Measurements_on_an_entangled_state EPR_paradox

the epr thought experiment, performed electron–positron pairs. source (center) sends particles toward 2 observers, electrons alice (left) , positrons bob (right), can perform spin measurements.

alice measures spin along z-axis. can obtain 1 of 2 possible outcomes: +z or −z. suppose gets +z. according copenhagen interpretation of quantum mechanics, quantum state of system collapses state i. quantum state determines probable outcomes of measurement performed on system. in case, if bob subsequently measures spin along z-axis, there 100% probability obtain −z. similarly, if alice gets −z, bob +z.

there is, of course, nothing special choosing z-axis: according quantum mechanics spin singlet state may equally expressed superposition of spin states pointing in x direction. suppose alice , bob had decided measure spin along x-axis. ll call these states ia , iia. in state ia, alice s electron has spin +x , bob s positron has spin −x. in state iia, alice s electron has spin −x , bob s positron has spin +x. therefore, if alice measures +x, system collapses state ia, , bob −x. if alice measures −x, system collapses state iia, , bob +x.

whatever axis spins measured along, found opposite. can explained if particles linked in way. either created definite (opposite) spin every axis—a hidden variable argument—or linked 1 electron feels axis other having spin measured along, , becomes opposite 1 axis—an entanglement argument. moreover, if 2 particles have spins measured different axes, once electron s spin has been measured x-axis (and positron s spin x-axis deduced), positron s spin z-axis no longer certain, if (a) knows measurement has taken place, or (b) has definite spin already, second axis—a hidden variable. however, turns out predictions of quantum mechanics, have been confirmed experiment, cannot explained local hidden variable theory. demonstrated in bell s theorem.

in quantum mechanics, x-spin , z-spin incompatible observables , meaning heisenberg uncertainty principle applies alternating measurements of them: quantum state cannot possess definite value both of these variables. suppose alice measures z-spin , obtains +z, quantum state collapses state i. now, instead of measuring z-spin well, bob measures x-spin. according quantum mechanics, when system in state i, bob s x-spin measurement have 50% probability of producing +x , 50% probability of -x. impossible predict outcome appear until bob performs measurement.

here crux of matter:

you might imagine that, when bob measures x-spin of positron, answer absolute certainty, since prior hasn t disturbed particle @ all. turns out bob s positron has 50% probability of producing +x , 50% probability of −x, meaning outcome not certain. s if bob s positron knows alice has measured z-spin of electron, , hence positron s own z-spin must set, x-spin remains uncertain.

put way, how bob s positron know way point if alice decides (based on information unavailable bob) measure x (i.e., opposite of alice s electron s spin x-axis) , how point if alice measures z, since supposed know 1 thing @ time? copenhagen interpretation rules wave function collapses @ time of measurement, there must action @ distance (entanglement) or positron must know more s supposed know (hidden variables).

here paradox summed up:

it 1 thing physical measurement of first particle s momentum affects uncertainty in own position, measuring first particle s momentum affects uncertainty in position of other thing altogether. einstein, podolsky , rosen asked how can second particle know have precisely defined momentum uncertain position? since implies 1 particle communicating other instantaneously across space, i.e., faster light, paradox .

incidentally, bell used spin example, many types of physical quantities—referred observables in quantum mechanics—can used. epr paper used momentum observable. experimental realisations of epr scenario use photon polarization, because polarized photons easy prepare , measure.