However, in quantum computation and quantum communication, there are many practical scenarios in which the state of our qubits cannot be written down as linear combinations in a given basis, but instead must be expressed in terms of ensembles (statistical mixtures) of multiple states, each with an associated probability of occurrence. In this paper, an alignment-free MDI-QKD scheme is proposed with rotational-invariant state, which is immune to the collective noise induced . Output : Result[] An array of measurement results. Double line represents classical bit. 2.2 Most general quantum measurement The most general quantum measurement can be described . Mid-circuit measurements play two primary roles in computations. quantum tomography [7]. The CNOT gate performs the act of un-entangling the two previously entangled qubits. In mathe-matics texts it is usual to denote a random variable as a capital letter, say X, and the variable denoting one of the values it can take as the corresponding lower case letter, x. Lecture 6 . let us choose 0.5 << 1. QM postulate: quantum measurement is described by a set of operators {Mm} acting on the state space of the system. 2: Circuit for Bell measurements. Pick orthonormal basis jv 1i;:::;jv di. These are operators acting on the state space of the system being measured. POVM stands for positive operator valued measure.The outcomes of such a measurement are indexed by positive operators, and the word "measure" here is used because there can conceivably be an infinite number of such outcomes, in which case you need to . Characteristics of quantum circuits, no cloning theorem, measurement in different Basis, Bell basis We do not worry too much about operation MeasureEachZ (targets : Qubit[]) : Result[] Input targets : Qubit[] An array of qubits to be measured. Remarks. Step A. The standard basis for measurement here is { | 0 , | 1 }. Measurement is performed in the computational basis, unless otherwise noted. This property forms the basis of quantum cryptography where the presence of an eavesdropper necessarily alters the quantum state being transmitted. The notion arises by considering states diagonal in that basis and investigating whether probability distributions associated with different quantum measurements can be converted into one another by probabilistic postprocessing. In quantum mechanics, this question is not well-posed. Quantum gates (operators) are applied sequentially to qubit states, with result shown on the right. A 1 -qubit system, in general, can be in a state a | 0 + b | 1 where | 0 and | 1 are basis vectors of a two dimensional complex vector space. 13.1 Measurements in Quantum Mechanics Quantum System S Measuring Apparatus M Surrounding Environment E Figure 13.1: System S interacting with is estimated based on measurements on Typically, the system is composed of many particles, and the Hamiltonian is a sum of single-particle terms where acts on the kth particle. The original quantum cryptography . Mohammad Mirhosseini, Omar S. Magaa-Loaiza, Seyed Mohammad Hashemi Rafsanjani, . However, often we want to perform a measurement in some other basis, defined by a complete set of orthonormal states.
For example, when a qubit is in a superposition state of equal weights, a measurement will make it collapse to one of its two basis states Title:Measurement-based quantum computation. Commonly . State initialization in a specific basis can be done explicitly with the cQASM instructions prep_z, prep_y and prep_x, which prepare qubits in the \vert 0 \rangle 0 , \vert R \rangle R and - e.g. Execute the circuits on the quantum system. In this way, the widespread vagueness . 3. j i\collapses" to jv ii. In this example let's say we wish to find the best basis ($|\\psi\\rangl. Similar to a bit whose states can be 0 . Measurement-based quantum computing is one of the most promising quantum computing models. Past research has explored how to learn a quantum state with a small number of measurements by exploiting techniques from elds such as compressive sensing [8]. . circuit.measure maps the quantum measurement to classical bits.
