Quantum Computation and Quantum Information. Home · Quantum Computation and Quantum Information Author: Michael A. Nielsen | Isaac L. Chuang. Michael A. Nielsen, Isaac L. Chuang, Massachusetts Institute of Technology. Publisher: Cambridge . Frontmatter. pp i-viii. Access. PDF; Export citation. Jul 8, PDF | On Nov 1, , Manuel Vogel and others published Quantum Computation and Quantum Information, by M.A. Nielsen and I.L. Chuang.

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Michael A. Nielsen & Isaac L. Chuang o M. Nielsen and I. Chuang Preskill's review of quantum error-correction[Pre97], Nielsen's thesis on quantum . Quantum Computation and Quantum Information / Michael A. Nielsen and Isaac L. Chuang. p. cm. Includes bibliographical references and index. M. Nielsen, I. Chuang, Quantum Computation and Quantum Information ( Cambridge, ). • build scalable macroscopic quantum circuits. • control open .

Then we would expect that the resulting state. Particle physics and cosmology study phe- nomena which are. The primary purpose of Part I is to provide a pedagogical introduction and reference for concepts in the field. The field of science which studies these fundamental connections between physics and infor- mation processing has come to be known as the physics of information. They are doubly useful, however, as a diagnostic tool, since error correcting codes can be used to determine what types of noise occur in a system. One step along the way to this result is an elegant repre- sentation theorem which relates these abstract requirements for a quantum operation to an explicit formula: First Class Honours.

Each physical theory may be treated as the basis for a theory of information processing. Chapter 2 Quantum information: A and B. Before we begin the Chapter proper. All this adds up to a mess of notation. This model is an attempt to formulate a general framework for the description of quantum information processing.

For that reason. It is assumed that you are familiar with elementary quantum mechanics. Suppose we single out an orthonormal basis set in the state space of such a system. State vectors will be written in the standard bra-ket notation.

With these concrete examples in hand. The Chapter concludes with an overview of the challenges facing an experimentalist wishing to do quantum information processing in the laboratory. The standard notation for density operators is sometimes inappropriate when discussing composite systems. The Chapter begins with an introduction to the fundamental unit of quantum information.

Four standard operators acting on a single qubit are the Pauli sigma operators. It is possible. It is instructive to compare bits and qubits.

The states 0i and 1i are known as the computational basis states. In discussions of real physical systems implementing qubits. A qubit can be in a continuum of states. In abstract discussions of quantum information processing.

The Pauli operators form a basis set for the vector space of operators on a single qubit. This two dimensional quantum system is known as the quantum bit or qubit []. It is not possible. By contrast. A bit can be in one of two states. There are several items of terminology related to qubits which we ought to agree upon now.

Y or Z instead. They are merely reference states. The Z Pauli operator is often known as the phase flip gate. Can she achieve her goal? The controlled not gate.

In certain systems. For all these reasons. It is possible to formulate classical information processing in terms of trits. The X Pauli operator is often known as the quantum not gate. Suppose Alice is in possession of two classical bits of information which she wishes to send Bob.

There are many reasons the qubit is regarded as the fundamental unit of quantum infor- mation. These gates are defined. At present there is no widely accepted term for the Y operator. In this respect. Classical information processing is accomplished by various logic gates which act on the bits being processed. In quantum mechanics. Their goal is to transmit some classical information from Alice to Bob.

This example shows that there are information pro- cessing tasks which can be performed with qubits which do not have natural analogues in terms of bits.

Superdense coding involves two parties. Quantum gates are operations acting on a fixed number of qubits. It is the simplest quantum mechanical system. Two more quantum gates which are of great importance are the Hadamard and phase shift gates. Superdense coding is an example of how quantum and classical information can be combined in an interesting way. Notice that the Bell states form an orthonormal basis.

Because the reduced states are the same regardless of which state was prepared. The intercepted qubit contains essentially no classical information. By sending her single qubit to Bob. This remarkable prediction of quantum mechanics has been given a partial experimental validation by Mattle et al using entangled photon pairs []. Here is the procedure she uses. Suppose Alice sends her qubit to Bob. Note that this is a fixed state. Examining the four states 2. It is surprising enough that a two level quantum system can be used to transmit two bits of classical information.

In the experiment. By doing a measurement in the Bell basis Bob can determine which of the four bit strings Alice sent. Alice now sends her qubit to Bob. Eve can infer nothing about the information Alice is trying to send by examining the qubit she has intercepted. In Chapter 6 we will return to study the limits to superdense coding in a much more detailed fashion. It was only possible to send a trit. From the previous equation.

