After inserting both the particles present in different quantum states into the imaginary Cloning machine the resultant particles would be a completely new set of particles as due to non-unitary transformations the exactness of the electron would be lost in a higher amount and superposition of both particles may take place resulting into a new and different particle almost every time the experiment takes place (Ignoring the probabilities which coincide with the previous results). Keeping this perspective aside, and if we consider Non Unitary Operations and an electron as the particle to be cloned and a photon as the material over which the copying procedure takes place. 1 and 0 which means either both the quantum states represent the same state or they are perpendicular to each other which prove that a general cloning device is impossible. Therefore, this equation has two admissible solutions i.e. Scalar Product of equation 1 and 2 yields: Therefore, the copying process can be expressed as:Ĭonsidering another state of particle lØ> and applying the same copying procedure, We use the Unitary Operation to undergo the copying process. the material to be cloned is any arbitrary unknown state lφ>2 existing in Hilbert Space = H2 where, H1=H2. Suppose, an electron is in the Hilbert Space = H1 and quantum state lΨ>1 and the other particle i.e. In order to illustrate we will consider two different particles present in two different quantum states. The square amplitude $|\psi|^2$ only acts as a probability when the system concerned has undergone decoherence.According to the No-Cloning theorem, a perfect identical copy of the particle (Quantum) cannot be created without knowing every possible detail about the particle, which is technically not possible keeping Heisenberg's Uncertainty Theorem into consideration. You are also confused about what the amplitudes mean in quantum mechanics. In addition, if you have many systems prepared in the same way you can perform measurements to find the state in which the system was prepared to any desired degree of accuracy - this is called quantum tomography: The no-cloning theorem is perfectly consistent with being able to prepare two different systems in the same state to an arbitrarily good approximation since this doesn't involve copying. That is, no physical system can perform the operation $|\psi\rangle_S|0\rangle_C \to |\psi\rangle_S|\psi\rangle_C$ for an arbitrary state. There is no physical process that can take that system and copy its state to another system in a blank state $|0\rangle_C$. Suppose you have a quantum system S in an arbitrary state $|\psi\rangle_S$. You have misunderstood the no-cloning theorem. Put in other words, if I can't replicate a wave-function at all then how do I confirm in my laboratory that the probabilities obtained by $|\psi|^2$ actually represents the likely-hood of the collapse happening in a given region?Įdit Is it the catch that the No-Clone Theorem suggests only that a given state can't be evolved to a state identical to another state and it allows the two states being identical if they are so from eternity? This seems like a good enough argument to accept $|\psi|^2$ as the probability density of the wave-function collapsing in a given infinitesimal small region even if I have only one wave-function.īut the No-Clone Theorem suggests that it is fundamentally impossible to make two states that are completely identical and thus, to my understanding, it makes absolutely no sense to talk about those identical wave-functions that were used to project $|\psi|^2$ as the probability density. The way this probability is interpreted (at least in the text-book by Griffiths) is that if I prepare a large number of identical states and then perform a measurement on each of them then the probability associated with $|\psi|^2$ actually represents the statistical results of the measurements made (individually) on all these wave-functions of the entire ensemble. The standard interpretation of $|\psi|^2$ is taken as the probability density of the wave-function collapsing in the given infinitesimal small region.
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