Update Math401_T6.md
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Zheyuan Wu
@@ -8,18 +8,255 @@ If you are familiar with the linear algebra defined before, you can jump right i
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### Pure states
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#### Pure state and mixed state
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A pure state is a state that is represented by a unit vector in $\mathscr{H}^{\otimes N}$.
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A mixed state is a state that is represented by a density operator in $\mathscr{H}^{\otimes N}$. (convex combination of pure states)
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if $\rho_j=|\psi_j\rangle\langle\psi_j|$, then $\rho=\sum_{j=1}^N p_j\rho_j$ is a mixed state, where $p_j\geq 0$ and $\sum_{j=1}^N p_j=1$.
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#### Coset space
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Two non-zero vectors $u,v\in \mathscr{H}$ are said to represent the same state if $u=cv$ for some complex number $c$ with $|c|=1$.
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The set of states of a quantum system is called the **coset space** of $\mathscr{H}$, $u\sim v$ if $u=cv$ for some complex number $c$ with $|c|=1$.
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The coset space is called the projective space of $\mathscr{H}$, denoted by $P(\mathscr{H})\colon=(\mathscr{H}\setminus\{0\})/\sim$.
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Any vector in the form $e^{i\theta}|u\rangle$ for some $u\in \mathscr{H}$ and $\theta\in \mathbb{R}$ represents the same state as $|u\rangle$.
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Example: the system of a qubit has a Hilbert space $\mathbb{C}^2$, the coset space is $P(\mathbb{C}^2)\cong S^2$ is the Bloch sphere.
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### Composite systems
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#### Tensor product
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The tensor product of two Hilbert spaces $\mathscr{H}_1$ and $\mathscr{H}_2$ is the Hilbert space $\mathscr{H}_1\otimes\mathscr{H}_2$ with the inner product $\langle u_1\otimes u_2,v_1\otimes v_2\rangle=\langle u_1,v_1\rangle\langle u_2,v_2\rangle$.
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The tensor product of two vectors $u_1\in \mathscr{H}_1$ and $u_2\in \mathscr{H}_2$ is the vector $u_1\otimes u_2\in \mathscr{H}_1\otimes\mathscr{H}_2$.
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#### Multipartite systems
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For each part in a multipartite quantum system, each part is associated a Hilbert space $\mathscr{H}_i$. The total system is associated a Hilbert space $\mathscr{H}=\mathscr{H}_1\otimes\mathscr{H}_2\otimes\cdots\otimes\mathscr{H}_n$.
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The state of the total system has the form $u_1\otimes u_2\otimes\cdots\otimes u_n$ for some $u_i\in \mathscr{H}_i$.
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#### Entanglement (talk later)
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A state $|\psi\rangle$ is entangled if it cannot be expressed as a product state $v_1\otimes v_2$ for any single-qubit states $|v_1\rangle$ and $|v_2\rangle$. In other words, an entangled state is non-separable.
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Example: the Bell state $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ is entangled.
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Assume it can be written as $|\psi\rangle=|\psi_1\rangle\otimes|\psi_2\rangle$ where $|\psi_1\rangle=a|0\rangle+b|1\rangle$ and $|\psi_2\rangle=c|0\rangle+d|1\rangle$. Then:
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$$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$$
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Setting this equal to $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ gives:
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$$ac|00\rangle+ad|01\rangle+bc|10\rangle+bd|11\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$$
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This requires:
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$$ac=bd=\frac{1}{2}$$
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$$ad=bc=0$$
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This is a contradiction, so $|\psi^+\rangle$ is entangled.
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### Mixed states and density operators
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#### Density operator
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A density operator is a Hermitian, positive semi-definite operator with trace 1.
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The density operator of a pure state $|\psi\rangle$ is $\rho=|\psi\rangle\langle\psi|$.
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The density operator of a mixed state is given by the unit vector $u_1,u_2,\cdots,u_n$ in $\mathscr{H}$ with the probability $p_1,p_2,\cdots,p_n$, $p_i\geq 0$ such that $\sum_{i=1}^n p_i=1$.
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The density operator is $\rho=\sum_{i=1}^n p_i|u_i\rangle\langle u_i|$.
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#### Trace 1 proposition
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Density operator on the finite dimensional Hilbert space $\mathscr{H}$ are positive operators having trace equal to 1.