That is, using this language, "measure Y Y " is equivalent to applying H S H S and then measuring in the computational basis, where S is an intrinsic quantum operation sometimes called the "phase gate," and can be simulated by the unitary matrix S= [1 0 0 i]. measurement and general formulas A measurement is described by an Hermitian operator (observable) M M = m P m - P m is the projector onto the eigenspace of M with eigenvalue m - A fter the measurement the state will be with probability p(m) = | P m | . First, they can be thought of as Boolean tests for a property of a quantum state before the final measurement takes place. For example, one qubit can be one electron where information can be stored in its spin. The second part of the lecture went over the basics of the quantum circuit model. The second part of the lecture went over the basics of the quantum circuit model. The reverse of this circuit can be used to convert the Bell basis back to the computational basis as shown in Fig. We see that each qubit parameter is expressed as an Nduv (name, date, unit value) object containing the local time at which the parameter was updated, the parameter name, parameter units, and the actual numerical parameter value.. gates - system.properties().gates gives detailed information on each gate that the system supports executing. However, they are subject to measurement errors due to hardware imperfections in near-term quantum devices. Assuming that the evolution of a quantum system is given by a dynamical map in the Kraus representation, one can switch to the Heisenberg picture and define the measurements in the time domain. 1, where the nal D means a measurement in the standard basis with result m= 0 or 1. It thus represents the ultimate form of cryptography . The probability p of a measurement resultm occurring when the state is measured is the state of the system after the measurementis completeness: the sum over all measurement outcomes has to be unity 2.6.1 The quantum measurement . Linear algebra PDF | On Oct 30, 2015, Constantin V. Usenko published Complexity of Measurement as the Basis of Quantum Channel Security | Find, read and cite all the research you need on ResearchGate Given a pure state j i, a \simple measurement" is as follows. Enter the email address you signed up with and we'll email you a reset link. A measurement in quantum mechanics consists of a set of measurement operators {M m}n =1. In quantum computing we usually label the basis with some boolean name but note carefully that this is only a name. surement basis for qubit j can be specied by a single pa-rameter, the measurement angle j.Themeasurement direction of qubit j is the vector on the Bloch sphere which corresponds to the rst state in the measurement basis B j ( j). The predictions that quantum physics makes are in general probabilistic. This implies that you cannot collect any additional information about the amplitudes j by repeating the measurement. Let's take any of the Bell states matrix form using the C basis {|00 , |01 , |10 , |11 } and check. It has to do with the historical development of quantum mechanics: "measure- ment" was invoked to get rid of certain conceptual problems and paradoxes back in the days before there was a fully consistent probabilistic formulation of quantum theory. Assume the state of the system immediately preceding the measurement is |i. measurement of a qubit in the computational basis measuring . A qubit, or quantum bit, is the smallest unit of quantum information. Although various universal resource states have been proposed so far, it was open whether only two Pauli .
theoretically propose and experimentally demonstrate a method for directly measuring the density matrix of an unknown quantum system in the basis of azimuthal angle. 1. E() = m Z() H = m Figure 1: The equatorial measurement E() on the left corresponds to the measurement circuit on the right. This provides technical precision, since the concept of a . Perform measurement error mitigations on the result to improve the accuracy in the energy estimation. For example, . Direct measurement of the quantum density matrix in the basis of azimuthal angle. This book is the first comprehensive treatment of modern quantum measurement and measurement-based quantum control, which are vital elements for realizing quantum technology. Furthermore, in order to demonstrate an advantage of our hypergraph state, we construct a verifiable blind quantum computing protocol that requires only X and Z-basis measurements for . When you are measuring in this basis, with | a | 2 | a | 2 + | b | 2 100 % probability you will find that the state after . Dr. Gorshkov's creating strong interactions between photons gives a basis for technology using light rather than electrons to perform . Receive outcome \i" with probability jhv ij ij 2. Any quantum state of these two photons belongs to a four-dimensional space of which obvious basis vectors are: x_1 x_2, x_1 y_2, y_1 x_2, and y_1 y_2. As we shall see, this is one of the key features of quantum mechanics that gives rise to its paradoxical properties as well as provides the basis for the power of quantum computation. Measures each qubit in a given array in the standard basis. Prerequisite knowledge.
The index mrefers to the measurement outcome. Lecture 1: Quantum information processing basics Mark M. Wilde The simplest quantum system is the physical quantum bit or qubit. measurement the state in B collapses and Bob can only get one result moreover this still holds if the subsystems . "Quantum measurements are described by a collection f Mm g of measurement oper-ators. Measurement-based quantum computation. 2. Particles do not have trajectories, but rather take all paths simultaneously (in superposition). the observable properties of a quantum system can be described in quantum mechanics, that is in terms of Hermitean operators. If a CNOT gate is applied to qubits A and B, followed by a Hadamard gate on qubit A, a measurement can be made in the computational basis. Quantum Measurement Theory Before we begin, it is worth mentioning a few things about terminology. Quantum Computation and Quantum Information (10th Edition) Edit edition Solutions for Chapter 4 Problem 33E: (Measurement in the Bell basis) The measurement model we have specified for the quantum circuit model is that measurements are performed only in the computational basis. FIG. In particular E() = m, "equatorial measurement at yields the value m," corresponds to the measurement circuit in Fig. We illustrate quantum measurement cooling (QMC) by means of a prototypical two-stroke two-qubit engine which interacts with a measurement apparatus and two heat reservoirs at different temperatures.
