Entanglement and nonlocality in multi-particle systems
Entanglement, the Einstein-Podolsky-Rosen (EPR) paradox and Bell’s failure of local-hidden-variable (LHV) theories are three historically famous forms of “quantum nonlocality”. We give experimental criteria for these three forms of nonlocality in multi-particle systems, with the aim of better understanding the transition from microscopic to macroscopic nonlocality. We examine the nonlocality of separated spin systems. First, we obtain multipartite Bell inequalities that address the correlation between spin values measured at each site, and then we review spin squeezing inequalities that address the degree of reduction in the variance of collective spins. The latter have been particularly useful as a tool for investigating entanglement in Bose-Einstein condensates (BEC). We present solutions for two topical quantum states: multi-qubit Greenberger-Horne-Zeilinger (GHZ) states, and the ground state of a two-well BEC.
Keywords: entanglement, quantum nonlocality, multi-particle, two-well Bose-Einstein condensates (BEC)
PACS numbers: 03.65.Ta, 42.50.St, 03.65.Ud, 03.75.Gg
Nonlocality in quantum mechanics has been extensively experimentally investigated. Results to date support the quantum prediction, first presented by Bell, that quantum theory is inconsistent with a combination of premises now generally called “local realism” (1); (2).
However, the extent that quantum mechanics is inconsistent with local realism at a more mesoscopic or macroscopic level is still not well understood. Schrödinger presented the case that loss of realism macroscopically would be a concern, and raised the question of how to link the loss of local realism with macroscopic superposition states (3); (4); (5); (6).
The advent of entangled Bose-Einstein condensate (BEC) states leads to new possibilities for testing mesoscopic and macroscopic quantum mechanics. With this in mind, the objective of this article is to give an overview of a body of work that explores nonlocality in multi-particle or multi-site systems.
Three types of nonlocality are reviewed: entanglement (3), the Einstein-Podolsky-Rosen (EPR) paradox (7), and Bell’s nonlocality (1); (2). Examples of criteria to demonstrate each of these nonlocalities is presented, first for multi-site “qubits” (many spin particles) and then for multi-site “qudits” (many systems of higher dimensionality such as high spin particles).
The criteria presented in this paper are useful for detecting the nonlocality of the -qubit (or -qudit) Greenberger-Horne-Zeilinger (GHZ) states (8); (9). These states are extreme superpositions that were shown by GHZ to demonstrate a very striking “all or nothing” type of nonlocality. This nonlocality can manifest as a violation of a Bell inequality, and at first glance these violations, because they increase exponentially with , appear to indicate a more extreme nonlocality as the size of the system increases (10).
We point out, however, that the detection of genuine -body nonlocality, as first discussed by Svetlichny (11); (12), requires much higher thresholds. Genuine -party nonlocality (e.g. genuine entanglement) requires that the nonlocality is shared among all parties, or particles. The violations in this case do not increase with , and the detection over many sites is very sensitive to loss and inefficiencies.
Ii Three Famous types of nonlocality
The earliest studies of nonlocality concerned bipartite systems. Einstein-Podolsky-Rosen (EPR) (7) began the debate about quantum nonlocality, by pointing out that for some quantum states there exists an inconsistency between the premises we now call “local realism” and the completeness of quantum mechanics.
Local realism (LR) may be summarized as follows. EPR argued (7); (15) first for “locality”, by claiming that there could be no “action-at-a-distance”. A measurement made at one location cannot instantaneously affect the outcomes of measurements made at another distant location. EPR also argued for “reality”, which they considered in the following context. Suppose one can predict with certainty the result of a measurement made on a system, without disturbing that system. Realism implies that this prediction is possible, only because the outcome for that measurement was a predetermined property of the system. EPR called this predetermined property an “element of reality”, though most often the element of reality is interpreted as a “hidden variable”. The essence of EPR’s local realism assumption is that results of measurements made on a system at one location come about because of predetermined properties of that system, and because of their local interactions with the measurement apparatus, not because of measurements that are made simultaneously at a distant locations.
ii.1 EPR paradox
EPR argued that for states such as the spin singlet state
there arises an inconsistency of the LR premises with the quantum predictions. Here, we define and as the spin “up” and “down” eigenstates of for a system at location . For the state (1), the prediction of the spin component can be made by measurement of the component at . From quantum theory, the two measurements are perfectly anticorrelated. According to EPR’s Local Realism premise (as explained above), there must exist an “element of reality” to describe the predetermined nature of the spin at . We let this element of reality be symbolized by the variable , and we note that assumes the values (1).
