Questions tagged [projection-operator]

A projection operator is one which when acts upon a quantum state (which is an element of a Hilbert space), "projects" it onto a subspace or onto another element of the same Hilbert space.

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Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

Let us consider two quantum channels $\Phi:M_d\rightarrow M_{d_1}$ and $\Phi_c:M_d\rightarrow M_{d_2}$ that are complementary to each other, i.e., there exists an isometry $V:\mathbb{C}^d\rightarrow \...
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If $\rho \approx_{\varepsilon}\sigma$, how to find $\Pi\rho\Pi$ to ensure that $\text{supp}(\Pi\rho\Pi)\subset\text{supp}(\sigma)$?

Let $\rho$ and $\sigma$ be positive semidefinite operators with trace less than or equal to 1. Let $\rho\approx_{\varepsilon}\sigma$ i.e. they are close in some distance, such as the trace distance. ...
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Does $\frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}=\rho_{AB}$ hold for a state $\rho_{AB}$ and projector $\Pi_A$?

For some projector $\Pi_A$ and state $\rho_{AB}$, let $$\sigma_{AB} = \frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}$$ Is it the case that $\sigma_B = \rho_B$? It seems ...
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Find the unitary implementing the transformation $|0\rangle\to\frac1{\sqrt2}(|0\rangle+|1\rangle),|1\rangle\to\frac1{\sqrt2}(|0\rangle-|1\rangle)$ [closed]

I have found a question for finding the Unitary operator for the following transformation: I found the solution as well. But I didn't understand how they got the solution!
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Is a projective measurements over a superposition of eigenstates possible?

All observables admit a spectral decomposition in terms of projectors $P_m$ into the eigenspace corresponding to the eigenvalue $m$. So given for example a collection of kets $|0\rangle, |1\rangle,...,...
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How do I prove that $P_\pm=\frac12(1\pm U)$ if $U^2=I$?

Suppose I have an $n$-qubit Hermitian operator $U$ such that $U^2=I$. The projection operators with eigenvalue $+1$ and $−1$ are $P_+$ and $P_-$. How can I prove that $P_+=\frac{1}{2}(1+U)$ and $P_-=\...
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How to project a composite system down into a smaller subspace in Python?

If we have a composite system over five qubits ($|\psi\rangle = |a\rangle|b\rangle|c\rangle|d\rangle|e\rangle$), and I want to project into a specific subspace of the first three systems, I can build ...
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Can we characterise how correlated the expectation values associated with a pair of observables are?

Consider a state $\rho$ and two observables $P$ and $Q$. Is there a good way to characterise how correlated the associated expectation values are? Be it in terms of mutual information or something ...
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What can I conclude about $\langle \phi|\pi_1\pi_2|\phi\rangle$ if $\langle \phi|\pi_i|\phi\rangle\ge e$?

If I have two projectors $\pi_1, \pi_2$ such that for some $|{\phi}\rangle$: $\langle {\phi}| \pi_1 |{\phi}\rangle \geq e$ and $\langle {\phi}| \pi_2 | {\phi}\rangle \geq e$ What can I conclude about ...
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Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
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186 views

How to code a projector operator in qiskit?

I'm new to qiskit and I want to know how do I define a projector operator in qiskit? Specifically, I have prepared a 3 qubit system, and after applying a whole lot of gates and measuring it in a state ...
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217 views

What's the observable when measuring multiple qubits in the computational basis?

In Nielsen and Chuang, Quantum computing and quantum information, the following definition is given to a projective measurement : Projective measurements are described by an observable $M$ : $$M = \...
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Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
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Increasing the von Neumann entropy despite the measurement?

Background Assume we have a density matrix $\rho$ of a sub-ensemble. However, we have an imperfect measuring instrument. While it does perform a measurement, we do not know exactly when it performs ...
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Showing measurement of a Hermitian Unitary operator gives final states as eigenvectors

This is related to exercise 4.34, The operation described can be written as $(H \otimes I)C^1(U)(H \otimes I)(|0\rangle \otimes |\psi\rangle)$ I can get to the point where the state of the system is ...
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What's the interpretation of the eigenvalues of qubit's projective operators?

