Lectures 1 and 2: Axioms for Quantum Theory

Lectures 1 and 2: Axioms for Quantum Theory Joseph Emerson Course: AMATH 900/AMATH 495/PHYS 490 Foundations and Interpretations of Quantum Theory Course Instructors: Joseph Emerson and Ray Laflamme Hosted by: Institute for Quantum Computing, University of Waterloo, and Perimeter Institute for Theoretical Physics Jan 12 and Jan 14, 2010 J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 1 / 79

Outline 1 Introduction and Motivation 2 Axioms for Quantum Theory Ideal Preparations: Hilbert Space Vectors Ideal Measurements: Self-adjoint Operators Composite Systems: Tensor-Product Structure Ideal Transformations 1: Unitary Operators Ideal Transformations 2: Projections 3 Generalized Axioms for Quantum Theory Generalized Preparations: Density Operators Generalized Measurements: POVMs Generalized Transformations: CP maps Measurement as a Generalized Transformation Composite Systems and Entanglement J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 2 / 79

Introduction and Motivation Some Introductory Thoughts The purpose of this course is to gain a deeper understanding of what kind of theory quantum theory is, and to learn what it tells us about the world. A main goal is to address the question, What is a quantum state? Throughout this course you will hear a surprising variety of answers to this simply question. A related goal is to understand what we may or may not deduce about reality, or, to use a more philosophical term, about the fundamental ontology, in light of quantum theory. Of course, one option is to deny that there is any reality at all, and another is to say that there are infinitely many. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 3 / 79

Introduction and Motivation Some Introductory Thoughts The purpose of this course is to gain a deeper understanding of what kind of theory quantum theory is, and to learn what it tells us about the world. A main goal is to address the question, What is a quantum state? Throughout this course you will hear a surprising variety of answers to this simply question. A related goal is to understand what we may or may not deduce about reality, or, to use a more philosophical term, about the fundamental ontology, in light of quantum theory. Of course, one option is to deny that there is any reality at all, and another is to say that there are infinitely many. Somewhat amazingly, we will see that, if you accept that there is something really going on, ie some unique reality, then irrespective of what ontology you believe in, it must satisfy certain constraints, in particular, non-locality and contextuality. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 3 / 79

Introduction and Motivation Some Introductory Thoughts One of the first things we will learn is that you can still be a practical user of quantum theory, i.e., a practical quantum technician, without ever worrying about the above issues. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 4 / 79

Introduction and Motivation Some Introductory Thoughts One of the first things we will learn is that you can still be a practical user of quantum theory, i.e., a practical quantum technician, without ever worrying about the above issues. So, then, why worry about them? First, intellectual curiosity. Of course the main goal of science to predict and control phenomena. But we also want to understand how Nature works. Second, missed opportunities in research. Not surprisingly, users with a poor understanding of these interpretational issues will often be led to erroneous conclusions about what is, and is not, possible to achieve with quantum theory. There are many historical examples of this. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 4 / 79

Introduction and Motivation Some Introductory Thoughts One of the first things we will learn is that you can still be a practical user of quantum theory, i.e., a practical quantum technician, without ever worrying about the above issues. So, then, why worry about them? First, intellectual curiosity. Of course the main goal of science to predict and control phenomena. But we also want to understand how Nature works. Second, missed opportunities in research. Not surprisingly, users with a poor understanding of these interpretational issues will often be led to erroneous conclusions about what is, and is not, possible to achieve with quantum theory. There are many historical examples of this. On the flip side, major advances in the application of quantum theory, such as quantum information technology, were born out of concerns about the unusual ontological implications of quantum phenomena such as superposition and entanglement. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 4 / 79

Introduction and Motivation Some Introductory Thoughts Finally, after almost a century of efforts, no one has been able to understand how to combine quantum theory and general relativity to construct a single theoretical framework, capable of describing physical phenomena on all scales and with all known forces involved. Is this because too many researchers have neglected answering carefully the simple question: What is a quantum state? J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 5 / 79

