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Quantum Teleportation Teleportation is commonly understood as a fictional method for transferring an object between two locations by a process of dislocation, information transmission, and reconstruction The destruction of the original object and the creation of an exact replica The actual object does not traverse the intervening distance 1 Teleportation should be impossible because of the Heisenberg Uncertainty It is impossible to measure all the attributes of a quantum state exactly It is impossible to measure both position and momentum But it is possible to exploit another aspect of the quantum theory, the notation of the entangeled states and nonlocal influences to create an exact replica of an arbitrary quantum state, but only if the original state was destroyed 2 Only “quantum states” are teleported, not the object itself We can not use this scheme to teleport an electron from one place to another But we can teleport the spin orientation of one electron at a particular location to another location The effect is nearly the same The state of the particle is destroyed and reincarnated on another particle at the destination without the original particle traversing the intervening distance If we can teleport a quantum state between two locations, and we use quantum states to encode qubits, then we can teleport a qubit between two locations 3 Quantum teleportation relies upon quantum phenomenon known as the EPR effect EPR named after Albert Einstein, Boris Podolsky, Nathan Rosen EPR effect describes the enduring interconnection between pairs of quantum systems that had in some earlier time interacted with one another Difference between local and nonlocal interactions A local interaction is one that involves direct contact or employees an intermediary that is direct contact friction, gravity Photons travel at the speed of light which is a finite speed Influence cannot propagate faster than the speed of light 4 If two events occur in the regions of space time such that no signal could ever reach one region from another, these events ought to be completely independent of one another Events are called to be spacelike separated Local interactions: They are mediated by another entity such as a particle or field They propagate no faster than the speed of light Their strength drops of with distance Locality predicts that events in spacelike separated regions ought to be independent of one another 5 All known forces in the Universe are local What is left to be nonlocal? • Collapse of the state vector • Nothing explains in quantum theory the mechanism of the collapse • Collapse of the state vector into a sharp state (eigenstate) involves no forces Nonlocal interaction is not limited by speed of light, not mediated by anything, and does not drop off in strength with distance • Conflict with Einstein’s Theory of Special Relativity • Nothing can travel faster than ligth • Problems with time Reality is non local because of the collapse of the state vector Measurement of a pairs of entangled states whose components particles are spacelike saparated 6 A state z ∈ H4 of a two-qubit system is decomposable if z can be written as a product of states in H2 z=x"y A state that is not decomposable is entangled ! The state 12 ( 00 + 01 + 10 + 11 ) Is decomposable because ! 1 00 + 01 + 10 + 11 ) = ( 2 1 1 1 = (0 0 + 0 1 + 1 0 + 1 1)= 0 + 1 )" ( (0 + 1) 2 2 2 ! 7 1 The state 2 ( 00 + 11 ) Is entangled, to prove it we assume the contrary 1 ( 00 + 11 ) = (a0 0 + a1 1 )(b0 0 + b1 1 ) = 2 ! = a0b0 00 + a0b1 01 + a1b0 10 + a1b1 11 " a0b0 = 1 2 a0b1 = 0 a1b0 = 0 1 a1b1 = 2 contradiction ! 1 ( 00 + 11 ) If two qubits are entangled state 2 then observing one of them will give 0 or 1, both with probability 1/2 It is not possible to observe ! different values on the qubits Experiments have shown that this correlation can remain even if the qubits are separated more than 10 km 1 If ( 00 + 11 ) is an input to a Hadamard 2 operator, the output is the same again ! 8 States can become entangled during unitary evolution of two separated states into entanglement states Consider two separate quantum systems " = 1 ( "a + "b 2 $0 = " % # 0 = $0 = " , # 0 = ) #0 1 ( "a + "b ) % #0 2 1 ( " a + " b ), #0 2 ! "(t) = U(T) "(0) 1 U ( #a , $0 + #b , $0 2 1 "(t) = ( #a , $a + #b , $b ) 2 "(t) = ! ) The structure of the state vector can no longer be written as a simple product We can only entangle them by applying the inverse unitary operator 9 The entangled bits or qubits of a state are called an ebit An ebit is a shared resource An ebit is allways disrtributed between two particles (qubits) 1 2 ( 00 + 11 ) An ebit provides a channel for communication Once either particle comprising the ebit is measured,!the states of both particles become definite However the channel is restricted in the sense that one cannot send an intentional message (deliberate sequence of 0’s and 1’s) The measuring process is random However a way was found... 10 Action at Distance Suppose a quantum system such as an atom emits a pair of photons Polarization of states are entangled Neither photon has a definite value for its polarization until its polarization is actually measured When are the polarized states actually determined? Second photon is determined at the instant that the polarization of the first photon is measured • No of the 4 forces in the Universe are involved • Nonlocal influence What would happen if you measured the polarization of both photons simultaneously, that no signal could travel from one photon to another No influence can propagate faster than light Correlation in the measured polarization of the photons hinted the existence of “hidden variables” that determined the polarization states of the photons from the outset 11 Bell’s Theorem Alice and Bop each have a polarizer oriented angle θ1 and θ2 Each angle can have a (v) vertical or (h) horizontal value The angle between their polarizers is θ12= θ1 - θ2 Suppose we add a third measuring polarizer θ3 from the same common reference as the two first From classical optics we arrive at Bell’s inequality states in correspondence to the “triangle inequality” 1 2 1 1 sin (" 2 # "1 ) + sin 2 (" 3 # " 2 ) $ sin 2 (" 3 # "1 ) 2 2 2 If reality is local, the inequality should be always true ! 