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tanglemenEntTelepoAlicytileFidrtate andBobNonlocaiol influencesn(a) Paranormal phenomena(b) Men are from Mars. Women are from Venus(c) Physics
Classical phase-space descriptions ofcontinuous-variable teleportationCarlton M. Caves and Krzysztof WodkiewiczPRL 69, 040506 (2004), quant-ph/0401149)Fidelity of Gaussian channelsCarlton M. Caves and Krzysztof Wodkiewicz“Fidelity of Gaussian channels,” Open Syst & Inf Dynamics 11, 309 (2004)Teleportation fidelity andsub-Planck phase-space structureCarlton M. Caves and Andrew J. Scottin preparationUNM Information Physicsinfo.phys.unm.edu
This photo shows JeremyCaves walking faster thanthe shutter speedsomewhere in Australia.Where is it?
This photo shows ascene somewhere inNew Mexico.Where is it?
Continuous-variable teleportationFurasawa et al., Science 282, 706 (1998)Bowen et al., PRA 67, 032302 (2003)Zhang et al., PRA 67, 033792 (2003)Teleportation ofcoherent states
Continuous-variable teleportationGibberishAliceBobTwo units of noiseUncertainty-principle limits on simultaneous measurements of x and p require
Continuous-variable teleportationMeasureCommunicateGibberishNo cloningDisplaceWhoa! This is a classicalphase-space description(local hidden-variable model)
Wigner function: A phase-space quasidistributionMarginals of the Wigner function give the statistics of measurements of thephase-space variables, x or p, or any linear combination of x and p.Zurek’s “compass” state: Superposition of four coherent states12 Φρ 10πQ(π/2)Wρρ0.55p00 5 10 0.5(a) 10(b) 50510 10(c) 50x510 10 50510 1
Wigner function: A phase-space quasidistributionMarginals of the Wigner function give the statistics of measurements of thephase-space variables, x or p, or any linear combination of x and p.Random superposition of first 100 number states12 Φρ 15πQρ(π/2)Wρ100.55p00 5 0.5 10 15(a) 15 10 5(b)0510 15 15 10 5(c)0x510 15 15 10 50510 15 1
Sub-Planck structureCompass state: Superposition of four coherent states12 Φρ 10πQ(π/2)Wρρ0.55p00 5 10 0.5(a) 10Measure(b) 50510 10 50x(c)510 10 50510 1
Sub-Planck structureRandom superposition of first 100 number states12 Φρ 15πQρ(π/2)Wρ100.55p00 5 0.5 10 15(a) 15 10 5(b)0510 15 15 10 5(c)0x510 15 15 10 50510 15 1
Continuous-variable teleportation: Wigner functionsCommunicateMeasureDisplace
Continuous-variable teleportation: Wigner functionsFor Gaussian input states (this includes coherent states), all of which havepositive Wigner functions, the Wigner function provides a classical phasespace description (local hidden-variable model). The hidden variables arethe phase-space variables for the three modes. The protocol runs on theclassical correlations between the phase-space variables, described by theWigner function.Yet if Alice and Bob share an unentangled state in modes A and B, themaximum fidelity for teleporting a coherent state using the standardprotocol is 1/2. For teleporting coherent states, a fidelity greater than 1/2requires a shared resource that quantum mechanics says is entangled, butwhich is used in a way that can be accounted for by classical correlationsof local hidden variables.
Continuous-variable teleportation: Wigner functionsCommunicateMeasureDisplace
Continuous-variable teleportation: Input and entanglementSqueeze parameter rMeasure of EPR correlations
Continuous-variable teleportation: Output and fidelity
Teleportation fidelity and sub-Planck structureCompass state: Superposition of four coherent states12 Φρ 10πQ(π/2)Wρρ0.55p00 5 10 0.5(a) 10(b) 50510 10(c) 50x5 1010 50510 1Random superposition of first 100 number states1215 Φρ πQρ(π/2)Wρ100.55p00 5 0.5 10 15(a) 15 10 5(b)0510 15 15 10 5(c)0x510 15 15 10 50510 15 1
Teleportation fidelity and sub-Planck structureCompass state: Superposition of four coherent states12 Φρ 10πQρ(π/2)Wρ0.55p00 5 10 0.5(a) 10(b) 5054510 10(c) 50x510 10 50510 110.80.6F0.40.200123t6Our measure of small-scale phase-structurefinds an operational significance in terms ofthe difficulty of teleportation.
Teleportation of non-Gaussian pure statesAll non-Gaussian pure states have Wigner functions thattake on negative values, so they cannot be incorporatedin the classical phase-space description.“Kicking and cheating” protocol Alice kicks the input state randomly with a Gaussian kicking strength tchosen so that the kicked state has a positive Wigner function and thus canbe incorporated within the classical phase-space description. The minimumrequired kicking strength is t 1 for all non-Gaussian pure states. Alice and Bob cheat by teleporting the kicked state with perfect fidelity.The fidelity achieved by the “kicking and cheating” protocol is the t 1 fidelitywithin the standard quantum protocol.Gold standard for continuousvariable teleportation ofnon-Gaussian pure states?
?Turn the overlap question forone mode into an expectationvalue for a pair of modes.This same technical result shows that if Alice and Bob sharean unentangled state, the maximum fidelity for teleporting acoherent state using the standard protocol is 1/2.
The “kicking with added communication” protocolAlice communicates her actual kick to Bob, who removes it aftercompleting the protocol, thereby achieving the quantum fidelity atthe cost of additional classical communication from Alice to Bob.
There is a classical phase-space description (local hidden-variablemodel) for teleportation, with arbitrary fidelity, of all Gaussianpure states, including coherent states, using the standard protocol.If Alice and Bob share an unentangled state, they cannot teleport acoherent state using the standard protocol with fidelity greaterthan 1/2. For teleporting coherent states, a fidelity greater than1/2 requires a shared resource that quantum mechanics says isentangled, but which is used in a way that can be accounted for byclassical correlations of local hidden variables.SummaryThere is no classical phase-space description of the teleportation ofany non-Gaussian pure state for fidelities greater than or equal to2/3, even if one allows Alice and Bob to cheat by using a protocolwith perfect fidelity.Alice and Bob can achieve the quantum fidelity within a “kicking withadded communication” protocol in which Alice communicates the kick,at the cost of upping the classical communication rate by 50%.
This photo shows JeremyCaves walking faster thanthe shutter speedsomewhere in Australia.Where is it?Echidna GorgeBungle Bungle RangePurnululu NPWestern Australia2004 June 28
This photo shows ascene somewhere inNew Mexico.Where is it?Sawtooth RangeWest of VLA2003 August 31
E n t a n gle. m e n t T e l e p o r t a N o n lo c a l i n fl ue n c e s ti o n A l i c e a nd. B o b F i d e l i t y (a) Paranormal phenomena (b) Men are