Pauli values are used primarily to specify the basis for a measurement. A quantum circuit is a computational routine consisting of coherent quantum operations on quantum data, such as qubits, and concurrent real-time classical computation. it is easy to implement a partial measurement, very useful in quantum information . POVM stands for positive operator valued measure.The outcomes of such a measurement are indexed by positive operators, and the word "measure" here is used because there can conceivably be an infinite number of such outcomes, in which case you need to . Today, we first talked about POVM measurements. In Quantum measurement scenario, a measurement operator is essentially a matrix (rather a carefully chosen matrix) that mathematically manipulates the initial state of the system. of origin and conceptual status of the textbook 'observable operators' in a way that can be understood even on the basis of textbook quantum theory. In any case there is nothing wrong with mentioning measurements. 2. pioneers of quantum mechanics1 it is the basis of the celebrated Einstein-Podolsky-Rosen paper2 which argued that its predictions are incompatible with locality 1Schrdinger E (1935). W e w ork within the standard form ulation of ortho do x (non-relativistic) quan tum mec hanics, 5 The separate \(t\bar{t}H\) and tH measurements lead to an observed (expected) upper limit on tH production of 15 (7) times the standard model prediction at the 95% confidence level (CL), with a . While the basic formalism of quantum mechanics was developed between 1925 and 1927, the standard interpretation of quantum measurement is attributed to von Neumann s theory presented in his book in 1932 (von Neumann, 1932). The key distinction between quantum physics and classical physics is that the results of individual quantum measurements cannot be predicted; quantum mechanics gives us only the probabilities of . QUANTUM MEASUREMENT THEORY P(y|x) y =0 y =1 x =0 1 x =1 1 Table 1.1: The likelihood function for a simple "two-outcome" measurement. program: The Program to execute.. shots: A positive integer that specifies the number of times the program measurement evaluation is to be repeated.. modes: An optional list of integers that specifies which modes we wish the backend to return for the quantum state.If the state is a mixed state represented by a density matrix, then the backend will . While the basic formalism of quantum mechanics was developed between 1925 and 1927, the standard interpretation of quantum measurement is attributed to von Neumann s theory presented in his book in 1932 (von Neumann, 1932). The qubit is a two-level quantum system|example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. If quantum measurements are one day taken from the human brain, they could be compared against our results to definitely decide whether consciousness is a classical or a quantum phenomenon. A number of models of quantum computation exist, including the now well-studied quantum circuit model. The experiment showed that the effect of the measurement on the velocity of the particles continued long after the particles had cleared the measurement device itself, as far as 5 metres away from it. We apply our . As we are preparing the . Device-independent quantum key distribution (DIQKD) is the art of using untrusted devices to distribute secret keys in an insecure network. . 2 Quantum Measurement the Emergence of POVMs and State Transformers 2.1 Groundwork 2.1.1 Motivation 2.1.2 Pointer States 2.2 Measurement(-like) Process . The measurement taken at line 5 will measure the qubit at index 0 in the z-basis, and store the result (a classical 0 or 1) in the classical register at index 0. Lecture 6 . The Result type specifies the result of a quantum measurement. Consequently, starting with an incomplete set of positive operators, one can . @misc{etde_20799479, title = {Quantum cryptography without switching of measurement basis} author = {Weedbrook, C, Lance, A M, Bowen, W P, Symul, T, Lam, P K, and Ralph, T C} abstractNote = {Full text: Quantum cryptography is a form of secret communication between two parties that guarantees absolute security. The approach and material are based upon previous presentations of spin systems by Feynman, Sakurai, Cohen-Tannoudji, and Townsend. For every single-qubit gate listed in the system basis . S = [ 1 0 0 i]. In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. in what is known as the quantum measurement problem. explaining its contributions to the question of measurement in quantum mechanics . Measurement. I wish to find a basis state for the quantum measurement of two states which provides the maximum possible distinguishability. Thus, the measurement angle j is the an-gle between the measurement direction at qubit j and the . The eng.run method accepts the arguments:. Download PDF Abstract: In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The control of individual quantum systems promises a new technology for the 21st century - quantum technology. The postulates of quantum mechanics are illustrated through their We quantify the probability that QMC occurs when the measurement basis is chosen randomly, and find that it can be very large as compared to . States of systems vs states of ensemb les of systems. Measurement is indicated by a box with a symbolic measurement device inside. Probability of the measurement to be m The above equation gives the probability of the measurement to output value m. Now assume that we initially know nothing about x, so that our . The Quantum Measurement Division (QMD) provides the physical foundation for the International System of Units (Systme International d'Units or SI), colloquially referred to as the metric system. Initialization and measurement bases By default, all qubits are initialized in the |0\rangle 0 state in the z-basis. 