12 However if we fix θ1=0 1 2 1 1 sin (" 2 ) + sin 2 (" 3 # " 2 ) $ sin 2 (" 3 ) 2 2 2 Is not true any more Hidden variable theories are wrong ! More than theoretical result, has been demonstrated experimentally using detection of photons Teleport of a qubit How to teleport a qubit? Let Alice be the sender and Bob the receiver A qubit is in a superposition between 0 and 1 Alice does not know the details about the qubit she wants to transfer.... 13 Prior to Alice attempting to teleport a quantum state to Bob, Alice and Bob must conspire to establish a kind of "quantum communication channel" This is done by Alice creating a pair of entangled particles, shipping one member of the pair off to Bob and keeping the other member of the pair for herself The entangled particles serve as the two ends of an ebit Their states, although individually ambiguous, are highly correlated with one another in the sense that measuring one end of the ebit instantaneously determines the value of a similar experiment performed at the other end of the ebit This "influence" however does not propagate through the space separating the ends of the ebit in the conventional sense Rather there is a lingering correlation between the states of the particles at either end of the ebit that becomes manifest when either particle is measured. 14 Below we describe a simulation of the operation of a teleportation "circuit" devised by Canadian computer scientist Gilles Brassard This is not a circuit in the usual sense because some of the lines passing through the circuit are actually severed However, this state does not pass through the circuit in the conventional sense the information in the unknown state is disassembled into a classical part and a quantum part, the two parts are transmitted through separate channels and re-combined by the receiver, Bob, to re-incarnate the original state 15 The circuit is composed of five different gates, L, R, S, T, and a XOR gate L, R, S and T are all gates that act on single qubits L and R perform rotations, and S and T perform phase shifts Rotation and Phase shift 1 #1 "1& 1 1 0 + 1 );L 1 = ( (" 0 + 1 ) % (;L 0 = 2 $1 1 ' 2 2 1 # 1 1& 1 1 L= ( 0 " 1 );L 1 = 2 ( 0 + 1 ) % (;R 0 = 2 $"1 1' 2 # i 0& S =% (;S 0 = i 0 ;S 1 = 1 $ 0 1' #"1 0 & T =% (;T 0 = " 0 ;T 1 = "i1 $ 0 "i' L= ! 16 XOR operator "1 $ 0 XOR = $ $0 $ #0 ! 0 0 0% ' 1 0 0' 0 0 1' ' 0 1 0& We embed these gates in a larger circuit so that we can create versions of these quantum operations that act on selected lines in a quantum circuit Preparation (Alice's job) For Alice to teleport an arbitrary, unknown, quantum state to Bob, she begins by creating a pair of particles whose quantum states are highly correlated with one another Alice keeps one of these particles and sends the other to Bob (via the bottom line of the teleportation "circuit") Although the two particles become physically remote from one another, the correlation between their states persists so long as neither particle is measured nor interacts with its environment in any way Such correlated particles are referred to as "ebits" in the jargon of quantum computing 17 To create the ebits, Alice pushes two standard states (two particles each in state |0>) through the following circuit. The action of the circuit is a state 1 ( 00 + 11 ) 2 ! 18 Entanglement (also Alice's job) Next, suppose Alice wants to teleport a particular quantum state to Bob Alice need not know what this state is in order to teleport it successfully, so without loss of generality we can say that the state is "unknown” To teleport the state, Alice entangles it with one of the ebits she created in Step 1 19 Measurement (Alice's last job) Next Alice measures the bits on lines 1 and 2 (the top and middle line) Top and middle output Tells what bits she measured by a conventional channel (2-bit channel) Bob's job Bob receive's Alice's 2-bit classical message and immediately converts those bits to corresponding kets for input into the quantum circuit shown below 20 ..in the upper and middle lines That even although Alice does not measure the output state at the bottom line of her circuit, it is affected by her measurements This is because it is entangled with the state on the middle line The state entering Bob’s portion of circuit contain the complete information about the unknown state " Portioned between a classical and quantum channel ! 21 Bob’s circuit can be thought of as applying a different rotation to the state on the bottom input to his circuit depending on the answer Alice send him A possible design of teleportation Computer Simulation 22 unknownState = .5 ket[0] + Sqrt[1-.5^2] ket[1] 0.5 ket[0] + 0.866025 ket[1] The joint state of the 3 qubits after Alice has pushed the 3 inputs through her circuit: 0.25 ket[0, 0, 0] + 0.433013 ket[0, 0, 1] + 0.433013 ket[0, 1, 0] + 0.25 ket[0, 1, 1] - 0.25 ket[1, 0, 0] + 0.433013 ket[1, 0, 1] + 0.433013 ket[1, 1, 0] - 0.25 ket[1, 1, 1] Next Alice measures the bits on lines 1 and 2 {{1, 1}, 0.866025 ket[1, 1, 0] - 0.5 ket[1, 1, 1]} Bit of line 1 and 2 are both one 23 All possible values could be: Bob receive's Alice's 2-bit classical message {1, 1} and immediately converts those bits to corresponding kets for input into the quantum circuit • 0.5 ket[1, 1, 0] + 0.866025 ket[1, 1, 1] • 0.5 ket[0] + 0.866025 ket[1] 0 24 25