Calculation shows that there is a similar prediction of a perfect anti-correlation for the other spin component pairs. Therefore, according to LR, each of the spin components and can also be represented by an element of reality, which we denote and respectively. A moment’s thought tells us that the if there is a state for which all three spins are completely and precisely predetermined in this way, then this “state” cannot be a quantum state. Such a “state” is generally called a “local hidden variable (LHV) state”, and the set of three variables are “hidden”, since they are not part of standard quantum theory. Hence, EPR argued, quantum mechanics is incomplete.
Since perfect anticorrelation is experimentally impossible, an operational criterion for an EPR paradox can be formulated as follows. Consider two observables and , with commutators like position and momentum. The Heisenberg Uncertainty Principle is , where and are the variances of the outcomes of measurements for and respectively. The EPR paradox criterion is (16)
where is the “variance of inference” i.e. the variance of conditional on the measurement of an observable at a distant location . The is defined similarly where is a second observable for measurement at . This criterion reflects that the combined uncertainty of inference is reduced below the Heisenberg limit. Of course, the reduced uncertainty applies over an ensemble of measurements, where only one of the conjugate measurements is made at a time. This criterion is also applicable to optical quadrature observables, where it has been experimentally violated, although without causal separation. With spin commutators, other types of uncertainty principle can be used to obtain analogous inferred uncertainty limits.
The demonstration of an EPR paradox through the measurement of correlations satisfying Eq. (2) is a proof that local realism is inconsistent with the completeness of quantum mechanics (QM). Logically, one must: discard local realism, the completeness of QM, or both. However, it does not indicate which alternative is correct.
ii.2 Schrödinger’s Entanglement
Schrödinger (3); (4); (5) noted that the state (1) is a special sort of state, which he called an an entangled state. An entangled state is one which cannot be factorized: for a pure state, we say there is entanglement between and if we cannot write the composite state (that describes all measurements at the two locations) in the form , where is a state for the system at only.
For mixed states, there is said to be entanglement when the density operator for the composite system cannot be written as a mixture of factorizable states (17). A mixture of factorizable states is said to be a separable state, which where there are just two sites, is written as
If the density operator cannot be written as (3), then the mixed system possesses entanglement (between and ). More generally, for sites, full separability implies
If the density operator cannot be expressed in the fully separable form (4), there is entanglement between at least two of the sites.
We consider measurements , with associated outcomes , that can be performed on the -th system (). For a separable state (4), it follows that the joint probability for outcomes is expressible as
where we have replaced for convenience of notation the index by , and used a continuous summation symbolically, rather than a discrete one, so that . The subscript represents “quantum”, because there exists the quantum density operator for which . In this case, we write , where the subscript reminds us that this is a quantum probability distribution. The model (5) implies (3) (18); (19), and has been studied in Ref. (20), in which it is referred to as a quantum separable model (QS).
We can test nonlocality when each system is spatially separated. We will see from the next section that LR implies the form (5), but without the subscripts “Q”, that is, without the underlying local states designated by necessarily being quantum states. If the quantum separable QS model can be shown to fail where each is spatially separated, one can only have consistency with Local Realism if there exist underlying local states that are non-quantum. This is an EPR paradox, since it is an argument to complete quantum mechanics, based on a requirement that LR be valid.
The EPR paradox necessarily requires entanglement (15); (21). The reason for this is that for separable states (3-5), the uncertainty relation that applies to each of the states and will imply a minimum level of local uncertainty, which means that the noncommuting observables cannot be sufficiently correlated to obtain an EPR paradox. In other words, the entangled state (1) can possess a greater correlation than possible for (3).