Introduction and Motivation Some Introductory Thoughts Finally, after almost a century of efforts, no one has been able to understand how to combine quantum theory and general relativity to construct a single theoretical framework, capable of describing physical phenomena on all scales and with all known forces involved. Is this because too many researchers have neglected answering carefully the simple question: What is a quantum state? Before we get to these issues, we first we have to be clear that we know how to be practical quantum users... J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 5 / 79

Axioms for Quantum Theory Ideal Preparations: Hilbert Space Vectors Ideal Preparations: Hilbert Space Vectors Axiom 1. An ideal preparation procedure is described by a Hilbert space vector ψ H. Ideal preparations are often called pure states. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 6 / 79

Axioms for Quantum Theory Ideal Preparations: Hilbert Space Vectors Ideal Preparations: Hilbert Space Vectors Axiom 1. An ideal preparation procedure is described by a Hilbert space vector ψ H. Ideal preparations are often called pure states. In discussions of interpretation, calling preparations states can lead to confusion, because the word state can connote ontological status. In finite dimensions H = C d. Normalization implies ψ = 1, which prescribes a hypersphere S 2d 1 in a 2d-dimensional real vector space. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 6 / 79

Axioms for Quantum Theory Ideal Preparations: Hilbert Space Vectors Ideal Preparations: Hilbert Space Vectors Axiom 1. An ideal preparation procedure is described by a Hilbert space vector ψ H. Ideal preparations are often called pure states. In discussions of interpretation, calling preparations states can lead to confusion, because the word state can connote ontological status. In finite dimensions H = C d. Normalization implies ψ = 1, which prescribes a hypersphere S 2d 1 in a 2d-dimensional real vector space. Because state vectors have a complex phase which is physically insignificant, distinct preparations are in one-to-one correspondence with elements of the complex projective space CP d 1. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 6 / 79

Axioms for Quantum Theory Ideal Measurements: Self-adjoint Operators Ideal Measurements: Self-adjoint Operators Let Â be a self-adjoint operator with discrete eigenvalues a l and eigenvectors { a l, m l }, where m l indexes an orthogonal set of vectors spanning any degenerate subspaces. An important representation of a self-adjoint operator is its spectral decomposition. In the case of a discrete spectrum we have Â = l a l ˆP al, where we introduce projectors onto the (possibly degenerate) eigenspaces associated with distinct eigenvalues P al = m l a l, m l a l, m l. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 7 / 79

Axioms for Quantum Theory Ideal Measurements: Self-adjoint Operators Ideal Measurements: Self-adjoint Operators Axiom 2. An ideal measurement procedure is represented by a self-adjoint operator Â. (a) The set of observable outcomes is given by the eigenvalues {a l } of Â. (b) The probability of finding outcome a l, given preparation ψ, is Pr(a l ) = Tr( ψ ψ P al ). Axiom 2.a) is responsible for the novel structural aspects of quantum theory. Observables can have a discrete spectrum, are quantized, meaning that the possible observable outcomes are discrete. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 8 / 79

Axioms for Quantum Theory Ideal Measurements: Self-adjoint Operators Ideal Measurements: Self-adjoint Operators Axiom 2. An ideal measurement procedure is represented by a self-adjoint operator Â. (a) The set of observable outcomes is given by the eigenvalues {a l } of Â. (b) The probability of finding outcome a l, given preparation ψ, is Pr(a l ) = Tr( ψ ψ P al ). Axiom 2.a) is responsible for the novel structural aspects of quantum theory. Observables can have a discrete spectrum, are quantized, meaning that the possible observable outcomes are discrete. Axiom 2.b) is the Born rule. It provides the statistical/probabilistic/indeterministic character of quantum predictions. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 8 / 79

Axioms for Quantum Theory Ideal Measurements: Self-adjoint Operators Ideal Measurements: Self-adjoint Operators We can replace the term probability (as it occurs in the Born rule) with the phrase relative frequency. After all, we can only ever empirically compare the RHS of the Born rule to observed outcome sequences. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 9 / 79

Axioms for Quantum Theory Ideal Measurements: Self-adjoint Operators Ideal Measurements: Self-adjoint Operators We can replace the term probability (as it occurs in the Born rule) with the phrase relative frequency. After all, we can only ever empirically compare the RHS of the Born rule to observed outcome sequences. In this way can we banish the term probability from the formulation of quantum theory altogether, and cancel two weeks of lectures? J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 9 / 79

