Spin entanglement, decoherence and Bohm's EPR paradox

We obtain criteria for entanglement and the EPR paradox for spin-entangled particles and analyse the effects of decoherence caused by absorption and state purity errors. For a two qubit photonic state, entanglement can occur for all transmission efficiencies. In this case, the state preparation purity must be above a threshold value. However, Bohm's spin EPR paradox can be achieved only above a critical level of loss. We calculate a required efficiency of 58%, which appears achievable with current quantum optical technologies. For a macroscopic number of particles prepared in a correlated state, spin entanglement and the EPR paradox can be demonstrated using our criteria for efficiencies {\eta}>1/3 and {\eta}>2/3 respectively. This indicates a surprising insensitivity to loss decoherence, in a macroscopic system of ultra-cold atoms or photons.

In the development of quantum information science, entanglement is central.It is at the heart of the Einstein-Podolsky-Rosen (EPR) paradox (author?) [1], which demonstrated the incompatibility between local realism (LR) and the completeness of quantum mechanics.The unambiguous demonstration of the failure of local realism has proved elusive, although many experimental groups are actively pursuing it.However, we show that an important step in that direction is within reach of current experimental technique.Here we obtain signatures of either entanglement or an EPR paradox, purely from measured correlations without assumptions about the quantum state.We obtain simple criteria for twin photon sources, which require eciencies that are now achievable with current detectors.We also extend these benchmarks to macroscopic quantum states and show their application to nondegenerate parametric down-conversion.These criteria are expected to have practical applications in areas of quantum information science such as quantum cryptography.
The issue of how to deal with decoherence -the result of losses -is vital in understanding both entanglement and the EPR paradox.In practice, one measures these eects by counting photons.Both coincident and noncoincident photo-counts occur.Some counts are lost due to detector ineciency, and some counts are added from background noise.One way to deal with this is to ignore the non-coincident counts, but this leads to an ambiguity: was the count that was ignored really part of the original quantum state or not?For unambiguous signatures of entanglement and EPR, all of the actual results given by the quantum measurement should be included(author?)[4].
Here we establish signatures of entanglement and EPR paradox for lossy photon correlation experiments, and give the required levels of detection eciency.In the ideal case of an initial Bell state, we nd that entangle- This level of eciency has been reported in a high-ux photon-pair experiment (author?) [5,6].Unambiguous evidence of an EPR paradox is more demanding, requiring η EP R = 1/ √ 3.While higher than that required for entanglement, this is a milestone that also is achievable with current detectors.
We start by recalling the case of the Bell inequality.
Bell(author?) [2] showed that quantum mechanics predicts results which cannot be obtained with any local realistic theory.It is hard to overemphasize the importance of this result, as it allows one to distinguish quantum from classical reality.It is known that detector eciency is a vital issue for the Bell inequality: there is a minimum detector eciency needed in order to violate the original Bell inequalities, which would completely rule out all classical local theories.
In the Bell case, it is assumed that a strongly entangled photon pair is produced in an ideal Bell state |Ψ S .
This can be visualized as a pair of particles with anticorrelated spin vectors J A and J B , as depicted in Fig ( 1), corresponding to the singlet quantum state: : that holds for any quantum state ρ A at Alice's site A.
Here the spin operators are dened as: The total observed photon number operator at the given location is N A = a † + a + + a † − a − .The notation a † ± indicates the creation operator for pho- tons at site A , with polarization ±.At Bob's location, B , J B x , J B y , J B z and N B are dened in terms of b ± .
The entanglement condition follows by assuming separability, ie, that the composite system ρ can be represented as a mixture of factorizable states of type ρ A ρ B .Introducing collective spin operators, dened as J i = J A i + J B i , a sucient condition for entanglement is (author?) [7]: where the particle number operator is This condition is extremely useful in dealing with nonideal states that are found in real experiments.In the laboratory, there are both additional randomly polarized photons, and various forms of loss that mean that only one photon is detected, instead of two.We therefore start with a non-ideal state composed of a singlet Bell state (1) with probability p, and a state with a photon of random polarization at each detector, with probability 1−p.This is customarily called a Werner state(author?)[18], ρ W = pρ S + (1 − p)I 2 .Here I 2 indicates an identity operator in this two-photon subspace.Next, to include losses, we suppose the overall detection eciency is η, so the mean number of counts per observer is N/2 = η.
The left-hand side of Eq (3) is thus: This gives unambiguous evidence of an entangled state for ηp > 1/3, including background counts.For η = 1, Due to the simplicity with which correlated photon states can be generated, it is the Bohm version(author?)[9] , with correlated photon polarizations, which is most often cited as an illustration of the EPR paradox.However, just as with entanglement, one must ask what one must measure to demonstrate conclusive evidence for the situation depicted in the EPR paradox.Intriguingly, we will show that there is a close relationship between the entanglement and EPR criterion.The EPR criterion involves a similar, but stronger inequality to the entanglement case, and hence a higher eciency threshold.From the experimental perspective, the EPR target, though more demanding than unambiguous entanglement, is still within reach of current detector technology.