Usually, while conducting a measurement on a qubit we are using two projectors, namely $P_0 = |0\rangle \langle 0|$ and $P_1 = |1\rangle \langle 1 |$. For the case of $P_0$ we have two possible ...
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Help in understanding the usage of eigenvalues in the definition of the projective measurement

Recently I was reading about the projective measurement in "Quantum Computation and Quantum Information" by Nielsen & Chuang, where they describe the projective measurement as follows: ...
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Prove that $P_{M_1}P_{M_2}= P_{M_2}P_{M_1}$ implies $\text{Pr}(\text{span}[M_1, M_2]) = \text{Pr}(M_1)+\text{Pr}(M_2)−\text{Pr}(M_1\cap M_2)$

Prove that if $\text{Proj}_{M_1}\text{Proj}_{M_2}= \text{Proj}_{M_2}\text{Proj}_{M_1}$ then $\text{Pr}(\text{span}[M_1, M_2]) = \text{Pr}(M_1) + \text{Pr}(M_2) − \text{Pr}(M_1 \cap M_2)$. In ...
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Quantum error correction using bit-flip code for the amplitude damping channel

I do not understand the error correction process that uses quantum codes for amplitude damping channel. I will take three bit-flip code for example. The logical state of a three bit-flip code is $|0\...
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Interpretation of specific Hamiltonian operator

In the paper https://arxiv.org/abs/1909.05820 the authors introduce several Hamiltonians. For example they define $$ H_G = A^\dagger \left( \mathbb{I} - \vert b \rangle \langle b \vert \right) A $$ in ...
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Implications of commuting within the code space

The question: I have a Hilbert space $\mathcal{H}=\mathcal{H}_A\otimes \mathcal{H}_B$, and a codespace $\mathcal{H}_{code}\subset \mathcal{H}$, so that $\mathcal{H}=\mathcal{H}_{code}\oplus\mathcal{...
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What does “measuring a state” mean?

I have been reading about secret sharing schemes, and they regularly come up with a line that says 'that a person upon receiving a state measures the state in one of the basis say the computational ...
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Confusion about the relation between POVMs and projective measurements

I'm a little confused about the terminology of measurement. So say that we have the single qubit state $|\phi \rangle=c_0|0\rangle+c_1|1\rangle$. If we perform the projective measurement $P_0=|0\...
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Projecting $\lvert ++ \rangle$ on Bell Basis

I understand that, projecting $\lvert 00\rangle$ on the Bell states would produce $\lvert\Phi^+\rangle$. Because, $$ CNOT(H\lvert0\rangle \otimes \lvert0\rangle) = \frac{1}{\sqrt{2}}(\lvert00\rangle +...
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Quantum channel representation of projective measurement

Let $P$ be a projector and $Q = I-P$ be its complement. How to find probability $p$ and unitaries $U_1, U_2$ such that for any $\rho$, $P\rho P + Q\rho Q = p U_1\rho U_1^\dagger + (1-p)U_2\rho U_2^\...
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105 views

Weeding out qubit states with leftmost qubit as 1

Need help! I was working on a project when I required to use a projection operator. For an example case, I have the Bell state, $$|\psi\rangle = \frac1{\sqrt2}\left(\color{blue}{|0}0\rangle+|11\rangle\...
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Error syndromes and recovery procedure in bit flip code

This question relates to exercise 10.4 in Nielsen and Chuang. For syndrome diagnosis, the textbook provides an example where one has four projectors, by which, you can identify where a one qubit ...
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91 views

What does it mean to perform a measurement in correspondence with different projections?

In error correction, like the bit flip, you perform a measurement which corresponds to different projections so that the outcomes can teach you about the error. What does it mean? How do you actually ...
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Projection operators and positive operators

I recently came across the concepts of operators. However with current my knowledge I am unable to solve the following problem.Given an operator $$\vec{A}=\frac{1}{2}(I+\vec{n}.\vec{\sigma})$$ where $\...
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Confusion regarding projection operator

Suppose we have a qutrit with the state vector $|\psi\rangle = a_0|0\rangle + a_1|1\rangle + a_2|2\rangle$, and we want to project its state onto the subspace having the basis $\{|0\rangle,|2\rangle\}$...