Axioms for Quantum Theory Composite Systems: Tensor-Product Structure Composite Systems: Tensor-Product Structure Suppose we have two systems, A and B, with state spaces H A and H B, both of which are separable Hilbert spaces and hence possess orthonormal bases { a k } and { b l } respectively. We wish to describe these systems jointly by a Hilbert space H AB. How is this Hilbert space related to the Hilbert spaces of the subsystems? J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 10 / 79

Axioms for Quantum Theory Composite Systems: Tensor-Product Structure Composite Systems: Tensor-Product Structure Suppose we have two systems, A and B, with state spaces H A and H B, both of which are separable Hilbert spaces and hence possess orthonormal bases { a k } and { b l } respectively. We wish to describe these systems jointly by a Hilbert space H AB. How is this Hilbert space related to the Hilbert spaces of the subsystems? Axiom 3. The Hilbert space of a composite system is given by the tensor product of the subsystem Hilbert spaces, H AB = H A H B. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 10 / 79

Axioms for Quantum Theory Composite Systems: Tensor-Product Structure Composite Systems: Tensor-Product Structure Remark that H AB is spanned by { a k b l }, k, l, and dim(h A H B ) = dim(h A )dim(h B ). Hence an arbitrary state vector ψ AB H AB of the joint system can be obtained from linear combinations of the joint-basis states, ψ AB = k,l ψ kl a k b l, ψ kl = ( a k b l ) ψ AB C. In the finite-dimensional case, where H A = C M and H B = C N, we have H AB = H A H B = C MN. Terminology: The state associated with a composition of two subsystems is called bipartite; similarly the state associated with a composition of three subsystems is called tripartite, and so on. The state space of each subsystem is called a factor space of the full Hilbert space. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 11 / 79

Ideal Transformations Axioms for Quantum Theory Ideal Transformations 1: Unitary Operators During the time-interval between the preparation procedure and the measurement procedure the system evolves under a dynamical transformation: J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 12 / 79

Ideal Transformations Axioms for Quantum Theory Ideal Transformations 1: Unitary Operators During the time-interval between the preparation procedure and the measurement procedure the system evolves under a dynamical transformation: Axiom 4. Ideal dynamical transformations are generated by a linear operator U, ψ(t 2 ) = Û(t 2, t 1 ) ψ(t 1 ) satisfying, i Û(t 2, t 1 ) t 2 = Ĥ(t 2 )Û(t 2, t 1 ), where Ĥ(t) is a self-adjoint operator representing the system energy function and t R is time, subject to the initial condition Û(t 1, t 1 ) = 11. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 12 / 79

Ideal Transformations Axioms for Quantum Theory Ideal Transformations 1: Unitary Operators From Axiom 4 we can deduce Schrödinger s equation, i d ψ(t) = Ĥ(t) ψ(t). dt J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 13 / 79

Axioms for Quantum Theory Ideal Transformations 2: Projections What is the state after an ideal filtering measurement? In order to understand how to describe the state of a quantum system after an ideal filtering measurement, von Neumann considered the Compton experiment where photons are scattered off electrons that are initially at rest. The quantum unitary evolution predicts a statistical dispersion in the final momenta of the particles, but we know that the final momenta of the electron and photon are correlated due to conservation of energy and momentum. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 14 / 79

Axioms for Quantum Theory Ideal Transformations 2: Projections What is the state after an ideal filtering measurement? von Neumann then imagined a scenario where there is a finite time difference t between the measurement of the electron and of the photon, i.e., a time-delay between the detection times of these particles. Forward reference: note the similarity to the EPR and Bell-type arguments involving entangled states. Because, after measurement of the electron, the photon s momentum can be predicted with certainty (within some finite precision) this means that, after the electron momentum has been observed, the state describing the photon must be somehow updated (to within the same finite precision) to a new state that is consistent with the observed outcome for the electron momentum. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 15 / 79

