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Probabilistic Cloning: Difference between revisions
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The probabilistic cloning machine is a quantum cloner which is only able to produce copies of a limited set of linearly-dependent states with a probability of success. Despite deterministic protocols which usually consist of performing a unitary transformation, probabilistic cloning protocols are continued by a measurement. Two cases for probabilistic cloning are discussed below, general case and special case of two qubits. The two-qubit case means that our probabilistic machine is able to copy two non-orthogonal qubits efficiently. A probabilistic cloning protocol has three stages: blank or ancillary state preparation, unitary evolution and, measurement. The protocol can be described as follows | The probabilistic cloning machine is a quantum cloner which is only able to produce copies of a limited set of linearly-dependent states with a probability of success. Despite deterministic protocols which usually consist of performing a unitary transformation, probabilistic cloning protocols are continued by a measurement. Two cases for probabilistic cloning are discussed below, general case and special case of two qubits. The two-qubit case means that our probabilistic machine is able to copy two non-orthogonal qubits efficiently. A probabilistic cloning protocol has three stages: blank or ancillary state preparation, unitary evolution and, measurement. The protocol can be described as follows | ||
[[File:Probabilisticcloner.jpg|right|thumb|1000px| Probabilistic cloner circuit for a single-qubit state (from set of two nonorthogonal but linearly independent state): Qubits <math>x</math>, <math>y</math> and <math>z</math> are the original qubit, blank qubit, and assistant qubit, respectively.<math>\circ</math> and <math>\bullet</math> respectively mean that the control states are <math>|0\rangle</math> and <math>|1\rangle</math>. <math>\otimes</math> indicates the operational qubit of the CNOT gate. <math>U_1</math> and <math>U_2</math> denote the unitary operations controlled by qubit x (explained in details in the Pseudo-code section). <math>H</math> is the Hadamard gate. <math>PM</math> represents the measurement.]] | [[File:Probabilisticcloner.jpg|right|thumb|1000px| Probabilistic cloner circuit for a single-qubit state (from set of two nonorthogonal but linearly independent state): Qubits <math>x</math>, <math>y</math> and <math>z</math> are the original qubit, blank qubit, and assistant qubit, respectively.<math>\circ</math> and <math>\bullet</math> respectively mean that the control states are <math>|0\rangle</math> and <math>|1\rangle</math>. <math>\otimes</math> indicates the operational qubit of the CNOT gate. <math>U_1</math> and <math>U_2</math> denote the unitary operations controlled by qubit x (explained in details in the Pseudo-code section). <math>H</math> is the Hadamard gate. <math>PM</math> represents the measurement.]] | ||
*'''General case:''' First, an orthonormal set of states should be prepared. These states will be used as ancillary states for the protocol. Then a unitary transformation will take the input states to a [[superposition]](linear combination of quantum states) of a successful two identically copied states and unsuccessful two-qubit states along with probes states to be measured later. The last stage is measurement. | *'''General case:''' First, an orthonormal set of states should be prepared. These states will be used as ancillary states for the protocol. Then a unitary transformation will take the input states to a [[superposition]](linear combination of quantum states) of a successful two identically copied states and unsuccessful two-qubit states along with probes states to be measured later. The last stage is the measurement. The output is measured in the basis of the ancillary orthonormal states. With some probability of success, the result of the measurement is the desired outcome and the final state collapses to the two identical desired copies. Otherwise, the protocol is aborted. | ||
*'''Two qubit case:''' This special case is a more practical and clear example of the above general probabilistic cloner. One is | *'''Two qubit case:''' This special case is a more practical and clear example of the above general probabilistic cloner. One is only interested in two non-orthogonal qubits to be effectively copied by the probabilistic cloning machine. Here, the orthonormal bases are <math>|0\rangle</math> and <math>|1\rangle</math>. At the first stage of the protocol, only two blank states (<math>|0\rangle|0\rangle</math>) are needed. At the second stage, the unitary transformation takes the states to a superposition of two identical states along with the state <math>|0\rangle</math> and a two-qubit state (wrong state) along with the qubit <math>|1\rangle</math>. At the final stage, the third qubit is be measured in basis <math>\{|0\rangle,|1\rangle\}</math>. If the result of the measurement is <math>|0\rangle</math> the protocol is successful. Otherwise, the round is discarded. This construction can be shown more clearly in the quantum circuit below. | ||
==Notations Used== | ==Notations Used== | ||