PHY.8:5.2 - Recover collective fluctuations from a prepared joint state
A readout measures the z component of N=100 spin-1/2 systems along one common axis. Write each normalized outcome as s_i=+1 or -1; the physical angular momentum is (hbar/2) s_i. The collective normalized signal is M=sum_i s_i.
Three ideal preparations give every individual spin equal probabilities of +1 and -1:
| Preparation during readout | Joint property used | Mean M | Variance of M |
|---|---|---|---|
| Independently prepared maximally mixed spins | Outcomes along the common axis are independent | 0 | 100 |
| Fifty independent singlet pairs | The two outcomes in every pair are opposite | 0 | 0 |
| With equal probabilities, all spins prepared up or all prepared down | Every outcome shares the same prepared sign | 0 | 10000 |
For a singlet pair, the state is (|up down>-|down up>)/sqrt(2). Its ideal common-axis measurements give opposite results, so each pair contributes zero to M. For the last preparation, M itself is +100 or -100 with equal probabilities. The variance entries follow from these joint properties and from addition of independent variances in the first preparation.
A readout designed only from the individual mean would predict the same zero signal in all three cases. Their root-mean-square collective signals are 10, 0 and 100. If saturation occurs when |M| exceeds 50, the last preparation always saturates. The first has probability at most 100/50²=0.04 by the variance bound, and the ideal paired preparation never saturates.
Now retain the paired preparation but read only one spin from each pair. The sum of those fifty outcomes has variance 50, because different pairs were prepared independently. The zero-variance conclusion applied to a complete-pair sum, not to an arbitrary selected subset.
This calculation uses joint states and a stated measurement. Common-axis anticorrelation alone would also be compatible with other preparations; it does not identify the singlet uniquely. Recovering entanglement would require a different question and suitable measurements. Detector errors or correlations between pairs would change the recording law or the preparation premise.