Axioms for Quantum Theory Another kind of Transformation Ideal Transformations 2: Projections The transformation taking the quantum state accorded just before measurement of the electron to that just after measurement is not consistent with the usual unitary (Schrodinger) evolution. We will prove this later, but for now consider that: After measurement, the state update must be indeterministic - it is conditional on the random outcome of the measurement of the electron momentum. After measurement, the state update must be discontinuous - an instantaneous update of the photon s state is required. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 16 / 79

Axioms for Quantum Theory Another kind of Transformation Ideal Transformations 2: Projections The transformation taking the quantum state accorded just before measurement of the electron to that just after measurement is not consistent with the usual unitary (Schrodinger) evolution. We will prove this later, but for now consider that: After measurement, the state update must be indeterministic - it is conditional on the random outcome of the measurement of the electron momentum. After measurement, the state update must be discontinuous - an instantaneous update of the photon s state is required. This practical consideration of the ideal filtering type measurements that were possible with entangled systems forced von Neumann to formally introduce a second kind of dynamical transformation into quantum theory: the projection postulate. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 16 / 79

Axioms for Quantum Theory Ideal Transformations 2: Projections Evolution under Sequential Measurements von Neumann realized from his analysis of sequential measurements that two kinds of transformation were required in quantum mechanics (von Neumann, 1932): Process 1. After observation/measurement of an outcome a k, the system is left in the eigenstate a k associated with the detected eigenvalue a k. We have the map, ψ = k c k a k a k. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 17 / 79

Axioms for Quantum Theory Ideal Transformations 2: Projections Evolution under Sequential Measurements von Neumann realized from his analysis of sequential measurements that two kinds of transformation were required in quantum mechanics (von Neumann, 1932): Process 1. After observation/measurement of an outcome a k, the system is left in the eigenstate a k associated with the detected eigenvalue a k. We have the map, ψ = k c k a k a k. Process 2. The normal Schrödinger (unitary) evolution. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 17 / 79

Projection Postulate Axioms for Quantum Theory Ideal Transformations 2: Projections Axiom 5. After observation/measurement of an outcome a k, the system is left in the eigenstate a k associated with the detected eigenvalue a k, that is, ψ = k c k a k a k. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 18 / 79

Projection Postulate Axioms for Quantum Theory Ideal Transformations 2: Projections Axiom 5. After observation/measurement of an outcome a k, the system is left in the eigenstate a k associated with the detected eigenvalue a k, that is, ψ = k c k a k a k. Terminology: the projection postulate is also known as the reduction of the wavepacket and the collapse of the wavefunction. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 18 / 79

Axioms for Quantum Theory Ideal Transformations 2: Projections Projection Postulate Axiom 5. After observation/measurement of an outcome a k, the system is left in the eigenstate a k associated with the detected eigenvalue a k, that is, ψ = k c k a k a k. Terminology: the projection postulate is also known as the reduction of the wavepacket and the collapse of the wavefunction. von Neumann imagined Process 1 as an essential randomness in nature, and he considered it grounds for abandoning the principle of sufficient cause, which I take to mean causal determinism. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 18 / 79

Axioms for Quantum Theory Ideal Transformations 2: Projections Projection Postulate Axiom 5. After observation/measurement of an outcome a k, the system is left in the eigenstate a k associated with the detected eigenvalue a k, that is, ψ = k c k a k a k. Terminology: the projection postulate is also known as the reduction of the wavepacket and the collapse of the wavefunction. von Neumann imagined Process 1 as an essential randomness in nature, and he considered it grounds for abandoning the principle of sufficient cause, which I take to mean causal determinism. More on this later! J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 18 / 79

Generalized Preparations: Density Operators General states as mixtures of pure states Suppose we want to describe a quantum system which is prepared according to one procedure, represented by state ψ 1, with probability p 1 and according to a distinct procedure, represented by state ψ 2, with probability p 2. How can we do this? If we are measuring the operator A = a aˆp a which possesses non-degenerate eigenvalues a R associated with orthogonal eigenspaces ˆP a, then the probability of obtaining outcome a given preparation ψ 1 is and similarly for preparation 2. Pr(a ψ 1 ) = Tr(ˆP a ψ 1 ψ 1 ), J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 19 / 79

