Qubits entangled states

In this equation, C andT refer to control and target qubits, respectively. The resulting state (output of the qugate) is said to be an entangled state of the two qubits, that is, a state that cannot be written as a product of states for each qubit [30]. The occurrence of such entangled states is another characteristic trait of QC, at the basis of secure quantum communication or cryptography. It also implies that, as opposed to what happens with a classical bit, an arbitrary quantum bit cannot be copied (the COPY classical operation is, in fact, based on the application of a succession of classical CNOT gates) [4]. 

Indeed, in current liquid-state NMR experiments e 3 x 10-5 on n < 10 qubits. So we may conclude that in all liquid-state NMR experiments to-date no entangled state has been accessed [Braunstein 1999]. So that resolves one issue. Our first intuition was that entanglement is everything in quantum computation, but recall that the Jozsa-Linden theorem does not actually say that for mixed states. Maybe one can still obtain a speed-up without entanglement It turns out that this was no less controversial than the first question. 

To experimentally demonstrate that the gate works, we first verify that we obtain the desired CNOT (appropriately conditioned) for the input qubits in states HH, HV, VH and VV. In Fig. 4 we compare the count rates of all 16 possible combinations. Then, it was proven that the gate also works for a superposition of states. The special case where the control input is a 45° polarized photon and the target qubit is a H photon is very interesting we expect that the state H + V)ai H)a2 evolves into the maximally entangled state ( HH)b11,2 + VV)b1 b2)- We input the state I ) first we measure the count rates of the 4 combinations of the output polarization (HH,..., VV) and then after going to the +), —) linear polarization basis a Ou-Hong-Mandel interference measurement is possible this is shown in Fig. 5. 

Thus, atomically doped carbon nanotubes offer another, alternative way to generate the qubit entanglement by using quasi-ID atomic polariton states formed by the atoms (ions) located close to or encapsulated inside CNs. Here we show that small-diameter metallic nanotubes indeed result in sizable amounts of the two-qubit atomic entanglement for sufficiently long times. 

The fast decoherence of locally prepared entangled states in condensed media studied here is compared with decoherence (in the 10 to 10 s range) in objects studied in quantum optics in high vacuum, with the disappearance of the superposition state in NHj or ND3 molecules in dilute gases, and with the lifetime of superconducting qubits in solids (10 s) at low temperature. 

It turned out that in the Bohr-Einstein controversy, Bohr was right. The Einstein-Podolsky-Rosen paradox resulted (in agreement with Bohr s view) in the concept of entangled states. These states have been used experimentally teleport a photon state without violating the Heisenberg uncertainty principle. Also, the entangled states stand behind the idea of quantum computing with a superposition (qubit) of two states a 0) + b l) instead of 0) and 1) as information states. 

The information contained in qubits and entangled states includes the phase any error in the phase has important implications (e.g., changing an entangled to a product state). To perform complex quantum computations, we need to reliably prepare a delicate superposition of states of a relatively large quantum system, which cannot be perfectly isolated from the environment hence the superpositions always decay. The decoherence of an entangled state is even faster the nonlocal correlations are extremely fragile and decay very rapidly. Also, applications of unitary transformations to qubits will not be flawless, and errors can accumulate. 

Quantum mechanics tells us that if a measurement is made on either qubit in the state given in (1.6.3), there will be 50% of chance to find it on 0) and 50% to find it on 11). But, if we find one of the qubits in, for instance, 0), it means that, after the measurement, the second qubit will also be in 0), even if no measurement is made over it In other words, the measurement of the state of one qubit in an entangled state, affects the state of the other qubit, independent on how distant they can be from each other ... 

There are other possible entangled states of a two qubit system ... 

Entangled states such as the cat state have a perfect correlation between the observables of the individual qubits of the system. Eor instance, the expected values of Xa and Xb for the cat state, V +>, are zero ( Xa) = 0 and Xb) = 0) ... 

This tells us that, in spite of the uncertainty about the sates of the individual qubits, they are perfectly correlated This correlation is responsible for the non-local action of entangled states. For instance, suppose a measurement is made on the first qubit, a, represented by the operators Mq = 0)(0 (g) 1 and = 1)(1 1. The probability of finding 0 is the same of finding 1 ... 

Therefore, a measurement of a qubit in an entangled state, defines the state of the other, upon which no measurement was performed If the measurement is made in a different basis, for instance, Mf = +) (- - and M = -) (-1, and the state - -) was found for qubit a, after the measurement, the quantum state of the system would be -l—1-), implying that the qubit b is also in the same - -) state. For non-correlated systems, for instance... 

The influence of the measurement result of a qubit affecting the state of another, as happens in an entangled state, is called non-locality. This strange property was pointed out for the first time in a very influential paper, published in 1935, by Albert Einstein, Boris Podolsky and Nathan Rose [13]. The paper aimed to demonstrate that Quantum Mechanics was an incomplete theory. According to the authors, a theory to be considered complete should contain what they defined reality elements. A reality element would be, still according to the authors, any physical quantity whose value could be predicted before performing a measurement on the system. For example, when a measurement of the observable cty is performed on a qubit, in an entangled cat state, the result determines the state of the other qubit, which could then be predicted before a measurement. Hence the observable Uy is a reality element. However, before the measurement is performed on the first qubit. 