This method has two major disadvantages. Quantum teleportation is a method for moving quantum states from one location to another which suffers from none of these problems. Suppose Alice is living in London and wishes to send a single qubit to Bob. This method also suffers from two major disadvantages. There are many different ways Alice could do this.

The channel used to do so may degrade over time. The situation then is even worse. More explicitly. Z and XZ. An even more remarkable effect. The cost of transmitting these classical bits may be considerable. There is no way she can send her system to Bob by sending Bob a classical description.

Suppose Alice performs a measurement on the two qubits in her possession. For quantum systems of many qubits it requires a huge number of classical bits to specify the state to reasonable accuracy. A second method is to physically move the quantum system from London to New York.

One method is for Alice to send a description of her state to Bob. The quantum circuit is read from left to right. The second line represents the qubit which Alice uses to share the initial entanglement with Bob. Circuit for quantum teleportation. Even more remarkably. This is despite the fact that in general it takes an infinite amount of classical information to describe the state to be teleported.

The controlled not gate is a unitary gate whose action is to flip the data qubit if the control 3 In general. The second gate is known as the controlled not gate. Quantum teleportation can be recast in the language of quantum gates which we met briefly earlier in this Chapter. A quantum circuit implementing teleportation is shown in figure 2. As can be seen from the figure.

This completes the teleportation process. The three lines traversing the circuit from left to right represent the three qubits involved in teleportation. In this particular gate. The top line represents the initial state which Alice wishes to teleport.

It is interesting to note that teleportation involves the transmission of only two bits of classical information. We shall refer to it as the data qubit. The rotation is accomplished by performing a controlled not from the data qubit to the ancilla qubit. The effect of these transformations is as follows: This completes our description of quantum teleportation in the language of quantum circuits. Notice that this sequence of operations corresponds exactly to the sequence of operations necessary for quantum teleportation.

The way this is accomplished is to do two gates which rotate the Bell states into the computational basis. Quantum teleportation is an important elementary demonstration of quantum information theory. It is interesting to note further that the measurement step can be removed from the circuit and the state of the data qubit will still be transferred to the target qubit.

Making use of this fact. After the controlled not is applied. Later in this Chapter we discuss the experimental implementation of quantum teleportation. The next step of teleportation is to perform a measurement on the data and ancilla qubits in the Bell basis. This defines the action of the controlled not on a basis. These and many other uses emphasize the role quantum teleportation has as an exemplar useful for the study of more complex forms of quantum information processing.

It is assumed that it is possible to perform the controlled not gate on any pair of qubits in the quantum computer. Ability to prepare states in the computational basis: It is assumed that any computa- tional basis state x1. Without further ado. We sometimes write xi for a computational basis state. The state space is thus a 2n dimensional complex Hilbert space. Product states of the form x1.

Recall that these gates are defined in the computational basis as follows: In principle. Classical resources: The quantum computer consists of two parts.

Ability to perform quantum gates: A suitable state space: We assume that the quantum part of the computer consists of some number. Ability to perform measurements in the computational basis: Measurements may be performed in the computational basis of one or more of the qubits in the computer. While classical computations can always be done. This section describes a single model of quantum computation. As an example.

This model of computation is equivalent to many other models of computation which have been proposed. In what ways may the quantum circuit model of computation be criticized? How might it be modified? Perhaps my sharpest criticism of the quantum circuit model is that its basis. That is. Then given an input for the problem. At a less trivial level. At a more practical level. Might there be anything to be gained by using systems whose state space is infinite dimensional?

What about the assumption that the starting state of the computer is a computational basis state? Everything is phrased in terms of finite dimensional state spaces. A very desirable goal for the future is to use fundamental physics to demonstrate or refute the following modern version of the Church-Turing thesis see also [58]: Any physically reasonable model of computation can be simulated in the quantum circuit model with at most polynomial overhead in physical resources.

Algorithm to build the quantum circuit: Suppose we wish to solve a problem using the quantum circuit model of computation.

The basic assumptions underlying the model are ad hoc. It might be that having access to certain states allows particular computations to be done much more easily than if we are constrained to start in the computational basis. Without imposing this im- portant requirement. This requirement — that the structure of the quantum circuit be specified by a classical algorithm — is known as the uniformity requirement for quantum computation.