Schrödinger also pointed to two paradoxes (3); (4); (5) in relation to the EPR paper. These gedanken-experiments strengthen the apparent need for the existence of EPR “elements of reality”, in situations involving macroscopic systems, or spatially separated ones. The first is famously known as the Schrödinger’s cat paradox, and emphasizes the importance of EPR’s “elements of reality” at a macroscopic level. Reality applied to the state of a cat would imply a cat to be either dead or alive, prior to any measurement that might be made to determine its “state of living or death”. We can define an “element of reality” , to represent that the cat is predetermined to be dead (in which case ) or alive (in which case ). Thus, the observer looking inside a box, to make a measurement that gives the outcome “dead” or “alive”, is simply uncovering the value of . Schrödinger’s point was that the element of reality specification is not present in the quantum description of a superposition of two macroscopically distinguishable states.
The second paradox raised by Schrödinger concerns the apparent “action at-a-distance” that seems to occur for the EPR entangled state. Unless one identifies an element of reality for the outcome , it would seem to be the action of the measurement of that immediately enables prediction of the outcome for the measurement at . Schrödinger thus introduced the notion of “steering”.
While all these paradoxes require entanglement, we emphasize that entanglement per se is a relatively common situation in quantum mechanics. It is necessary for a quantum paradox, but does not by itself demonstrate any paradox.
ii.3 Bell’s nonlocality: failure of local hidden variables (LHV)
EPR claimed as a solution to their EPR paradox that hidden variables consistent with local realism would exist to further specify the quantum state. It is the famous work of Bell that proved the impossibility of finding such a theory. This narrows down the two alternatives possible from a demonstration of the EPR paradox, and shows that local realism itself is invalid.
Specifically, Bell considered the predictions of a Local Hidden Variable (LHV) theory, to show that they would be different to the predictions of the spin-half EPR state (1). Following Bell (1); (2), we have a local hidden variable model (LHV) if the joint probability for outcomes of simultaneous measurements performed on the spatially separated systems is given by
Here are the “local hidden variables” and is the probability of given the values of , with being the probability distribution for . The factorization in the integrand is Bell’s locality assumption, that depends on the parameters , and the measurement choice made at only. The hidden variables describe a local state for each site, in that the probability distribution for the measurement at is given as a function of the . The form of (6) is formally similar to (4) except in the latter there is the additional requirement that the local states are quantum states. If (6) fails, then we have proved a failure of all LHV theories, which we refer to as a Bell violation or Bell nonlocality (20).
The famous Bell-Clauser-Horne-Shimony-Holt (CHSH) inequalities follow from the LHV model, in the case. Bell considered measurements of the spin components and . He then defined the spin product and showed that for the LHV model, there is always the constraint
The quantum prediction for an entangled Bell state (1) is and the inequality is violated for the choice of angles
for which the quantum prediction becomes . Tsirelson’s theorem proves the value of to be the maximum violation possible for any quantum state (22). We note that experimental inefficiencies mean that violation of the CHSH inequalities for causally separated detectors is difficult, and has so far always required additional assumptions in the interpretation of experimental data.
ii.4 Steering as a special nonlocality
Recently, Wiseman et al (WJD) (18); (19) have constructed a hybrid separability model, called the Local Hidden State Model (LHS), the violation of which is confirmation of Schrödinger’s “steering” (Figure 1). The bipartite local hidden state model (LHS) assumes
Thus, for one site which we call “Alice”, we assume a local hidden variable (LHV) state, but at the second site , which we call “Bob”, we assume a local quantum state (LQS). The violation of this model occurs iff there is a “steering” of Bob’s state by Alice (23).