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#### Pure state lemma
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A state is pure if and only if $Tr(\rho^2)=1$.
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For any mixed state $\rho$, $Tr(\rho^2)<1$.
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[Proof ignored here]
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#### Unitary freedom in the ensemble for density operators theorem
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Let $v_1,v_2,\cdots,v_l$ and $w_1,w_2,\cdots,w_l$ be two collections of vectors in the finite dimensional Hilbert space $\mathscr{H}$, the vectors being arbitrary (can be zero) except for the requirement that they define the same density operator $\rho$.
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$$
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\sum_{i=1}^l |v_i\rangle\langle v_i|=\sum_{i=1}^l |w_i\rangle\langle w_i|
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$$
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Then there exists a unitary matrix $U=(\mu_{ij})_{1\leq i,j\leq l}$ such that:
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$$
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v_i=\sum_{j=1}^l \mu_{ij}w_j
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$$
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The converse is also true.
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If $\rho$ is a density operator on $\mathscr{H}$ given by: $\sum_{i=1}^l |w_i\rangle\langle w_i|$ and vector $v_i$ is given by: $v_i=\sum_{j=1}^l \mu_{ij}w_j$, then $\rho_1=\sum_{i=1}^l |v_i\rangle\langle v_i|$ is the density operator of the subsystem $\mathscr{H}_1$.
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[Proof ignored here]
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### Density operator of subsystems
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#### Schmidt Decomposition theorem
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Let $|u\rangle\in \mathscr{H}_1\otimes\mathscr{H}_2$ be a unit vector (pure state), then there exists orthonormal bases $|v_i\rangle$ of $\mathscr{H}_1$ and $|w_j\rangle$ of $\mathscr{H}_2$ and $\{\lambda_k\},k\leq r$, where $r$ is the Schmidt rank of $|u\rangle$, such that:
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$$
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|u\rangle=\sum_{k=1}^r \lambda_k|v_k\rangle\otimes|w_k\rangle
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$$
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where $\lambda_k$ are **non-negative real numbers**. such that $\sum_{k=1}^r \lambda_k^2=1$.
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[Proof ignored here]
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**Remark**: non-zero vector $u\in \mathscr{H}_1\otimes\mathscr{H}_2$ decomposes as a tensor product $u=u_1\otimes u_2$ if and only if the Schmidt rank of $u$ is 1. **A state** that cannot be decomposed as a tensor product is called **entangled**.
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#### Reduced density operator
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In $\mathscr{H}_1\otimes\mathscr{H}_2$, the reduced density operator of the subsystem $\mathscr{H}_1$ is:
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$$
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\rho_1=\operatorname{Tr}_2(\rho)=\sum_{k=1}^r \lambda_k^2|v_k\rangle\langle v_k|
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$$
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where $\rho$ is the density operator in $\mathscr{H}_1\otimes\mathscr{H}_2$.
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Example:
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Let $\rho=\frac{1}{2}(|01\rangle+|10\rangle)\in \mathbb{C}^2\otimes\mathbb{C}^2$,
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Expand the expression of $\rho$ in the basis of $\mathbb{C}^2\otimes\mathbb{C}^2$:
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$$
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\rho=\frac{1}{2}(|01\rangle\langle 01|+|01\rangle\langle 10|+|10\rangle\langle 01|+|10\rangle\langle 10|)
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$$
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then the reduced density operator of the subsystem $\mathbb{C}^2$ in first qubit is:
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$$
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\begin{aligned}
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\rho_1&=\operatorname{Tr}_2(\rho)\\
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&=\frac{1}{2}(\langle 1|1\rangle|0\rangle\langle 0|+\langle 1|0\rangle|0\rangle\langle 1|+\langle 0|1\rangle|1\rangle\langle 0|+\langle 0|0\rangle|1\rangle\langle 1|)\\
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&=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)\\
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&=\frac{1}{2}I
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\end{aligned}
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$$
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### State purification
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Every mixed state can be derived as the reduction of a pure state on an enlarged Hilbert space.
