= ( 1) b b in .NET Compose USS Code 128 in .NET = ( 1) b b

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= ( 1) b b use .net barcode standards 128 integration togenerate code 128a for .net Windows Forms = exp(i b b).. (11.3). The expecta USS Code 128 for .NET tion value of this operator with respect to the state of Eq. (11.

2) is easily found to be . = out out = cos (2 ). (11.4). which depen .NET code128b ds not just on but on 2 . If we once again use the calculus of error propagation, we will obtain, rather generally, the uncertainty in the phase measurements as.

= . (11.5). 11.2 Entanglement and interferometric measurements 2 where 1 .NET Code 128A b 2 owing to the fact that b = I , the identity operator. In our b= particular case, we obtain = 1/2.

If we compare this result with the optimal coherent state result for the same average photon number at the input, = 1/ 2 for n = 2, we see an improvement in the phase sensitivity, the resolution, by a factor of 1/ 2. This improvement is the result of the entanglement of the state inside the interferometer. In fact the state, the state of Eq.

(6.17), is a special type of state known as a maximally entangled state (MES). It is maximal in the sense that it is a superposition of states where all the photons are in one beam and where they are all in the other and maximizes the correlations between the modes.

A more general form of such a state for a total of N photons is. N . 1 = (. N 2 . + ei 0 a . N a ),. (11.6). where N is USS Code 128 for .NET some phase that may depend on N. If such states could be generated, and that s a big if , one can show that phase resolutions of the exact form = 1/N will result, an improvement of the optimal coherent state result with light of equivalent average photon number n = N .

This form represents, in fact, the tightest measure allowed by quantum mechanics for the interferometric measurement of phase shifts [4], and is referred to as the Heisenberg limit (HL): HL 1/N . Of course, the maximally entangled states, sometimes called the NOON states for what we hope are obvious reasons, must be realized experimentally somehow. For the case of N = 2, as we have seen, the twin single-photon states at the input of a 50:50 beam splitter are suf cient to generate the two-photon maximally entangled state.

But for arbitrary N > 2, a beam splitter will not generate maximally entangled states. However, a number of schemes have been proposed for generating such states, some involving various kinds of state-reductive measurements [5] and others involving nonlinear media with giant susceptibilities [6]. However, the necessary parity measurements require photon detectors able to distinguish photon numbers at the level of a single photon.

But detectors with such resolutions are becoming available as we write [7]. As another tack, instead of generating maximally entangled states, one could instead feed twin number states . N a N b into a VS .NET code-128c Mach Zehnder interferometer [8]. At the output, the differences of the average photon numbers vanish as in the case for a maximally entangled state, but, once again, parity measurements can be performed on one of the output beams.

It has been shown that the Heisenberg limit in the form HL = 1/(2N ), the total number of photons being 2N in this case, is approached as N becomes large, and is substantially below the standard quantum limit SQL = 1/ 2N for all N, at least if the phase shift is small [9]. But leaving aside all the details of these considerations, which are beyond the scope of this book, the essential point of this discussion is that entanglement is not merely a curiosity, but does have practical applications; in this case the improvement of interferometric measurements..

Applications of entanglement 11.3 Quantum teleportation Another app lication of entanglement, this one much more in the context of quantum information processing, is quantum teleportation whereby an unknown quantum state is transferred from one point to another, these points possibly being widely separated. We wish to emphasize that it is the unknown quantum state that is to be teleported, not the particle or particles in such states. We shall restrict ourselves to a simple and rather basic description of teleportation, as originally given by Bennett et al.

[10]. We shall use states denoted . 0 and 1 as usual, but we will not necessarily think of them in terms of photon number states but rather as states of some sort of two-level system, such as the polarization states of single photons. Furthermore, in the quantum information business, the symbols . 0 and 1 and super positions of these states represent the quantum bits of information, the qubits. Qubits are the quantum information theoretic name for the states of any two-level system and a single qubit refers to one system whose states can be represented, in general, by superpositions of the states . 0 and 1 . Qubits could be realized also by two-level atoms. As this text is concerned in the main with photons, one might wish to interpret the states .

0 and 1 as vertic ally and horizontally polarized states, . V and H respectiv ely, but the procedures described are rather general. We require two participants in the teleportation procedure (or protocol) usually given the names Alice (A) and Bob (B). We suppose that Alice is given a photonic mode in the quantum state .

= c0 . 0 + c1 1 and that visual .net Code 128C she wishes to teleport the state to Bob so that he can re-create it in a photonic mode in his possession. The state is unknown to Alice, i.

e. she does not know the coef cients c0 and c1 . In fact, if she did know, the state the protocol about to be described would be unnecessary as she could simply convey the information to Bob over a classical channel (e.

g. via telephone). We further suppose that some source of light can produce an entangled state of the form.

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