WJD pointed out the association of steering with the EPR paradox (18). The EPR criterion is also a criterion for steering, as defined by the violation of the LHS model. An analysis of the EPR argument when generalized to allow for imperfect correlation and arbitrary measurements reveals that violation of the LHS model occurs iff there is an EPR paradox (20); (15). As a consequence, the violation of the LHS model is referred to as demonstration of a type of nonlocality called “EPR steering” (20). EPR steering confirms the incompatibility of local realism with the completeness of quantum mechanics, just as with the approach of EPR in their original paper (7).
The notion of steering can be generalized to consider sites, or observers (24). The multipartite LHS model is (Figure 1)
where here we have quantum states, and LHV local states. We use the symbol to represent the quantum states, since these are the“trusted sites” in the picture put forward by WJD (18). This refers to an application of this generalized steering to a type of quantum cryptography in which an encrypted secret is being shared between sites. At some of the sites, the equipment and the observers are trusted, while at other sites this is not the case.
In this picture, which is an application of the LHS model, an observer wishes to establish entanglement between two observers Alice and Bob. The violation of the QS model is sufficient to do this, provided each of the two observers Alice and Bob can be trusted to report the values for their local measurements. It is conceivable however that they report instead statistics that can give a violation of LHS model, so it seems as if there is entanglement when there is not. WJD point out the extra security present if instead there is the stronger requirement of violation of the LHV model, in which the untrusted observers are identified with a LHV state.
Cavalcanti et al (24) have considered the multipartite model (10), and shown that violation of (10) where is sufficient to imply an EPR steering paradox exists between at least two of the sites. Violation where is proof of Bell’s nonlocality, and violation where is a confirmation of entanglement (quantum inseparability).
ii.5 Hierarchy of nonlocality
WJD established formally the concept of a hierarchy of nonlocality (18); (19). Werner (25) showed that some classes of entangled state can be described by Local Hidden Variable theories and hence cannot exhibit a Bell nonlocality. WJD showed that not all entangled states are “steerable” states, defined as those that can exhibit EPR steering. Similarly, they also showed that not all EPR steerable states exhibit Bell nonlocality. However, we see from the definitions that all EPR steering states must be entangled, and all Bell-nonlocal states ( defined as those exhibiting Bell nonlocality) must be EPR steering states. Thus, the Bell-nonlocal states are a strict subset of EPR steering states, which are a strict subset of entangled states, and a hierarchy of nonlocality is established.
Iii Multiparticle Nonlocality
Experiments that have been performed on many microscopic systems support quantum mechanics. Those that test Bell’s theorem (1); (2), or the equivalent, are the most useful, since they directly refute the assumption of local realism. While these experiments still require additional assumptions, it is generally expected that improved technology will close the remaining loopholes.
There remains however the very important question of whether reality will hold macroscopically. Quantum mechanics predicts the possibility of superpositions of two macroscopically distinguishable states (6), like a cat in a superposition of dead and alive states. Despite the apparent paradox, there is increasing evidence for the existence of mesoscopic and macroscopic quantum superpositions.
As with microscopic systems, there is a need to verify the loss of reality for macroscopic superpositions in an objective sense, by following Bell’s example and comparing the predictions of quantum mechanics with those based on premises of local realism. The first steps toward this have been taken, through theoretical studies of nonlocality for multi-particle systems. Two limits have been rather extensively examined. The first is that of bipartite qudits. The second is multipartite qubits. Surprisingly, while it may have been thought that the violation of LR would diminish or vanish at a critical number of particles, failure of local realism has been shown possible according to quantum mechanics, for arbitrarily large numbers of particles. The third possibility of multipartite qudits has not been treated in as much detail.
iii.1 Bipartite qudits
The simplest mesoscopic extension of the Bell case (1) is to consider bipartite qudits: two sites of higher dimensionality. The maximally entangled state in this case is
where is abbreviation for , and is the dimensionality of the systems at and . In this case at each site and the possible outcomes are . This system can be realized by two spin systems, for which the outcomes are given by , so that , and of Eq. (11) is where is the outcome of spin. It can also be realized by multi-particle systems.
It was shown initially by Mermin, Garg and Drummond, and Peres and others (26); (27); (28); (29); (30); (31) that quantum systems could violate local realism for large . The approach was to use the classic Bell inequalities derived for binary outcomes.