Generalized Preparations: Density Operators If we do not know which preparation took place then the net probability of finding outcome a is simply Pr(a) = p 1 Pr(a ψ 1 ) + p 2 Pr(a ψ 2 ). By linearity of the trace we deduce that, where Pr(a) = Tr(ˆP a ρ) ρ = p 1 ψ 1 ψ 1 + p 2 ψ 2 ψ 2 is non-negative operator called a density operator satisfying the normalization condition Tr(ρ) = 1 (which ensures that probabilities are conserved). J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 20 / 79

Generalized Preparations: Density Operators General states as mixtures of pure states In this way we can construct general quantum states from probabilistic mixtures (convex combinations) of pure states as follows: (i) Discrete case: ρ = i p i ψ i ψ i with i p i = 1 and p i 0. (ii) Continuous case: ρ = dλp(λ) ψ(λ) ψ(λ) for λ R, with dλp(λ) = 1 and p(λ) 0. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 21 / 79

General states from the partial trace Generalized Preparations: Density Operators Suppose we have a quantum state (density operator) ρ = ρ AB on a composite Hilbert space H AB, where in general ρ need not correspond to a pure state ρ = ψ ψ but may be a probabilistic mixture of pure states. How does one describe the state of subsystem A alone (with a state ρ A ) or B alone (with a state ρ B )? The relationship between ρ A and ρ AB is generated by the partial trace operation: ρ A = Tr B (ρ AB ). The state ρ A is called the reduced state associated with ρ AB. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 22 / 79

Generalized Preparations: Density Operators This relationship can be deduced from physical consistency of demanding that Â 11 B = Â for all Hermitian operators Â and states ˆρ AB. Hence, (ρ A ) ll = k (ρ AB ) lkl k, which gives us an explicit matrix representation of ˆρ A in terms of the matrix elements of ρ AB via the partial trace. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 23 / 79

General states from the partial trace Generalized Preparations: Density Operators Definition The partial trace over a subsystem B of an operator O acting on the composite space H AB, Ô A = Tr B [Ô AB ], can be defined in terms of the matrix representation, (Ô A ) ll = l Ô A l = k l k Ô AB l k. It should be understood that the operation Tr B ( ) takes as input any linear operator on H AB (not necessarily a density operator) and generates a linear operator on H A. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 24 / 79

Generalized Preparations: Density Operators Generalized states Generalized Axiom 1: The preparation of a system is represented by a positive semidefinite operator ρ subject to the normalization constraint Tr(ρ) = 1. An operator P is positive semi-definite iff it is self-adjoint and satisfies u P u 0 for every vector u in the Hilbert space. A positive semidefinite operator ( i.e., a non-negative operator) is often just called a positive operator. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 25 / 79

Generalized Preparations: Density Operators Pure States vs Mixed States For a state operator ˆρ subject to the normalization condition Tr(ˆρ) = 1 there are three equivalent definitions of purity: i) ˆρ 2 = ˆρ, which means that ρ is projector. ii) Tr(ˆρ 2 ) = 1. iii) ˆρ = ψ ψ, defining a projector onto a one-dimensional subspace of H. Definition If ρ can not be expressed in the form ρ = ψ ψ for any ψ H, i.e., if ρ is not a pure state, then it is called a mixed state. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 26 / 79

Pure States vs Mixed States Generalized Preparations: Density Operators General states obtained via partial trace are sometimes called improper mixtures, whereas the term proper mixtures refers to general states obtained from probabilistic mixing of pure states. These two conceptually distinct classes of mixed states are mathematically (and operationally) indistinguishable, as is evident from the following theorems: Theorem Any mixed state can be expressed as a convex combination of pure states. Theorem Any mixed state can be realized as the reduced state obtained from an (entangled) pure state on an extended Hilbert space. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 27 / 79

Generalized Preparations: Density Operators Non-uniqueness of purifications Any pure state on the extended Hilbert space H AB = H A H B associated with a reduced state ρ A is called a purification of ρ A. If the reduced state has rank m, then the ancilla factor space H B must have dimension greater than or equal to m. The purification of a mixed state is never unique. Indeed any state ψ = (11 U) ψ, where U is an arbitrary unitary operator acting on H B, provides a valid purification of ρ A. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 28 / 79