10 CHAPTER 1. The nature and behavior of matter and energy at that level is sometimes referred to as quantum physics and quantum mechanics. Let the quantum system be prepared in a state represented by the state vector . . The Spin and Quantum Measurement course is an introduction to quantum mechanics through the analysis of sequential Stern-Gerlach spin measurements.
For example, when a qubit is in a superposition state of equal weights, a measurement will make it collapse to one of its two basis states Title:Measurement-based quantum computation. Commonly . State initialization in a specific basis can be done explicitly with the cQASM instructions prep_z, prep_y and prep_x, which prepare qubits in the \vert 0 \rangle 0 , \vert R \rangle R and - e.g. Execute the circuits on the quantum system. In this way, the widespread vagueness . 3. j i\collapses" to jv ii. In this example let's say we wish to find the best basis ($|\\psi\\rangl. Similar to a bit whose states can be 0 . Measurement-based quantum computing is one of the most promising quantum computing models. Past research has explored how to learn a quantum state with a small number of measurements by exploiting techniques from elds such as compressive sensing [8]. . circuit.measure maps the quantum measurement to classical bits.
That is, using this language, "measure Y Y " is equivalent to applying H S H S and then measuring in the computational basis, where S is an intrinsic quantum operation sometimes called the "phase gate," and can be simulated by the unitary matrix S= [1 0 0 i]. measurement and general formulas A measurement is described by an Hermitian operator (observable) M M = m P m - P m is the projector onto the eigenspace of M with eigenvalue m - A fter the measurement the state will be with probability p(m) = | P m | . First, they can be thought of as Boolean tests for a property of a quantum state before the final measurement takes place. For example, one qubit can be one electron where information can be stored in its spin. The second part of the lecture went over the basics of the quantum circuit model. The second part of the lecture went over the basics of the quantum circuit model. The reverse of this circuit can be used to convert the Bell basis back to the computational basis as shown in Fig. We see that each qubit parameter is expressed as an Nduv (name, date, unit value) object containing the local time at which the parameter was updated, the parameter name, parameter units, and the actual numerical parameter value.. gates - system.properties().gates gives detailed information on each gate that the system supports executing. However, they are subject to measurement errors due to hardware imperfections in near-term quantum devices. Assuming that the evolution of a quantum system is given by a dynamical map in the Kraus representation, one can switch to the Heisenberg picture and define the measurements in the time domain. 1, where the nal D means a measurement in the standard basis with result m= 0 or 1. It thus represents the ultimate form of cryptography . The probability p of a measurement resultm occurring when the state is measured is the state of the system after the measurementis completeness: the sum over all measurement outcomes has to be unity 2.6.1 The quantum measurement . Linear algebra PDF | On Oct 30, 2015, Constantin V. Usenko published Complexity of Measurement as the Basis of Quantum Channel Security | Find, read and cite all the research you need on ResearchGate Given a pure state j i, a \simple measurement" is as follows. Enter the email address you signed up with and we'll email you a reset link. A measurement in quantum mechanics consists of a set of measurement operators {M m}n =1. In quantum computing we usually label the basis with some boolean name but note carefully that this is only a name. surement basis for qubit j can be specied by a single pa-rameter, the measurement angle j.Themeasurement direction of qubit j is the vector on the Bloch sphere which corresponds to the rst state in the measurement basis B j ( j). The predictions that quantum physics makes are in general probabilistic. This implies that you cannot collect any additional information about the amplitudes j by repeating the measurement. Let's take any of the Bell states matrix form using the C basis {|00 , |01 , |10 , |11 } and check. It has to do with the historical development of quantum mechanics: "measure- ment" was invoked to get rid of certain conceptual problems and paradoxes back in the days before there was a fully consistent probabilistic formulation of quantum theory. Assume the state of the system immediately preceding the measurement is |i. measurement of a qubit in the computational basis measuring . A qubit, or quantum bit, is the smallest unit of quantum information. Although various universal resource states have been proposed so far, it was open whether only two Pauli .