Later, Kaszlikowski et al (32) showed that for maximally entangled states (11), the strength of violation actually becomes stronger for increasing . A new set of Bell inequalities for bipartite qudits was presented by Collins et al (CGLMP) et al (33); (34) and it was shown subsequently by Acin et al (35) that greater violations can be obtained with non-maximally entangled states, and that the violations increase with . Chen et al (36) have shown that the violation of CGLMP inequalities increases as to a limiting value.
We wish to address the question of how the entanglement and EPR steering nonlocalities increase with . Since Bell nonlocality implies both EPR steering and entanglement, these nonlocalities also increase with However, since there are distinct nested classes of nonlocality, the violation could well be greater, for an appropriate set of measures of the nonlocalities, and this problem is not completely solved for the CGLMP approach. We later investigate alternative criteria that show differing levels of violation for the different classes of nonlocality.
iii.2 Multipartite qubits: MABK Bell inequalities
The next mesoscopic - macroscopic scenario that we will consider is that of many distinct single particles the multi-site qubit system. The interest here began with the Greenberger-Horne-Zeilinger (GHZ) argument (8), which revealed a more extreme “all-or-nothing” form of nonlocality for the case of three and four spin particle (three or four qubits), prepared in a so-called GHZ state. The N qubit GHZ state is written
where and in this case are spin up/ down eigenstates. Mermin then showed that for this extreme superposition, there corresponded a greater violation of LR, in the sense that the new “Mermin” Bell inequalities were violated by an amount that increased exponentially with (10). These new multipartite Bell inequalities of Mermin were later generalized by Ardehali, Belinski and Klyshko, to give a set of MABK Bell inequalities (37); (38).
The MABK inequalities define moments like , where and , , , are the standard quantum spin operators. In the MABK case of qubits, Pauli operators are used, so that the spin outcomes are normalized to . The are redefined accordingly. The moments are defined generally by
where and and . A LHV theory expresses such moments as the integral of a complex number product:
where and where is the expected value of outcome for measurement made at site given the local hidden state . The is a complex number product, which Mermin (10) showed has the following extremal values: for odd, a magnitude at angle to real axis; for even, magnitude aligned along the real or imaginary axis. With this algebraic constraint, LR will imply the following inequalities, for odd :
For even , the inequality will hold. However, it is also true, for even , that (37)
The Eqns (15-16) are the MABK Bell inequalities (38). Maximum violation of these inequalities is obtained for the -qubit Greenberger-Horne-Zeilinger (GHZ) state (12) (39). For optimal angle choice, a maximum value
can be reached for the left -side of (15), while for a different optimal angle choice, the maximum value
can be reached for the left-side of (16). MABK Bell inequalities became famous for the prediction of exponential gain in violation as the number of particles (sites), , increases. The size of violation is most easily measured as the ratio of left-side to right-side of the inequalities (15,16), seen to be for the MABK inequalities. Werner and Wolf (39) showed the quantum prediction to be maximum for two-setting inequalities.
iii.3 MABK-type EPR steering and entanglement inequalities for multipartite qubits
Recently, MABK-type inequalities have been derived for EPR steering and entanglement (24). Entanglement is a failure of quantum separability, where each of the local states in (10) are quantum states (). EPR steering occurs when there is failure of the LHS model with . To summarize the approach of Ref. (24), we note the statistics of each quantum state must satisfy a quantum uncertainty relation
As a consequence, for every quantum local state ,
which implies the complex number product can have arbitrary phase, leading to the new nonlocality inequalities, which apply for all , even or odd, and :
For , these inequalities if violated will imply entanglement, as shown by Roy (40). If violated for , there is EPR steering. As pointed out in (24), the exponential gain factor of the violation with the number of particles increases for increasing : the strength of violation as measured by left to right side ratio is , but for both inequalities (21-22).