It is not my purpose here to do a detailed examination of the physics underlying the models used for quantum computation. I wish merely to raise in your mind the question of the completeness of the quantum circuit model.

We will give a proof by contradiction that such an algorithm cannot exist.. A key point made by Turing is that his programs can be numbered 0. An algorithm to compute a function is expressed in terms of a program. More formally. Is there an algorithm which computes the halting function? Strictly speaking. Let us return to the question asked in section 1. This random number generator can be called as part of the algorithm. The point of view we take is that the quantum circuit model provides an essentially complete account of the information processing tasks.

We begin with the halting problem. It is not difficult to see that even in this slightly generalized model of computation. As noted in the previous section.

It is only in the next section that the question of the completeness of the quantum circuit model of quantum information processing will be an important issue.

We have already done that. Turing invented the modern concept of a programming language for his computers. We make use of the algorithm for HALT to construct another program. What observables may be realized as measurements on a quantum system? In this section we discuss this problem from a somewhat different point of view than was done earlier. In pseudocode: Perhaps there really exist in nature quantum processes which can be used to compute functions which are clas- sically non-computable.

By carrying out the simulation to a high enough level of accuracy. Having demonstrated that there is no algorithm capable of computing the halting function. The algorithm is very simple: Recognizing such a process poses some problems. If we assume that the quantum circuit model provides a complete description of the class of information processing tasks which may be performed in quantum mechanics then we are left to conclude that physical law does not allow measurement of the halting family of observables.

Given n. Suppose that. The answer is no. How could we verify that a process com- putes the halting function or any other non-computable function? Because of the algorithmic unsolvability of the halting problem.

We will outline an algorithm for a Turing machine that will compute the halting function. It is far-fetched. To see why not. What is the value of h t? It is intriguing to consider the consequences if it were possible to measure the halting ob- servable or. We are left to conclude that it is not possible.

These same optical methods are of little use in their present form for more general quantum information processing tasks. Given this impressive progress. I believe this would be a long and difficult task. Optical methods have been used to successfully implement an impressive variety of quantum information processing tasks of this high precision-small size type. We then discuss in some detail the approach to quantum computing based upon liquid state nuclear magnetic resonance.

Related restrictions apply for unitary dynamics []. If such a result could be established. This section reviews the requirements that must be met in order to do interesting quantum information processing tasks. The specific requirements which must be met by a system which is to do quantum information processing depend upon the task which the system is to perform. Let us now turn to the more immediately practical topic of experimental quantum information processing.

In many ways it is a digression from the main stream of the Dissertation. Purely optical methods do not appear to scale very well. The section concludes with an account of the use of nuclear magnetic resonance to accomplish quantum teleportation. These limitations may go considerably beyond the familiar limits of the type discovered by Heisenberg.

The section begins with a discussion of some of the general principles to be met by quantum information processors5. That concludes our discussion of realizable quantum measurements. Given sufficient empirical evidence of this sort.

In what future directions may this line of thinking be taken? The most obvious is to clarify the extent to which the quantum circuit model of computation is a complete framework for the description of quantum information processing. Two of these proposals stand out as they have led to the successful implementation of simple quantum logical operations.

In Chapter 9 we will investigate quantum error correcting codes which. It is also worth noting that a third technology. If the goal is to implement the quantum circuit model described in the previous section. In the near term. These proposals are based on the linear ion trap. The performance of each of the above tasks will inevitably be imperfect. These problems make it seem unlikely that photons will be the primary basis for large scale quantum information processors.

What general requirements are desirable in a system which is to be used for large scale quantum information processing? In this subsection we focus on a description of the NMR approach. In our setup. The scheme has since been applied to do numerous interesting quantum information processing tasks [ This molecule consists of two Carbon atoms. In the liquid state. Schematic representation of the labeled TCE molecule. The sample is placed in a large. The field is as large as can be made with current technology for reasonable cost.

Methods for doing quantum information processing using liquid state NMR were proposed independently at about the same time by Cory.

The structure of the molecule is shown in figure 2. Fahmy and Havel [51]. The molecules are prepared in such a way that the Carbon atoms are actually the 13 C isotope. In this limit. The NMR method is unusual in that it makes use of a model of quantum information processing that is significantly different to the quantum circuit model of quantum computation.

The liquid state NMR approach to quantum information processing makes use of a large number of molecules dissolved in a solvent such as chloroform. C1 or C2. Similar observations may be made about the other possible rotations. In the absence of externally applied rf fields.