Generalized Measurements Generalized Measurements: POVMs Recall from standard Axiom 2 that the primitives of a measurement, associated with a self-adjoint operator A, are the orthogonal projectors P a (onto distinct, possibly degenerate, eigenspaces of A) in the spectral decomposition of A. Any set of orthogonal projectors {P a }, satisfying a P a = 11 is called a projector valued measure, or PVM for short. We can construct measurements that have a more general structure than a PVM in two different ways. First, we can build up a more general measurement by considering classical probabilistic mixtures of PVM measurements. Second, we can consider what structure occurs when look at the reduced measurement obtained from different kinds of PVM measurement on an extended Hilbert space. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 29 / 79

Generalized Measurements: POVMs Mixtures of Projector Valued Measures Consider two distinct PVMs, each given by a discrete set of D rank-one orthogonal projectors: {P i } with i = 1,..., D and { P j } with j = 1,..., D, satisfying i P i = 11 and j P j = 11, where the orthogonality implies P i P i = P i δ ii and P j P j = P j δ jj Note that in general the elements P i and P j are non-orthogonal. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 30 / 79

Generalized Measurements: POVMs Mixtures of Projector Valued Measures Suppose we have a device which performs the first PVM at random with probability p and the second with probability 1 p. Given a preparation ρ on a D-dimensional Hilbert space, from Axiom 2 we know that we can represent the probability of each of the 2D possible outcomes as follows: Pr(i) = ptr(p i ρ) Pr(j) = (1 p)tr( P j ρ). J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 31 / 79

Generalized Measurements: POVMs Mixtures of Projector Valued Measures Let E ν = pp i for ν = i and E ν = (1 p) P j for ν = D + j. Then we can describe the probabilities of the 2D possible outcomes with the simple formula Pr(ν) = Tr(E ν ρ), where these new operators satisfy: E ν = 11 ν E ν foreachν. Note that when p (0, 1) the operators {E ν } are not projectors. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 32 / 79

Generalized Measurements: POVMs PVMs on an Extended Hilbert Space Suppose instead now that we have a composite system represented by the state ρ A ρ B and we perform a joint measurement of both systems. This is represented by a PVM {P ν } acting on H A H B, with the usual properties P ν P ν = P ν δ νν and ν P ν = 11, and where the Greek index run from 1 to K MN = dim(h A ) dim(h B ). The probability of outcome ν is Pr(ν) = Tr[P ν (ρ A ρ B )] = M ij=1 kl=1 N (P ν ) ik,jl (ρ A ) ji (ρ B ) lk, where (P ν ) ik,jl = i k (P ν ) j l. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 33 / 79

PVMs on a Composite System Generalized Measurements: POVMs The above measurement can be expressed in the form Pr(ν) = Tr[E ν ρ A ] where (E ν ) ij = kl (P ν) ik,jl (ρ B ) lk is an operator acting on H A. It is easy to see that the measurement operators E ν satisfy E ν = 11 A ν E ν 0. which are the same conditions we found for mixtures of PVMs. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 34 / 79

Generalized Measurements: POVMs PVMs on a Composite System Note that the measurement operators E ν are not necessarily orthogonal (this is an important difference from a PVM) and hence that the number of elements in the set {E ν } may be greater than the Hilbert space dimension. Indeed the measurement operators {E ν } can also form a continuous set. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 35 / 79

Generalized Measurements: POVMs PVMs on a Composite System Another important measurement paradigm is the following: In order to measure a property of system A, prepared in state ρ A, we allow it to interact in a controlled way with another system B, which is initially prepared in some known state ρ B = 0 B 0 B. We then perform a measurement on the system B alone. This paradigm models the important case of coupling the system to an apparatus which is, in turn, observed directly. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 36 / 79

Generalized Measurements: POVMs PVMs on a Composite System In this measurement method, not only do we gain information about the initial state of system A, but we can deduce also something about the state of system B after the measurement. That is, this paradigm provides a filtering type-measurement of system A, which is a method of preparing a known state. Note this paradigm is a model for the kind of measurement von Neumann considered (ie, the Compton experiment set-up) when he deduced the necessity of introducing the projection postulate as a dynamical process associated with measurement. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 37 / 79