iii.4 CFRD Multipartite qudit Bell, EPR steering and entanglement inequalities
We now summarize an alternative approach to nonlocality inequalities, developed by Cavalcanti, Foster, Reid and Drummond (CFRD) (41); (42); (43); (44); (45). These hold for any operators, and are not restricted to spin-half or qubits. We shall apply this approach to the case of a hierarchy of inequalities, with some quantum and some classical hidden variable states. Consider
We can see that for any LHV, because the variance is always positive, one can derive an inequality for any operator
but then for a quantum state in view of the uncertainty relation (19), it is the case that for qubits (spin-1/2)
For the particular case of qubits, the outcomes are so that simplification occurs, to give final bounds based on local realism that are identical to (21-22). We note that at , there is also a CFRD Bell inequality, but it is weaker than that of MABK, in the sense that the violation is not as strong as is not predicted for . Since this approach holds for any operator, we now can generalize to arbitrary spin.
The expression (25-26) also holds for arbitrary spin, for which case we revert to the usual spin outcomes (rather than the Pauli spin outcomes of ). The LHV result for arbitrary spin is constrained by (25). The quantum result however requires a more careful uncertainty relation that is relevant to higher spins. In fact, for systems of fixed dimensionality , or fixed spin , the “qudits”, the following uncertainty relation holds
Entanglement is verified if ()
An EPR-steering nonlocality is verified if ()
Bell inequality (). The criterion to detect failure of the LHV theories is
These criteria will be called the “ CFRD nonlocality criteria, and allow investigation of nonlocality in multisite qudits, where the spin is fixed.
We investigate predictions for quantum states that are maximally entangled, or not so, according to measures of entanglement that are justified for pure states. Maximally-entangled, highly correlated states for a fixed spin are written
where is the eigenstate of and (eigenvalue for ), defined at site , and the dimensionality is . This state is the extension of (11) for multiple sites. We follow (35) however and consider more generally the non-maximally entangled but highly correlated spin states of form
where , . Here we will restrict to the case of real parameters symmetrically distributed around . The amplitude can be selected to optimize the nonlocality result. It is known for example, with sites and a spin- system that the optimized state:
will give improved violation over the maximally entangled state (for which the amplitudes are uniform) for some Bell inequalities (35).
With the optimization described above, we summarize the results explained in Ref. (47) that a growth of the violation of the nonlocality inequalities for increasing number of spin sites is maintained with arbitrary . This is shown in Figure 2 for qudits and (spin and ), but means for higher that one can obtain in principle a violation of inequalities for arbitrary by increasing . Thus, quantum mechanics predicts that at least for some states, increasing contradiction with separable theories is possible, as the number of sites increases, even where one has at each site a system of high spin. These results are consistent with those obtained by other authors (48); (50); (49).
Iv Genuine Multiparticle Nonlocality: qubit example
Svetlichny (11) addressed the following question. How many particles are genuinely entangled? The above nonlocality inequalities can fail if separability/ locality fails between a single pair of sites. To prove all sites are entangled, or that the Bell nonlocality is shared between all sites, is a more challenging task, and one that relates more closely to the question of multi-particle quantum mechanics.
To detect genuine nonlocality, one needs to construct different criteria. For example where , to show genuine tripartite entanglement, we need to exclude that the statistics can be described by bipartite entanglement i.e., by the models
where can be any density operator for composite system and . These models can fail only if there is genuine tripartite entanglement. Thus, to show there is a genuine tripartite Bell nonlocality, one needs to falsify all models encompassing bipartite Bell nonlocality, i.e..
and the permutations. In the expansion (36), locality is not assumed between and , but is assumed between a composite system , and . This model allows bipartite entanglement between and , but not tripartite entanglement. To test genuine nonlocality or entanglement, it is therefore useful to consider hybrid local-nonlocal models. What is a condition for genuine partite entanglement?