Our route to an unambiguous EPR paradox is via an inference argument together with the known quantum uncertainty principle (2) for spins -the same uncertainty principle used in the entanglement criterion.We thus present an EPR criterion, which simply requires that the inferred variance of Bob's measurements must be less than any possible quantum state can have; that is: Measurement schemes for detection of all the quantities in ( 5), following methods demonstrated in a number of recent experiments(author?) [12] , are depicted in Fig- Here inf J B φ proceeds as in (author?) [11,12], and N B ( S B 0 in (author?) [11]) is measured by taking the sum rather than the dierence current at B. PBS is a polarising beam splitter; θ,φ represent the choice to measure J A θ , J B φ , and can be achieved by half and quarterwave plates (author?) [13,14,15].A single down-conversion crystal can be used for this.measurement J B x given an outcome J A x .This inferred uncertainty is nothing more than the average error associated with the inferred result for a remote measurement J B x , given measurement of J A x .To prove the EPR cri- terion (5), one considers the conditional distributions as predictions for B given A. If local realism holds, the predetermined prediction for J B x means there is a cor- responding localised state ρ B at B. This is because if the systems are causally separated, according to LR, the measurement at B does not induce immediate change to A. In other words, LR would tell us that any predetermined knowledge of the result of a remote measurement can only occur because the remote system is already in a predetermined state admitting these predictions.EPR called such predetermined states elements of reality.
In the case of Bohm's EPR paradox, the assumption of local realism means that elements of reality exist for each of the spins J B x , J B y , J B z .The variances associ- ated with the prediction for each of them are respectively, Where we satisfy (5), EPR's elements of reality defy the quantum uncertainty relation (2).That is, it is impossible to represent Einstein's proposed element of a reality as a quantum state ρ B .
In this way the contradiction of LR with the completeness of quantum mechanics is able to be experimentally demonstrated.This is an important conceptual boundary, which demonstrates the inadequacy of the classical concept of local realism in dealing with quantum states.
However, real detectors are not 100% ecient, and this eect increases the uncertainties associated with the inference of measurements at Bob's location, given Alice's results.We take the Bell state and calculate the EPR inequality including the eect of the detection ineciency.The RHS of (5 Finally, and possibly most importantly, we will show that these unambiguous entanglement and EPR measures can be used for macroscopic states with more than one particle per site.This gives a much more powerful test of quantum measurement theory than the simple Bell state, as it tests features of quantum reality in domains that become meso-or macroscopic.In this domain, a number of alternatives to quantum mechanics have been suggested, where quantum superpositions are prevented from forming via novel mechanisms such as couplings to gravitational eects(author?)[10].To test for quantum eects in such macroscopic cases, we rst consider the way in which the relevant states would be generated in practice.We consider the macroscopic version of the Bell state (1), using the Schwinger representation An example of a physical system of interest is shown in Fig. (2).This generates the states of Eq. ( 6) using two parametric ampliers as modeled by the interaction Hamiltonian With an initial vacuum state, the solution after a time t is a superposition of the |ψ j .Predictions of a particular |ψ j could be tested by restricting to the subensemble with a xed outcome of N B .However, we propose instead a second experimental regime involving the entire ensemble.in order to test entanglement and EPR-Bohm correlations in the limit of large N B .In this regime, higher detection eciencies (η ≈ 0.9) can be achieved, although a precise photon count, which would enable a test of Bell's inequality for this state (author?) [16,17,18], is not possible.The solutions are readily obtained to The eect of detection ineciency can be analysed using a standard beam splitter model which adds vacuum terms so that nal outputs after loss become a where the a ±,0 and b ±,0 represent independent vacuum inputs.With this we get (J A Z ) 2 = (1/2)ηsinh 2 r(1 and N = 4ηsinh 2 r, where r = |κ|t.For all N , eciencies η > 1/3 are enough to demonstrate unambiguous entanglement via (3).A dierent result is obtained for the case of the spin EPR correlations.Here the eciency required to satisfy (5) increases with particle number N .We calculate the inference variances for this Gaussian system using a linear estimate approach (author?) [20], where the estimate for the result of the remote measurement J B x is simply J B θ,est = gJ A θ , so that the average inference vari- We note that since the true best estimate is the mean of the conditional, these linear estimates, for non-Gaussian states, give variances greater than or equal to those dened by the conditonal variances.Nonetheless, they are often easier to measure, and their substitution into (5) will still allow a test of Bohm's spin EPR paradox.We calculate the linear inference variance for (7) by selecting g and N B = 2ηsinh 2 r.Fig. 2 plots the minimum eciency η required for satisfaction of ( In summary, unambiguous entanglement is provable with an overall eciency of η = 1/3.This is now feasible (author?) [5,6], thus clearly providing a route towards unambiguous entanglement measurements.Further improvements should enable a denitive demonstration of EPR correlations for causally separated photo-detectors, which is the rst step towards demonstrating an unambiguous Bell inequality violation.This requires an efciency threshold of η = 1/ √ 3.Even higher eciencies of η > 2/3 would allow the unambiguous detection of macroscopic EPR entanglement, in photonic or massive(author?)[10,21,22] atomic systems, which would test quantum mechanics in completely new regimes.

Figure 1 :
Figure 1: Schematic diagram of an experiment with correlated spins at spatially-separated locations A and B.

Figure 2 :
Figure 2: Apparatus to detect an EPR paradox of Bohm's type for spin.Measurement of ∆ 2
calculation agrees with previously known results for the Werner state.For a pure Bell state with p = 1, the minimum eciency requirement is η EN T = 1/3.
must allow a local state to be inferred at Bob's location -assuming local realism.If this inferred state has a lower uncertainty than allowed by quantum mechanics, the situation of the EPR paradox is obtained.Thus, either quantum mechanics is incomplete (it fails to fully describe the inferred state, since this violates the uncertainty principle) or local realism is false (local prediction does not imply a local element of reality).This logic is central to the EPR argument applied to real experiments.
5), to indicate a test of macroscopic EPR for large N B and η > 2/3.suitable to causal experiments with large separations between the detectors.Another potential application is to atomic entanglement, either via internal spin or spatial modes of ultra-cold atoms, where once again there are large advantages in making entanglement or EPR measurements with no local oscillator, thus eliminating a major source of phase decoherence.
been achieved with Bell states.The spin EPR correlations proposed here are closely linked to polarization squeezing measurements, which do not require a separate local oscillator.Therefore, these types of measurement appear most