Generalized Measurements: POVMs PVMs on a Composite System How can we represent this process as a measurement operator acting on system A alone? Applying Axiom 2, we represent the direct measurement of the apparatus system with the PVM {P m }, where m = 1,..., K with K N = dim(h B ). This measurement can be expressed on the joint system as the PVM {11 A P m }. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 38 / 79

Generalized Measurements: POVMs PVMs on a Composite System Let U be an arbitrary unitary operator that couples the two systems. Then the probability of outcome m is Pr(m) = Tr[(11 A P m )U(ρ A ρ B )U ] = i i A m B U 0 B ρ A 0 B U i A m B = Tr[A m0 ρ A A m0 ] = Tr[E mρ A ], where we have used the cyclic property of the trace and defined E m A m0 A m0. The operators E m are positive (semi-definite) operators that act only on H A and satisfy the properties: (i) E m 0 and (ii) m E m = 11 A. These are the same conditions on the operators E ν that we found previously. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 39 / 79

Generalized Measurements: POVMs Generalized Measurement: Discrete Case We ve seen three measurements paradigms which motivate the following definition and axiom: Definition (Discrete POVM) A discrete positive operator valued measure (POVM) is a set of operators {E ν } satisfying: (i) E ν 0 for each ν {1, 2,... }. (ii) ν E ν = 11. Generalized Axiom 2 (Discrete Case): A measurement procedure with discrete outcomes is represented by a discrete POVM {E ν }, and the probability of observing outcome ν, given any preparation ρ, is Pr(ν) = Tr(E ν ρ). J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 40 / 79

Generalized Measurements: POVMs Continuous Outcome POVMs We can generalize the preceding to the case of continuous outcomes: Let Ω be a non-empty set and F be a σ-algebra of subsets of Ω so that (Ω, F) forms a measure space. Those unfamiliar with measure spaces can just think of Ω as a space of possible outcomes, e.g., the real line, and of F as the measurable subsets of Ω, e.g., arbitrary intervals on the real line. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 41 / 79

Generalized Measurements: POVMs POVM - general definition Definition A positive operator valued measure (POVM) E : F L(H) is defined by the properties: (i) E(X ) 0 for all X F (ii) E(Ω) = 11 (iii) E( i X i) = i E(X i) for all disjoint sequences {X i } F J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 42 / 79

Generalized Measurements: POVMs POVM as a continuous PVM If the POVM elements satisfy E(X ) = E(X ) 2 for all X F then the POVM reduces to a PVM: in which case the set Ω may be taken without loss of generality to be the real line R and the σ-algebra consists of the B(R), the Borel subsets of R. As a result we recover a continuous PVM as a one-parameter family of projection operators. That is, in terms of the Borel sets we can define a PVM E : B(R) L(H) by the conditions: (i) E(X ) = E 2 (X ) for all X B(R) (ii) E(R) = 11 (iii) E( i X i) = i E(X i) for all disjoint sequences {X i } B(R), Note that i) implies E(X Y ) = E(X )E(Y ) for all X, Y F and also implies that E(X ) = E (X ). J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 43 / 79

Generalized Measurements Generalized Measurements: POVMs This gives a more general version of Axiom 2: Generalized Axiom 2: Any measurement procedure can be represented by a POVM E : F L(H), and tor any preparation ρ, the probability of observing an outcome X F is Pr(X ) = Tr(E(X )ρ). You can think of outcome X as corresponding to a question like: Was the position q found to be within the interval X R? J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 44 / 79

Non-uniqueness of purifications Generalized Measurements: POVMs So we have seen that physically realizable cases of generalized measurement correspond to a POVM measurement. But does every POVM measurement correspond to some physically realizable measurement, and, in particular, to some realizable PVM measurement? J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 45 / 79

Generalized Measurements: POVMs Neumark s Theorem The answer to this question is given by Neumark s theorem (actually a simplified version of it): Theorem (Neumark) For any POVM {E} acting on a Hilbert space H A there exists a PVM {P} acting on H A H B and a state φ φ acting on H B such that Tr[(ρ φ φ )P(X )] = Tr[E(X )ρ] for any state ρ acting on H A and any X F. The PVM can always be expressed in the form U (11 A P)U, i.e., the PVM P acts only on H B. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 46 / 79