In the TCE molecule. For our present purposes. The necessary interactions happen fast enough that the contribution from the ZZ coupling between spins may be neglected. This effect is known as the chemical shift. A clever technique known as refocusing allows this to be done. These couplings can be effectively removed by a technique known as refocusing. We will also ignore the Chlorines. Using these external rf fields it is possible to perform single qubit rotations on individual nuclei in the molecule.

In many situations. In addition to the uniform magnetic field. Note that the frequencies of C1 and C2 are not identical. For simplicity. In order to achieve, this, let t be any length of time. Suppose we cause the following sequence of operations to occur:.

The interaction between H and C1 has vanished; we say that it has been refocused. We will use single qubit rotations and spin-spin couplings to perform unitary dynamics on our nuclear spins. Whether this forms a universal set for quantum computation depends upon the details of the molecule being considered; see [69] for a discussion of this point. For our much less grandiose purpose of doing quantum teleportation the interactions available are certainly sufficient to implement the quantum circuit for teleportation.

The chief difficulty is perhaps that pulses applied to the two carbon nuclei are applied non-selectively. However, standard tricks based upon the chemical shift can be used to apply selective pulses to C2 [64].

Liquid state NMR involves bulk systems; typically, on the order of sample molecules occur in the sample being examined. The signal which is read out from the sample is an ensemble average over all those molecules, not a projective measurement which yields a single result, as in the quantum circuit model.

In an NMR machine, magnetic pick-up coils are used to determine the magnetization in the x-y plane. The signal read-out from the coils is then Fourier transformed to give a spectrum for the system. The number of observables whose ensemble average can be directly observed in this way is thus rather limited. However, by making use of reading pulses immediately before the final measurement, it is possible to greatly extend the range of observables which can be determined.

At room temperature, the initial state of the system is highly mixed. This state does not appear to be at all like the pure computational basis state which is used in the quantum circuit model of quantum computation. There is a clever idea which allows us to work around this problem, suggested independently by Cory, Fahmy and Havel [51], and Gershenfeld and Chuang [69].

Perhaps the simplest scheme to illustrate the basic idea is the following method, known as temporal labeling [96]. Suppose we have a molecule with n nuclei. The idea is to define a set of unitary operators which permute all the computational basis states, 0i,. In each experiment, the corresponding unitary operator is applied before the experiment begins. The net contribution due to the states 0i,. Thus we have performed a computation with an effectively pure state.

It is straightforward to efficiently implement such operations P using standard quantum gates [7, 11], so this can be done in NMR.

Suppose in each of these experiments we perform the unitary Uk , followed by some unitary operation U , and then observe some component of the spin, say hXi i. That is, the summed averages behave as if the pure state N ihN had been prepared, the unitary operation U applied to that pure state, and the average of Xi observed.

Similar remarks apply to other observations which may be made in NMR. This method is known as temporal averaging because it requires that the experiment be repeated many different times, and the results summed. Temporal averaging is only one possible means for performing state preparation in NMR quantum information processing. It is an especially easy method to explain, but in the laboratory other methods may be considerably better.

In our. The precise details of what was done are beyond our present scope, but the basic idea may be explained quite easily. Essentially what is done is to vary the strength of the magnetic field applied in the z direction across the sample. This causes nuclei at different locations in the sample to rotate around the z axis at different frequencies.

When applied for the appropriate length of time, the ensemble averaged values for the X and Y components of magnetization average to zero. That is, a gradient pulse applied to a single spin has the effect of setting the x and y components of the Bloch vector for the ensemble to zero, while leaving the z component of the Bloch vector untouched.

Cory et al [51] have described how a combination of gradient pulses, rf pulses, and delays may be combined to prepare effectively pure states, along similar lines to the temporal labeling method described above. We will not give further details of this method here. NMR-based approaches to quantum information processing have many attractive features. NMR is a well-developed technology, and a considerable amount of high quality, easy-to-use equip- ment has been developed for use off-the-shelf.

The noise timescale is typically on the order of a second, while the time to perform a two qubit gate is on the order of one to ten milliseconds, giv- ing a best-case estimate of about one thousand couplings possible, although there is no doubt that achieving this in a useful computation will be extraordinarily difficult.

