Thus, for the singlet state is violating this inequality for and
where the constant is for and for respectively. For larger the constant is determined numerically. For even the criterion (49) is maximally violated by the many-particle singlet state for any
For and for large it can be seen that the right-hand side of Eq. (50) equals A comparison with Eq. (49) shows that our condition (50) is strictly stronger in this case. For Eq. (50) is not strictly stronger any more, but still is more effective in detecting quantum states with a large
This seems to be the advantage of our inequality compared to Eq. (49): It has information not only about the variances in the and directions, but also about the second moment in the third direction.
Iv Spin- entanglement criteria transformed to higher spins
In this section, we present a method to map spin- entanglement criteria to criteria for higher spins. We use it to transform the original spin-squeezing parameter Eq. (2) to a spin-squeezing parameter for higher spins. We show that two of the optimal spin-squeezing inequalities are strictly stronger than the transformed original spin-squeezing criterion. We also convert some other spin- entanglement criteria to criteria for higher spins.
iv.1 The original spin-squeezing parameter for higher spins
Next, we present a mapping that can transform every spin-squeezing inequality for an ensemble of spin- particles written in terms of the first and the modified second moments of the collective spin operators to an entanglement condition for spin- particles, also given in terms of the first and the modified second moments.
Observation 5.—Let us consider an entanglement condition (i.e., a necessary condition for separability) for spin- particles of the form
where is a six-dimensional function. Then, the inequality obtained from Eq. (51) by the substitution
is an entanglement condition for spin- particles. Any quantum state that violates it is entangled.
Proof. Let us consider a product state of spin- particles
and define the quantities Then the first and modified second moments of the collective spin can be rewritten in terms of those quantities as
For the length of the single-particle Bloch vectors we have the constraints
Both the lower and the upper bound are sharp, and these are the only constraints for physical states for every Constraint (). Thus, the set of allowed values for and for product states of the form Eq. (53) are independent from This is also true for separable states since separable states are mixtures of product states. Let us now consider the range of
for separable states. We have seen that the set of allowed values for the arguments of the function
in Eq. (56) for separable states is independent of Thus, the range of Eq. (56) for separable states is also independent of
Hence the statement of Observation 5 follows concavity ().
Next, we will transform the spin-squeezing parameter to higher spins.
Proof. Let us first write down the entanglement condition for spin- particles based on the spin-squeezing parameter (2) in terms of the modified variance as
Then, we use Observation 5 to obtain
It is instructive to rewrite Eq. (58) as
Equation (59) can be further reformulated such that the second term depends only on the average single-particle density matrix, as
and is the single-particle reduced density matrix for the particle. Thus, in Eq. (60) we wrote down the new spin-squeezing parameter as the sum of the original parameter given in Eq. (2) and a second term that depends only on single particle observables and is related to single particle spin squeezing. For , this second term in Eq. (60) is zero. For it is nonnegative. Hence, for there are states that violate Eq. (2), but do not violate This is shown in a simple example with qutrits.
Example 1.—Let us consider a multi-particle state of the form
for For and for any the original spin-squeezing inequality (2) is violated by the state (62). On the other hand, no separable state can violate thus, it is the correct formulation of the original spin-squeezing inequality for
There is another interpretation on how to use the original spin-squeezing inequality (2) for the case. Equation (2) is inherently for ensembles of spin- particles. When used for higher spins, should be the number of spin- constituents rather than the number of spin- particles. Then, Eq. (2) detects entanglement between the spin- constituents of the particles, and cannot distinguish between entanglement among the spin- particles and entanglement within the spin- particles F08 ().
Observation 7.—The optimal spin-squeezing inequality with three variances, Eq. (9b), and the one with one variance, Eq. (9c), for are strictly stronger than the spin-squeezing inequality [ is defined in Eq. (5)], since they detect strictly more states.
Proof. To see this, let us rewrite Eq. (9c) for the particular choice of coordinate axes as