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#### State purification theorem
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Let $\rho$ be a mixed state in a finite dimensional Hilbert space $\mathscr{H}$, then there exists a unit vector $|w\rangle\in \mathscr{H}\otimes\mathscr{H}$ such that:
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$$
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\rho=\operatorname{Tr}_2(|w\rangle\langle w|)
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$$
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Hint of proof:
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Let $u_1,u_2,\cdots,u_d$ be an orthonormal basis of $\mathscr{H}$, $\sum_{i=1}^d p_i=1$, $p_i\geq 0$, then:
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$$
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\rho=\sum_{i=1}^d p_i|u_i\rangle\langle u_i|
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$$
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Let $w=\sum_{i=1}^d \sqrt{p_i}u_i\otimes u_i$.
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### Observables
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The observables in the quantum theory are self-adjoint operators on the Hilbert space $\mathscr{H}$, denoted by $A\in \mathscr{O}$
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In finite dimensional Hilbert space, $A$ can be written as $\sum_{\lambda\in \operatorname{sp}{(A)}}\lambda P_\lambda$, where $P_\lambda$ is the projection operator onto the eigenspace of $A$ corresponding to the eigenvalue $\lambda$. $P_\lambda=P_\lambda^2=P_\lambda^*$.
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### Effects and Busch's theorem for effect operators
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Below is a section on Topic 4, about Gleason's theorem and definition of states, and Born's rule for describing the states using density operators.
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#### Definition of states (non-commutative (_quantum_) probability theory)
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A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\mu:\mathscr{P}\to[0,1]$ such that:
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1. $\mu(O)=0$, where $O$ is the zero projection.
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2. If $P_1,P_2,\cdots,P_n$ are pairwise disjoint orthogonal projections, then $\mu(P_1\lor P_2\lor\cdots\lor P_n)=\sum_{i=1}^n\mu(P_i)$.
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Where projections are disjoint if $P_iP_j=P_jP_i=O$.
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#### Definition of density operator (non-commutative (_quantum_) probability theory)
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A density operator $\rho$ on the finite-dimensional Hilbert space $\mathscr{H}$ is:
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1. self-adjoint ($A^*=A$, that is $\langle Ax,y\rangle=\langle x,Ay\rangle$ for all $x,y\in\mathscr{H}$)
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2. positive semi-definite (all eigenvalues are non-negative)
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3. $\operatorname{Tr}(\rho)=1$.
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If $(|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle)$ is an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$, for the eigenvalue $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$.
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We can write $\rho$ as
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$$
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\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
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$$
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(under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal)
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Every basis of $\mathscr{H}$ can be decomposed to these forms.
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#### Theorem: Born's rule
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Let $\rho$ be a density operator on $\mathscr{H}$. then
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$$
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\mu(P)\coloneqq\operatorname{Tr}(\rho P)=\sum_{j=1}^n p_j\langle\psi_j|P|\psi_j\rangle
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$$
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Defines a probability measure on the space $\mathscr{P}$.
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[Proof ignored here]
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#### Theorem: Gleason's theorem (very important)
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Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}(\mathscr{H})$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that
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$$
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\mu(P)=\operatorname{Tr}(\rho P)
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$$
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for all $P\in\mathscr{P}(\mathscr{H})$. $\mathscr{P}(\mathscr{H})$ is the space of all orthogonal projections on $\mathscr{H}$.
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[Proof ignored here]
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Extending the experimental procedure in quantum physics, **many of the outcome probabilities are expectation of effects instead of projections.** (POVMs)
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#### Definition of effect
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An effect is a positive (self-adjoint) operator $E$ on $\mathscr{H}$ such that $0\leq E\leq I$.
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The set of effects on $\mathscr{H}$ is denoted by $\mathscr{E}(\mathscr{H})$.
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An operator $E$ is said to be the **extreme point** of the convex set $\mathscr{E}(\mathscr{H})$ if it cannot be written as a convex combination of two other effects.
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That is, If $E$ is an extreme point, then $E=\lambda E_1+(1-\lambda)E_2$ for some $0\leq \lambda\leq 1$ and $E_1,E_2\in \mathscr{E}(\mathscr{H})$ implies $E=E_1=E_2$.
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#### Effect operator lemma
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The set of orthogonal projections on $\mathscr{H}$, $\mathscr{P}(\mathscr{H})$, is the set of extreme points of $\mathscr{E}(\mathscr{H})$.
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#### Projection operator and effect operator
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### Measurements
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### Quantum operations and CPTP maps
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