Present experimental work in NMR quantum information processing usually involves on the order of ten couplings. With regard to the power of NMR quantum information processing from the point of view of computational complexity, and in comparison with the quantum circuit model, I will not essay an opinion here. A considerable amount of interesting discussion has taken place on or closely related to this topic and I refer the reader to, for example, [39, 69, 96, , ] for further discussion.

What does seem certain is that NMR provides a powerful means for conducting interesting investigations into small-scale quantum information processing. A few qubits may not be much, but it represents the current best we can do with our quantum information processors.

The essential idea of the scheme is to implement the quantum circuit for teleportation discussed in section 2. Our implementation of teleportation is performed using liquid state nuclear magnetic reso- nance NMR , applied to an ensemble of molecules of labeled trichloroethylene TCE, as discussed in the previous section.

To perform teleportation we make use of the Hydrogen nucleus H , and the two Carbon 13 nuclei C1 and C2 , teleporting the state of the second Carbon nucleus to the Hydrogen. Figure 2. The circuit has three inputs, which we will refer to as the data C2 , ancilla C1 , and target H qubits. The goal of the circuit is to teleport the state of the data qubit so that it ends up on the target qubit.

State preparation is done in our experiment using the gradient-pulse techniques described by Cory et al [51], and phase cycling [64, 74]. The unitary operations performed during teleportation may be implemented in a straightforward manner in NMR, using non-selective rf pulses tuned to the Larmor frequencies of the nuclear spins, and delays allowing entanglement to form through the interaction of neighboring nuclei, as described in the previous section. Commented pulse sequences for our experiment may be obtained on the world wide web [].

An innovation in our experiment was the method used to implement the Bell basis measure-. For this reason, we must modify the projective measurement step in the standard description of teleportation, while still preserving the remarkable teleportation effect. We use a procedure inspired by Brassard et al [31], who suggested a two-part procedure for performing the Bell basis measurement. Part one of the procedure is to rotate from the Bell basis into the computational basis, 00i, 01i, 10i, 11i.

We implement this step in NMR by using the natural spin-spin coupling between the Carbon nuclei, and rf pulses. Part two of the procedure is to perform a projective measurement in the computational basis.

As Brassard et al point out, the effect of this two part procedure is equivalent to performing the Bell basis measurement, and leaving the data and ancilla qubits in one of the four states, 00i, 01i, 10i, 11i, corresponding to the different measurement results.

We cannot directly implement the second step in NMR. Instead, we exploit the natural phase decoherence occurring on the Carbon nuclei to achieve the same effect. Recall that phase decoherence completely randomizes the phase information in these nuclei and thus will destroy coherence between the elements of the above basis. Its effect on the state of the Carbon nuclei is to diagonalize the state in the computational basis,. As emphasized by Zurek [], the decoherence process is indistinguishable from a measurement in the computational basis for the Carbons accomplished by the environment.

We do not observe the result of this measurement explicitly, however the state of the nuclei selected by the decoherence process contains the measurement result, and therefore we can do the final transformation conditional on the particular state the environment has selected. As in the scheme of Brassard et al, the final state of the Carbon nuclei is one of the four states, 00i, 01i, 10i, 11i, corresponding to the four possible results of the measurement.

In our experiment, we exploit the natural decoherence properties of the TCE molecule. The phase decoherence times T2 for the C1 and C2 are approximately 0. This implies that for delays on the order of 1s, we can approximate the total evolution by exact phase decoherence on the Carbon nuclei. The total scheme therefore implements a measure- ment in the Bell basis, with the result of the measurement stored as classical data on the Carbon nuclei following the measurement. We can thus teleport the information from the Carbon to the Hydrogen and verify that the information in the final state decays at the Hydrogen rate and not the Carbon one.

Examining figure 2. Experimentally, the use of multiple refocusing pulses ensures that the data qubit has effectively not interacted with the target qubit. Nevertheless, quantum mechanics predicts that we are still able to recover the complete system after this decoherence step, by quantum teleportation. Ex- perimentally, we determined the Larmor and coupling frequencies for the Hydrogen, C1 and C2 to be:. The coupling frequencies between H and C2, as well as the Chlorines to H, C1 and C2, are much lower, on the order of ten Hertz for the former, and less than a Hertz for the latter.

Experimentally, these couplings are suppressed by multiple refocusings, and will be ignored in the sequel. Note that the frequencies of C1 and C2 are not identical; they have slightly different frequencies, due to the different chemical environments of the two atoms. We performed two separate sets of experiments. In one set, the full teleportation process was executed, making use of a variety of decoherence delays in place of the measurement.