Generalized Measurements: POVMs Neumark s Theorem Recall in the case of generalized preparations, which were given by density operators, the sets of proper and improper mixtures were mathematically equivalent (and hence operationally indistinguishable). This is not the case for POVM measurements. That is, we can define proper POVMs as those obtained from convex combinations of PVMs. Similarly, we can define improper POVMs as those obtained from a PVM measurement on an extended Hilbert space. From Neumark s theorem we know that proper POVMs must be a subset of improper POVMs. However, they are a strict subset. This means that operationally implementing some POVM measurements requires access to (and control over) a larger Hilbert space. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 47 / 79

Generalized Measurements: POVMs Example of improper POVM Here is a simple example of an improper POVM: Example Consider the trine given by the set of three projectors χ ν χ ν acting on C 2 defined by: (σ n ν )χ ν = χ ν where n 1, n 2 and n 3 denote three unit vectors making angles of 120 degrees with each other. Let E ν = (2/3) χ ν χ ν. The trine is the smallest possible POVM that is not a PVM. It also can not be generated, mathematically or operationally, from taking convex combinations of PVMs. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 48 / 79

Generalized Transformations: CP maps Generalized Transformations As with measurements and states, there are two ways to construct generalized transformations: By taking convex combinations of unitary transformations. By considering a unitary acting on an extended Hilbert space and then tracing out the ancillary system. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 49 / 79

Mixtures of Unitary Operators Generalized Transformations: CP maps Consider a procedure whereby we subject a preparation ρ to transformation U j with probability p j. The effective transformation is then given by a convex combination of unitary operators Λ(ρ) = p j U j ρu j. j Clearly this map is in general non-unitary, but it always preserves the trace of the input state. Specifically, if ρ = Λ(ρ), from the linearity of the trace we see that Trρ = j p j Tr(U j ρ j U j ) = 1. By convexity, the output state will remain a positive (semi-definite) operator. Hence the map Λ is called positive. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 50 / 79

Generalized Transformations: CP maps Unitary acting on Extended Hilbert Space Consider the effect of a unitary operator on extended Hilbert space H A H B acting on an uncorrelated initial state ρ A (t) = Λ t (ρ A (0)) Tr B [U(t)ρ A (0) 0 B 0 B U (t)] = k A k ρ(0)a k where A k = k U(t) 0 is a linear operator acting on H A. By linearity, a decomposition of the same form is obtained also in the case that the initial environment state is an arbitrary mixed state ρ B. The requirement that the initial state is uncorrelated is strictly stronger than the requirement that the state be separable. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 51 / 79

Generalized Transformations: CP maps Unitary acting on Extended Hilbert Space Clearly this map preserves the trace of the output state: Tr[ k A k ρ A A k ] = Tr[Uρ A(0) 0 B 0 B U ] = 1. Because this holds for any ρ A, from the cyclic property of the trace we deduce that A k A k = 11 A. k Hence it is easy to see from the properties of the partial trace that this map also guarantees the positivity of the reduced state. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 52 / 79

Generalized Transformations: CP maps Kraus Decomposition Definition The expression Λ(ρ) = k A k ρ(0)a k subject to the constraint A k A k = 11 k is called a Kraus decomposition or an operator-sum decomposition of the map Λ, and the set of (bounded) linear operators {A k } are called Kraus operators. For a map Λ constructed from a mixture of unitary operators, one choice for the Kraus operators is the unitary operators weighted by the appropriate probabilities. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 53 / 79

Generalized Transformations Generalized Transformations: CP maps Definition Any linear map Λ taking linear operators to linear operators is called a superoperator. Definition Any superoperator Λ representing a dynamical transformation on the space of quantum states is called a quantum dynamical map. Remark: For some physicists these terms are used interchangeably. J. Emerson (IQC, Univ. of Waterloo) Axioms for Quantum Theory Jan 12 and Jan 14, 2010 54 / 79