In the year of 1964 John Bell discovered a remarkable result [14], which sets the rules for deciding experimentally if the non-locality is indeed a fact of entangled systems. The result is expressed in the form of an inequality, which establishes an upper limit for the local correlations in a two-partite system. Since then, it is known as the Bell s inequality. For two qubits, Bell s inequality says that a determined quantity S, essentially a correlation function between observables, should not overcome the value 2. However, quantum mechanics is non-local, and predicts the value S = 2 /2 2.83, for an entangled state, therefore violating the Bell s inequality. 

Teleport is a process through which the state of a qubit is transferred to another, using the non-local properties of entangled states [17]. Differently from superdense coding, no qubit is transferred in teleport, but only a quantum state. 

Remember that this form is motivated from a high temperature approximation for the NMR equilibrium density matrix, for which e = hcoL/2"kBT. But, whatever the situation, one must have Tr(l) = 2" and Tr(/0i) = 1. Consequently, Tr(/Oe) = 1, as it must be for density matrices. The matrix p can represent an equilibrium mixed state, or a pseudopure state. In particular, it can represent an entangled state. For instance, for two-qubits it could be the cat-state ... 

So, how to quantify entanglement For two qubits, the elements of the Bell basis represent maximally entangled states, but as the number of qubits increases, the quantification of entanglement becomes difficult. For an arbitrary number of qubits, nobody knows how to quantify entanglement. Notice that these difficulties are present to any physical system where noise is present or not, and by no means is exclusive to NMR. Indeed, any quantum system in the presence of white noise can be written in the form (6.1.3). The difference is that in experiments of liquid-state NMR made at room temperature, that form is intrinsic. For discussions about general aspects, characterization and quantification of entanglement, see [2,3]. 

The Peres criterium is a necessary and sufficient condition for entanglement in the case of two qubits, but for larger Hilbert spaces the partial transposition operation of entangled states can take the density matrix to other positive operators, that is, with no negative eigenvalues, and the criterium fails. 

The problem of determining bounds of e for n qubits, is rather complicated. A general Peres criterium can be obtained for n qubits, but since its applicability is restricted to a small number of qubits, it cannot be used to analyze NMR entanglement and the scaling problem. An alternative and more general analysis was made by Braunstein and co-workers in 1999 [1]. They found that a n-qubit pseudo-entangled state will be separable for... 

Entanglement transfer experiment in NMR quantum information processing - This paper of 2002 by Boulant and co-workers [18] describes an experiment of entanglement transfer by NMR. The aim is to transfer an entangled state of a pair of qubits to another pair of qubits, a process which was first demonstrated using photons in 1998, by Pan and collaborators. The authors used the four nuclei of crotonic acid as qubits. The process was followed by quantum state tomography and the efficacy of the experiment was quantified by a measured called attenuated correlation. A value of 0.65 for this measure at the end of the protocol, indicated that the pseudo-entangled state was indeed transferred from one pair of qubits to the other. 

Practical implementations of twirl operations - This paper, reported in 2005 by Anwar and co-workers [20], deals with a practical implementation of a proposal made by Bennett and co-workers in 1996 for the purification of entanglement from a mixed state. The typical situation would be that in which the qubits of an initially pure entangled state rjr ) = ( 01) - 110 V2 are sent through a noisy channel. The twirl operation is a step for the purification. This operation converts an arbitrary mixed state of two qubits into a pseudo-entangled state ... 

This state can not be written as a product of a state of qubit 1 times a state of qubit 2. Quantum states with this property are usually called entangled states. Such states have properties that are of purely quantum mechanical nature. If the two qubits are seperated and measurements are performed on both systems one encounters nonlocal correlations bewteen the systems. That is to say, a measurement on one system has an immediate influence on the possible measurement outcomes of the other systems even if they are light years apart from each other. This leads to correlations that can not be explained classically and it is often said that quantum mechanics is non-local (Bell 1964, Greenberger et al. 1990). As a si deremark we note that this does not lead to contradictions with Einsteins special theory of relativity. Entangled states play also an essential role for quantum computing. 

The interaction process of the QC with the environment can be understood as follows. We consider a single qubit interacting with its environment. The environment can be viewed as a high-dimensional quantum system. Initially the quantum bit and the environment are in a product (or non-entangled) state... 

Note that we can not only perform complete measurements. For example, we could measure just the first qubit of the 2-bit entangled state ( 0)i 0)2 + l)i l)2) /y/2 in basis 0), 1). If the outcome is 0(1) the state of the combined system is 0)i 10)2(11)111)2) after the measurement. Thus for entangled states a measurement on one subsystem influences the state of the other subsystem. 

Quantum computation exploits entanglement. The simplest kind of quantum computer is an n-qubit register, i.e., a system of n electrons. Each electron is a spin-1/2 particle so, by the analysis we did in Section 10.2, the state space is... 

Laflamme et al. s three-qubit system [103] is 26. In this molecule the two carbon nuclei are in different chemical environments and therefore represent two qubits, whilst the proton serves as the third. Compound 26 therefore represents the binary numbers 000, 001, 010, 011, 100, 110, and 111. A pulse of radio waves causes the nuclei to be thrown into the entangled superposition state, where they can act as qubits. The NMR machine then initiates the quantum computer program—a series of radiofrequency pulses that act like gates on the qubits and carry out the calculation. The superposition state is then collapsed to give an answer. 

---