The readout was performed on the Hydrogen nucleus, and a figure of merit — the dynamic fidelity — was calculated for the teleportation process.

The dynamic fidelity is a quantity in the range 0 to 1 which measures the combined strength of all noise processes occurring during the process, which we will study in detail in Chapter In particular, an dynamic fidelity of 1 indicates perfect teleportation, while an dynamic fidelity of 0.

Perfect classical transmission corresponds to an dynamic fidelity of 0. The second set of experiments was a control set. In those experiments, only the state preparation and initial entanglement of H and C1 were performed, followed by a delay for decoherence on C1 and C2. The readout was performed in this instance on C2, and once again, a figure of merit, the dynamic fidelity, was calculated for the entire process.

The results of our experiment are shown in figure 2. Errors in our experiment arise from the strong coupling effect, imperfect calibration of rf pulses, and rf field inhomogeneities.

These uncertainties are due primarily to rf field inhomogeneity and imperfect calibration of rf pulses. In order to determine the dynamic fidelities for the teleportation and control experiments, we performed quantum process tomography. This procedure, described in detail in section 3.

In particular, we will show in section 3.

By preparing a complete set of four linearly independent initial states, we were able to obtain a complete description of the quantum process. This experiment is not the first experimental implementation of quantum teleportation. The present NMR-based method illustrates some of the advantages of using NMR to do elementary quantum information processing. In conclusion. Our experimental observations are consistent with this prediction.

Dynamic fidelity is plotted as a function of decoherence time. Earlier experiments by Boschi et al [28] and Bouwmeester et al [29] used optical methods to achieve quantum teleportation. This description. The top curve represents the fidelity of the quantum teleportation process. Three elements ought to be noted in figure 2. Classical external control. Preshared entanglement can be used to transmit two classical bits with the transmission of only one qubit.

Ability to prepare states in the computation basis. An algorithm for applying quantum gates controlled-not and single qubit unitary gates and projective measurements in the computational basis to the system. A two level quantum system. Summary of Chapter 2: Quantum information: The fundamental unit of quantum information. Preshared entanglement can be used to transmit a qubit with the transmission of two classical bits.

The usual way to describe such a measurement is the following. The first type is the evolution of a closed quantum mechanical system. Under such an evolution. Suppose a measurement is performed which has outcomes labeled by m. Elementary quan- tum mechanics texts usually do this by separating the dynamics into two different types.

The system being measured is no longer a closed system. Chapter 3 Quantum operations Quantum mechanics describes the dynamics which can occur in physical systems. To make the idea of quantum operations more concrete. Suppose we have a single qubit quantum system. The interaction of the quantum system with an external world allows dynamics that are neither unitary nor described by the usual model of projective measurements. You may wonder how it is possible to go beyond the usual textbook description of state changes in terms of unitary transformations and projective measurements.

We will suppose this environment is also a single qubit system. The key observation is that many state changes of interest occur in open quantum systems. Left to themselves these systems will interact according to some unitary interaction U.

For instance. The theory of quantum operations can be used to describe a wide class of state changes that may occur in quantum systems. For the sake of definiteness we will suppose that U is the controlled not operation. The elementary material appearing here has its origins in earlier work by people such as Hellwig and Kraus [ That output state need not even be a state of the same system.

In addition to elementary review material. In the case where the process is deterministic. Choi [36] and Kraus []. We suppose some physical process occurs. Then we would expect that the resulting state. In this case. In places the Chapter contains rather detailed mathematics. The material relating quantum teleportation and the quantum operations formalism is based upon a collaboration with Caves []. To cope with the case of measurements. What requirements must the map E: We will enumerate a set of axioms which any such map must satisfy.

The formalism we develop shall. More concretely. A physical quantum operation is one that satisfies the requirement that probabilities never exceed 1.

Then two quantum operations are used to describe this process. It is amusing to speculate that in systems in which such selection rules exist it might be allowable for systems to undergo dynamics which are not completely positive. This requirement applies both to density operators on the system for which the dynamics is occurring. This requirement. By definition. To illustrate the importance of this point.

It will be convenient to use the same index. Surprisingly to me. Suppose we introduce a system. The map E is a linear map. Let A be any positive operator acting on the state space of an extended system. Let iR i and iQ i be orthonormal bases for R and Q. Our aim will be to find an operator-sum representation for E.