theoretically propose and experimentally demonstrate a method for directly measuring the density matrix of an unknown quantum system in the basis of azimuthal angle. 1. E() = m Z() H = m Figure 1: The equatorial measurement E() on the left corresponds to the measurement circuit on the right. This provides technical precision, since the concept of a . Perform measurement error mitigations on the result to improve the accuracy in the energy estimation. For example, . Direct measurement of the quantum density matrix in the basis of azimuthal angle. This book is the first comprehensive treatment of modern quantum measurement and measurement-based quantum control, which are vital elements for realizing quantum technology. Furthermore, in order to demonstrate an advantage of our hypergraph state, we construct a verifiable blind quantum computing protocol that requires only X and Z-basis measurements for . When you are measuring in this basis, with | a | 2 | a | 2 + | b | 2 100 % probability you will find that the state after . Dr. Gorshkov's creating strong interactions between photons gives a basis for technology using light rather than electrons to perform . Receive outcome \i" with probability jhv ij ij 2. Any quantum state of these two photons belongs to a four-dimensional space of which obvious basis vectors are: x_1 x_2, x_1 y_2, y_1 x_2, and y_1 y_2. As we shall see, this is one of the key features of quantum mechanics that gives rise to its paradoxical properties as well as provides the basis for the power of quantum computation. Measures each qubit in a given array in the standard basis. Prerequisite knowledge.
The index mrefers to the measurement outcome. Lecture 1: Quantum information processing basics Mark M. Wilde The simplest quantum system is the physical quantum bit or qubit. measurement the state in B collapses and Bob can only get one result moreover this still holds if the subsystems . "Quantum measurements are described by a collection f Mm g of measurement oper-ators. Measurement-based quantum computation. 2. Particles do not have trajectories, but rather take all paths simultaneously (in superposition). the observable properties of a quantum system can be described in quantum mechanics, that is in terms of Hermitean operators. If a CNOT gate is applied to qubits A and B, followed by a Hadamard gate on qubit A, a measurement can be made in the computational basis. Quantum Measurement Theory Before we begin, it is worth mentioning a few things about terminology. Quantum Computation and Quantum Information (10th Edition) Edit edition Solutions for Chapter 4 Problem 33E: (Measurement in the Bell basis) The measurement model we have specified for the quantum circuit model is that measurements are performed only in the computational basis. FIG. In particular E() = m, "equatorial measurement at yields the value m," corresponds to the measurement circuit in Fig. We illustrate quantum measurement cooling (QMC) by means of a prototypical two-stroke two-qubit engine which interacts with a measurement apparatus and two heat reservoirs at different temperatures.