Consider again the -qubit system. A recent analysis (24) follows Svetlichny (11) and Collins et al. (CGPRS) (12), to consider a hybrid local-nonlocal model in which Bell nonlocality can exist, but only if shared among or fewer parties. Separability must then be retained between any two groups and of and parties respectively, if , and one can write:
Violation of all such “ - nonlocality” models then implies the nonlocality to be genuinely “ partite”. We summarize Ref. (24) who use (37) to consider consequences of the hybrid model (37) for the three different types of nonlocality. Multiplying out reveals recursive relations , which imply algebraic constraints that must hold for all theories (10)
In order to investigate the other nonlocalities, for example the genuine multipartite steering, the authors of Ref. (24) suggest the hybrid approach of quantizing , the group of qubits, but not group . In this case, the extremal points of the hidden variable product of is constrained only by the algebraic limit (38), whereas the product for group is constrained by the quantum result (18). We note that a criterion for genuine -qubit entanglement is obtained by constraining both and to be quantum, leading to the condition
In short, genuine particle nonlocality can be confirmed using MABK Bell inequalities for qubits, but a higher threshold is required. The threshold is reached by the quantum prediction of the GHZ states, but the higher bound implies the level of violation is no longer exponentially increasing with . As a related consequence, the higher threshold also implies a much higher bound for efficiency, which makes multi-particle nonlocality difficult to detect for increasingly larger systems.
V Investigating Entanglement using Collective measurements: spin squeezing inequalities
While detection of individual qubits could be fulfilled in many systems, the demonstration of a large multi-particle nonlocality would likely require exceptional detection efficiencies if one is to detect a genuine multi-particle nonlocality for large . We thus review and outline a complementary approach, which is the measurement of the collective spin of a system.
v.1 Spin squeezing entanglement criterion
Consider identical spin- particles (Figure 3). One defines the collective spin operator
and similarly a and . Entanglement between the spin particles can be inferred via measurements of these collective operators. The concept of spin squeezing was pioneered by Kitagawa and Ueda (54), and Wineland et al (55).
To investigate entanglement, we note that for each particle, or quantum site , the Heisenberg uncertainty relation holds
If the system is fully separable (no entanglement) then
For a mixture, the variance is greater than the average of the variances of the components, which for a product state is the sum of the individual variances (56). Thus, separability implies
The next point to note is that for a fixed dimensionality spin- system, there is a constraint on the minimum value for the variance of spin. The constraint on the minimum arises because of the constraint on the maximum variance, which for fixed spin must be bounded by
This implies, by the uncertainty relation, the lower bound on the minimum variance for a spin J system
Then we can prove, using (44) to get the first line,
and the Cauchy Schwarz inequality to get the second to last line (use where and ). We can rewrite and use the Cauchy-Schwarz inequality again (this time, and ), to obtain
We can express the result as
where and . For , we obtain the result that for a fully separable state,
(). This result for spin was first derived by Sorenson et al (57), and is referred to as the “spin squeezing criterion” to detect entanglement. Failure of (50) reflects a reduction in variance (hence “squeezing”), and is confirmation that there is entanglement between at least two particles (sites). The criterion is often expressed in terms of the parameter defined by Wineland et al (55), that is useful measure of interferometric enhancement, as
The spin squeezing criterion has been used to investigate entanglement within a group of atoms in a BEC by Esteve et al, Gross et al and Riedel et al (58); (13); (14). In fact, spin squeezing is predicted for the ground state of the following two-mode Hamiltonian
which is a good model for a two-component BEC. Here denotes the conversion rate between the two components, and is a self interaction term. More details on one method of solution of this Hamiltonian and some other possible entanglement criteria are given in Ref. (59). To summarize, collective spin operators can be defined in the Schwinger representation:
The system is viewed as atoms, each with two-levels (components) available to it. For each atom, the spin is defined in terms of boson operators where the total number for each atom is , and the outcomes for and are and . The collective spin defined as can then be re-expressed in terms of the total occupation number sums of each level. Figure 4 shows predictions for the variance of the collective spins or , where the mean spin is aligned along direction , as a function of ratio , for a fixed number of atoms and a fixed intercomponent coupling. In nonlinear regimes, indicated by , we see is predicted, which is sufficient to detect entanglement. Heisenberg relations imply .