E is obviously linear. The domain of E is the real vector space of Hermitian operators on HQ. Suppose next that E is a quantum operation. This completes the first part of the proof. The range of E is contained in the real vector space of 0 Hermitian operators on HQ.

Let I denote the identity map on system R. One step along the way to this result is an elegant repre- sentation theorem which relates these abstract requirements for a quantum operation to an explicit formula: The operators Ei appearing in this expression are said to generate an operator-sum repre- sentation for the quantum operation E.

The map E is completely positive. A construction of this sort is given at the beginning of Chapter 5. It is a truly remarkable fact. The operator sum representation gives us a way of describing the dynamics of the principal system. One reason for our interest in the operator-sum representation is that it gives us a way of characterizing the dynamics of a system in terms of intrinsic quantities.

Slightly less obviously. By linearity of E and trB. QED This result allows us to give easy proofs that many interesting maps are quantum operations. To see 0 this.

Define a linear operator Ei: In terms of the operator-sum representation. P Similarly. Suppose we have a joint system AB. Let ji be a basis for system B. If it is only the dynamics of the principal system which are. Non-unitary behaviour of quantum system can only arise because of the action of external systems. An even more useful result is the observation that the partial trace is a quantum operation. By reasonable. We can relate the operator-sum representation picture of quantum operations to the idea of a quantum system interacting with other systems.

E Figure 3. We suppose that Q and E are initially independent systems. The first result shows how to determine the operator-sum representation appropriate for a quantum system interacting in a specified way with other quantum systems. The second result shows that for any quantum operation. We will denote this system by the letter Q.

The case where no measurement is made corresponds to the special case where there is only a single measurement outcome. Environmental model for a quantum operation. Adjoined to Q is another system which we will refer to variously as the ancilla or environment system. We will prove two results. After the unitary interaction a measurement may be performed on the joint system. Our aim is to determine the final state of Q as a function of the initial state.

The situation is summarized in figure 3. This measurement is described by projectors Pm. Let ii be an orthonor- mal basis set for E. The construction will only be given for quantum operations mapping the input space to the same output space. We now review a construction converse to this. Introduce an orthonormal basis kihk for the system E.

Thus the operator U can be extended to a unitary operator acting on the entire state space of the joint system. This is an affine map. Following the unitary U. In this representation. Recall from section 2. This method allows one to get an intuitive feel for the behaviour of quantum operations in terms of their action on the Bloch sphere.

P Conversely. Analogously to the earlier construction. For each m. A more interesting generalization of this construction is the case of a set of physical quantum operations. Simply introduce an extra operator. Introduce an environmental system. Now repeat the same construction as before to obtain a unitary operator U. The projection process above cannot increase the norm of the Bloch vector.

This picture can be used to obtain simple pictures of quantum operations on single qubits. This is but one example of the use of this geometric picture. Less trivially. Viewed this way. The meaning of the affine map equation 3. This geometric picture makes it very easy to verify certain facts about this quantum operation. To see this. When do two sets of states. These two apparently very different physical processes give rise to exactly the same system dynamics.

Understanding this question is important for at least two different reasons. As an example of the theorem. We To begin. In this section we study in more detail the question of when two sets of operators give rise to the same quantum operation.

Suppose we flipped a fair coin. This process corresponds to the first operator-sum representation for E. It turns out that the answer to this question has a surprising number of interesting and useful consequences. P Suppose Ej and Fk are two sets of operators. QED This result allows us to characterize the freedom in operator-sum representations. To avoid confusion. As usual. Smolin and Wootters [22]. This means that there is a one-to-one linear map from the state space of 3 onto the state space of 1.

Suppose Alice has possession of an input system. Though this map is not unique. Alice might also have access to another system. This result is surprisingly useful. We will use it. The work in this section is based upon a collaboration with Caves [].

Some of the ideas were arrived at independently about the same time by Bennett. Recall that teleportation involves a sender. I would especially like to thank Chris Fuchs.

We assume that systems 1 and 3 are identical and thus have the same state space. In that Chapter we will see that certain sets operators in the operator sum representation give more useful information about the quantum error correction process.

As discussed in section 2. In this section we show how quantum teleportation can be understood within the quantum operations formalism. Classical and Quantum Information. Quantum information and computing. Quantum Information Theory and Quantum Statistics. Classical and quantum information. Quantum Stochastics and Information. Topological quantum computation. Fundamentals of quantum optics and quantum information.

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