| No-cloning Theorem |
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Information AboutNo-cloning Theorem |
| CATEGORIES ABOUT NO CLONING THEOREM | |
| quantum information science | |
| physics theorems | |
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Note that the state of one system can be identically Entangled with the state of another system, such as by using a CNOT Gate , but this does not constitute cloning since the systems will always yield the same value upon measurement. The no cloning theorem describes the inability to make separately measurable states. PROOF Suppose the state of a quantum system A is a Qubit , which we wish to copy. The state can be written (see Bra-ket Notation ) as | ||
| The | "http://wwwinformationdelightinfo/encyclopedia/entry/complex_number" class="copylinks">Complex coefficients ''a'' and ''b'' are unknown to us In order to make a copy, we take a system B with an identical Hilbert Space and initial state <math>e
angle_B</math> (which must be independent of <math>\psi
angle_A</math>, of which we have no prior knowledge) The composite system is then described by the Tensor Product , and its state is |
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| There Are Only Two Ways To Manipulate The Composite System We Could Perform An Observation, Which Irreversibly | "http://wwwinformationdelightinfo/encyclopedia/entry/Wavefunction_collapse" class="copylinks">Collapses the system into some Eigenstate of the Observable , corrupting the information contained in the qubit This is obviously not what we want Alternatively, we could control the Hamiltonian of the system, and thus the Time Evolution Operator U(&Delta''t''), which is Linear We must fix a time interval &Delta''t'', again independent of <math>\psi
angle_A</math> Then U(&Delta''t'') acts as a copier provided |
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| :<math>U(\Delta T) 0 Angle A e Angle B | 0
angle_A 0
angle_B </math> |
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| :<math>U(\Delta T) 1 Angle A e Angle B | 1
angle_A 1
angle_B</math> |
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| Thus, <math>U(\Delta T) \psi Angle A e Angle B</math> Is Generally Not Equal To <math>\psi Angle A \psi Angle B</math>, As May Be Verified By Plugging In ''a'' | ''b'' = 2<sup>-1/2</sup>, so U(&Delta''t'') cannot act as a general copier QED |
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| If Alice Wishes To Transmit A "0", She Measures The Spin Of Her Electron In The '''z''' Direction, Collapsing Bob's State To Either <math>z+ Angle B</math> Or <math>z- Angle B</math> Bob Creates Many Copies Of His Electron's State, And Measures The Spin Of Each Copy In The '''z''' Direction Bob Will Know That Alice Has Transmitted A "0" If All His Measurements Will Produce The Same Result Otherwise, His Measurements Will Be Split Evenly Between +1/2 And -1/2 This Would Allow Alice And Bob To Communicate Across | "http://wwwinformationdelightinfo/encyclopedia/entry/space-like" class="copylinks">Space-like separations, potentially violating Causality |
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