Sorenson and Molmer (60) have evaluated the exact minimum variance of the spin squeezing for a fixed . Their result for agrees with (46) and also (50), but for there is a tighter lower bound for the minimum variance, which can be expressed as
where the functions are given in Ref. (60).
The above criteria hold for particles that are effectively indistinguishable. It is usually of most interest to detect entanglement when the particles involved are distinguishable, or, even better, causally separated. We ask how to detect entanglement between spatially-separated or at least distinguishable groups of spin . We examined this question in Section III, and considered criteria that were useful for superposition states with mean zero spin amplitude.
where we have for convenience exchanged the notation of and directions (compared to (54)). The expression applies when considering states which have a fixed spin , and could be useful where the mean spin is nonzero.
v.2 Depth of entanglement and genuine entanglement
We note from (54) that the minimum variance (maximum spin squeezing) reduces as increases. Sorenson and Molmer (SM) showed how this feature can be used to demonstrate that a minimum number of particles or sites are genuinely entangled (60). If
then we must have and so a minimum number of particles (where the maximum spin for a block of atoms is , we must have are involved, to allow the higher spin value.
It will be useful to summarize the proof of this result giving some detail as follows. Consider a system with the density matrix
We will consider for the sake of simplicity that the overall system has a fixed number of atoms and a fixed total spin . The density operator (57) describes a system in a mixture of states , with probability . For each possibility , there are blocks each with atoms and a total spin (note that ) (Figure 5).
We note that if the maximum number of atoms in each block is then the maximum spin for the block is . Also, if the total number of atoms is fixed, at , then , which implies that each has a definite number , meaning it cannot be a superposition of state of different numbers. Similarly, for a product state the total spin must be the sum of the individual spins (as readily verified on checking Clebsch-Gordan coefficients), which implies that if the total spin is fixed, then each has a fixed spin (that is, cannot be in a superposition state of different spins).
Using again that the variance of the mixture cannot be less than the average of its components, and that the variance of the product state is the sum of the variances of each factor state , we apply (54), that the variance has a lower bound determined by the spin . Thus we can write:
Now we can use the fact that the curves are nested to form a decreasing sequence at each value of their domain, as increases, as explained by Sorenson and Molmer. We then apply the steps based on the SM proof (lines (6) -(8) of their paper), which uses convexity of the functions . We cannot exclude that the total spin of a block can be zero, , for which , but such blocks do not contribute to the summation and can be formally excluded. We define the total spin for each but note that for fixed total spin this is equal to , and we also note that . In the later steps below, we define the total spin as and the collective spin operator .
The total spin is maximum at where is total number of atoms over both systems, but is assumed measurable. Thus, if the maximum number of atoms in each block does not exceed , then the inequality (59) must always hold. The violation of (59) is a demonstration of a group of atoms that are genuinely entangled (60).
The predictions of the model (52) are given in Figure 6, for a range of values of (the total number of atoms). In each case, there is a constant total spin, , given by where . We keep and the interwell coupling fixed, and note that the variance of decreases with increasing , while the variance in increases. Evaluation of the normalized quantities of the SM inequality (59) are given in the second plot of Figure 6. Comparing with the functions reveals the prediction of a full particle entanglement, where is an integer value.
We note this treatment does not itself test nonlocality, or even the quantum separability models (4-5) because measurements are not taken at distinct locations. However, it can reveal, within a quantum framework, an underlying entanglement, of the type that could give nonlocality if the individual spins could be measured at different locations. The great advantage however of the collective criteria is the reduced sensitivity to efficiency, since it is no longer necessary to measure the spin at each site. The depth of spin squeezing has been used recently and reported at this conference to infer blocks of entangled atoms in BEC condensates (13); (14).
To test nonlocality between sites, the criteria need will involve measurements made at the different spatial locations. How to detect entanglement between two-modes using spin operators (61); (62); (63); (64); (65), and how to detect a true Einstein-Podolsky-Rosen (EPR) entanglement (15); (16); (66); (67); (68);