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Use Quantum See also index module

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	== See also ==		== See also ==
			{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	* [[Quantum discord]]
	* [[Physics:~~Quantum process~~\|~~Quantum process]]~~
	* {{slink\|Quantum ~~entanglement#Entanglement of top quarks~~}}

	== References ==		== References ==

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 {{Short description|Reconstruction of quantum states based on measurements}}
-'''Quantum [[Tomography|tomography]]''' or '''quantum state tomography''' is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states.<ref>Quantum State Tomography. {{cite web|url=http://research.physics.illinois.edu/QI/Photonics/tomography/|title=UIUC}}</ref> The source of these states may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general [[Physics:Mixed state|mixed state]]s. To be able to uniquely identify the state, the measurements must be '''tomographically complete'''. That is, the measured [[Operator (mathematics)|operators]] must form an [[Operator (mathematics)|operator]] [[Basis (linear algebra)|basis]] on the [[Hilbert space]] of the system, providing all the information about the state. Such a set of observations is sometimes called a '''quorum'''. The term ''tomography'' was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography.<ref>{{Cite journal |last1=Smithey |first1=D. T. |last2=Beck |first2=M. |last3=Raymer |first3=M. G. |last4=Faridani |first4=A. |date=1993-03-01 |title=Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum |journal=Physical Review Letters |volume=70 |issue=9 |pages=1244–1247 |doi=10.1103/physrevlett.70.1244 |pmid=10054327 |bibcode=1993PhRvL..70.1244S |issn=0031-9007}}</ref>  [[File:Figure 1.PNG|frame|right|alt=Figure 1: One harmonic oscillator represented in the phase space by its momentum and position|Figure 1: One harmonic oscillator represented in the phase space by its momentum and position]] [[File:ManyHarmonicOscillatorsPhaseSpaceRepresentation.PNG|frame|right|alt=Figure 2: Many identical oscillators represented in the phase space by their momentum and position|Figure 2: Many identical oscillators represented in the phase space by their momentum and position]]
+{{Quantum book backlink|Measurement and information}}
 In '''quantum process tomography''' on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, '''quantum measurement tomography''' works to find out what measurement is being performed. Whereas, [[Randomized benchmarking|randomized benchmarking]] scalably obtains a figure of merit of the overlap between the error prone physical quantum process and its ideal counterpart.
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 This can be easily understood by making a classical analogy. Consider a [[Physics:Harmonic oscillator|harmonic oscillator]] (e.g. a pendulum). The [[Position (vector)|position]] and [[Finance:Momentum|momentum]] of the oscillator at any given point can be measured and therefore the motion can be completely described by the [[Phase space|phase space]]. This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a probability distribution in the [[Phase space|phase space]] (figure 2). This distribution can be normalized (the oscillator at a given time has to be somewhere) and the distribution must be non-negative. So we have retrieved a function <math>W(x,p)</math> which gives a description of the chance of finding the particle at a given point with a given momentum.
 For quantum mechanical particles the same can be done. The only difference is that the Heisenberg's [[Uncertainty principle|uncertainty principle]] mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time. The particle's momentum and its position are called quadratures (see [[Physics:Optical phase space|Optical phase space]] for more information) in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the [[Marginal distribution|marginal distribution]], <math>\mathrm{pr}(X)</math> or <math>\mathrm{pr}(P)</math> (see figure 3). In the following text we will see that this probability density is needed to characterize the particle's quantum state, which is the whole point of quantum tomography. [[File:MarginalDistribution.PNG|frame|right|alt=Figure 3: Marginal Distribution|Figure 3: Marginal Distribution]]
 == What quantum state tomography is used for ==

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