Pauli values are used primarily to specify the basis for a measurement. A quantum circuit is a computational routine consisting of coherent quantum operations on quantum data, such as qubits, and concurrent real-time classical computation. it is easy to implement a partial measurement, very useful in quantum information . POVM stands for positive operator valued measure.The outcomes of such a measurement are indexed by positive operators, and the word "measure" here is used because there can conceivably be an infinite number of such outcomes, in which case you need to . Today, we first talked about POVM measurements. In Quantum measurement scenario, a measurement operator is essentially a matrix (rather a carefully chosen matrix) that mathematically manipulates the initial state of the system. of origin and conceptual status of the textbook 'observable operators' in a way that can be understood even on the basis of textbook quantum theory. In any case there is nothing wrong with mentioning measurements. 2. pioneers of quantum mechanics1 it is the basis of the celebrated Einstein-Podolsky-Rosen paper2 which argued that its predictions are incompatible with locality 1Schrdinger E (1935). W e w ork within the standard form ulation of ortho do x (non-relativistic) quan tum mec hanics, 5 The separate \(t\bar{t}H\) and tH measurements lead to an observed (expected) upper limit on tH production of 15 (7) times the standard model prediction at the 95% confidence level (CL), with a . While the basic formalism of quantum mechanics was developed between 1925 and 1927, the standard interpretation of quantum measurement is attributed to von Neumann s theory presented in his book in 1932 (von Neumann, 1932). The key distinction between quantum physics and classical physics is that the results of individual quantum measurements cannot be predicted; quantum mechanics gives us only the probabilities of . QUANTUM MEASUREMENT THEORY P(y|x) y =0 y =1 x =0 1 x =1 1 Table 1.1: The likelihood function for a simple "two-outcome" measurement. program: The Program to execute.. shots: A positive integer that specifies the number of times the program measurement evaluation is to be repeated.. modes: An optional list of integers that specifies which modes we wish the backend to return for the quantum state.If the state is a mixed state represented by a density matrix, then the backend will . While the basic formalism of quantum mechanics was developed between 1925 and 1927, the standard interpretation of quantum measurement is attributed to von Neumann s theory presented in his book in 1932 (von Neumann, 1932). The qubit is a two-level quantum system|example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. If quantum measurements are one day taken from the human brain, they could be compared against our results to definitely decide whether consciousness is a classical or a quantum phenomenon. A number of models of quantum computation exist, including the now well-studied quantum circuit model. The experiment showed that the effect of the measurement on the velocity of the particles continued long after the particles had cleared the measurement device itself, as far as 5 metres away from it. We apply our . As we are preparing the . Device-independent quantum key distribution (DIQKD) is the art of using untrusted devices to distribute secret keys in an insecure network. . 2 Quantum Measurement the Emergence of POVMs and State Transformers 2.1 Groundwork 2.1.1 Motivation 2.1.2 Pointer States 2.2 Measurement(-like) Process . The measurement taken at line 5 will measure the qubit at index 0 in the z-basis, and store the result (a classical 0 or 1) in the classical register at index 0. Lecture 6 . The Result type specifies the result of a quantum measurement. Consequently, starting with an incomplete set of positive operators, one can . @misc{etde_20799479, title = {Quantum cryptography without switching of measurement basis} author = {Weedbrook, C, Lance, A M, Bowen, W P, Symul, T, Lam, P K, and Ralph, T C} abstractNote = {Full text: Quantum cryptography is a form of secret communication between two parties that guarantees absolute security. The approach and material are based upon previous presentations of spin systems by Feynman, Sakurai, Cohen-Tannoudji, and Townsend. For every single-qubit gate listed in the system basis . S = [ 1 0 0 i]. In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. in what is known as the quantum measurement problem. explaining its contributions to the question of measurement in quantum mechanics . Measurement. I wish to find a basis state for the quantum measurement of two states which provides the maximum possible distinguishability. Thus, the measurement angle j is the an-gle between the measurement direction at qubit j and the . The eng.run method accepts the arguments:. Download PDF Abstract: In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The control of individual quantum systems promises a new technology for the 21st century - quantum technology. The postulates of quantum mechanics are illustrated through their We quantify the probability that QMC occurs when the measurement basis is chosen randomly, and find that it can be very large as compared to . States of systems vs states of ensemb les of systems. Measurement is indicated by a box with a symbolic measurement device inside. Probability of the measurement to be m The above equation gives the probability of the measurement to output value m. Now assume that we initially know nothing about x, so that our . The Quantum Measurement Division (QMD) provides the physical foundation for the International System of Units (Systme International d'Units or SI), colloquially referred to as the metric system. Initialization and measurement bases By default, all qubits are initialized in the |0\rangle 0 state in the z-basis. 10 CHAPTER 1. The nature and behavior of matter and energy at that level is sometimes referred to as quantum physics and quantum mechanics. Let the quantum system be prepared in a state represented by the state vector . . The Spin and Quantum Measurement course is an introduction to quantum mechanics through the analysis of sequential Stern-Gerlach spin measurements.