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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Mutual information of shared state is larger than expectation values
Im trying to prove the following identity for a special case:
Alice and Bob share the Bell state
\begin{align*}
|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).
\end{align*}
Consider the ...
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How much complexity is required to implement $\text{C$_\Pi$NOT}$ gate?
The projector-controlled not gate $\text{C$_\Pi$NOT}$ is defined as
$$\text{C$_\Pi$NOT} \, \colon= \Pi \otimes X + (\mathbb{I}-\Pi)\otimes\mathbb{I}\,, \tag{1}$$ in András Gilyén et. al. (2018)[arXiv:...
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Construct of a quantum circuit for the projection $|0\rangle\langle0| + |1\rangle\langle1| $ and its generalizations
We can construct a projection over $|0\rangle \langle 0|$ using a quantum circuit with two qubits via the Hadamard test circuit
$$U = H_1 X_2 CZ_{1,2} X_2 H_1 X_1\,, \tag{1}$$
and by performing ...
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How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?
I am confused as to how we get $M_{0} = \sqrt{1-p} I$ and the following $M_{1}, M_{2}, M_{3}$.
The above notes say that we should partially trace over the environment in the $|{0}\rangle, |{1}\rangle, ...
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How is implemented the hamiltonian simulation of Hermitian operator multiplied by projection
The article "Quantum Topological Data Analysis with Linear Depth and Exponential Speedup" (Ubaru et al) discusses the implementation of the Hamiltonian $\Delta_\Gamma$, named the ...
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Motivation behind POVM and projective measurement
This is in reference to Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chung [page 90, 92].
Any POVM elements $E_{m}$ are defined as $E_{m} = M_{m}^{\dagger}M_{m}$. A ...
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A question on a subset of projectors onto symmetric subspace
Use $\text{perm}_t$ to denote the set of all permutations among $t$ items. For any particular subset $S\subseteq\{0,1\}^n$ and any $\sigma\in \text{perm}_t$, we define
\begin{align}
P_S(\sigma) = \...
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Can you project on an orthogonal basis for a multipartite system using only local measurements and classical communication?
Say Alice possesses one qubit, and Bob two, and that the joint state is $|\psi_{A, B_1, B_2}\rangle = \alpha|n_1\rangle + \beta |n_2\rangle$, where $|n_1\rangle$ and $|n_2\rangle$ are orthonormal ...
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How to implement projective measurement from multiple measurements?
In the following paper by Harrow et al.: https://arxiv.org/pdf/1607.03236.pdf, they want to implement a measurement operator that is the average of a set of measurement operators. On page 9, right ...
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Entanglement entropy of a projector
In this paper the author goes from the equation (13): $\rho_A = \frac{|G_A|}{2^{N_A}}\mathcal{P}^{(\overrightarrow k_A)}$, where $\mathcal{P}^{(\overrightarrow k_A)}$ is a projector acting on a ...
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Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement
The following question is exercise 2.62 from Nielsen and Chuang's "Quantum Computation and Quantum Information"
Show that any measurement where the measurement operators and the POVM ...
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How do we show that a measurement is a projective measurement
In order to show that a measurement is a projective measurement, is it sufficient to prove that the measurement operators $\{M_{m}\}$ satisfy the properties:
Hermitian: $M_{m}^{T*} = M_{m}$
...
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Symmetric subspaces and Haar averaging over the Unitary group
I am interested in the following Haar average over the unitary group
$D(x) = \int d\mathscr{U} ~(\mathscr{U})^{\otimes 2}(|\tilde{x}_{\mathscr{U}}\rangle\langle \tilde{x}_{\mathscr{U}}|)^{\otimes 2} (\...
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How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?
From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
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Prove that the twirling operation on a channel gives a decomposition $\int dU\, U^\dagger\mathcal E(U\rho U^\dagger)U=\alpha P+\beta Q$
The twirled operation of a quantum channel $\mathcal E$ is defined as
\begin{align}
\mathcal E_T(\rho)
&= \int dU U^\dagger \mathcal E(U \rho U^\dagger)U,
\end{align}
where the integral is over ...
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Optimal estimation of quantum state overlap - Circuit implementation?
I've been reading this paper, but don't understand what their optimal method really is, and how it can be realized as a quantum circuit.
The paper mentions the "Schur transform" which has a ...
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How does a quantum system identify hermitian and unitary matrices?
I am a beginner in quantum computing. I know that multiplying a state $|u\rangle$ with a hermitian matrix $M$ yields spectral decomposition and multiplying $|u\rangle$ with a unitary matrix yield an ...
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How to write post-measurement states when the measurement apparatus measures one of two observables?
If I want to measure an observable $A$ but the measurement apparatus has $(1-p)$ probability of measuring the observable $B$ and probability $p$ that a measurement of $A$ would be done. So how can I ...
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Projective measurement operation in Qiskit
I would like to implement the operation $\pi = \begin{bmatrix}
1 & 0 \\
0 & 0
\end{bmatrix}$
on qiskit but I don't know how to do that.
If I use the reset ...
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How to write down product operators acting on non-adjacent subsystems?
Given the following fusion gate (type-2) which is projecting 2 qubits to an even state
$$F_{ZZ}=(\langle00|+\langle|11|)$$
I would like to find the operator for the bigger space. For example, if I ...
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Qiskit implementation for projecting a hermitian operator and finding its eigenvalues
I'm brand new to quantum computing and have been learning Qiskit for a few weeks now. I am attempting to find the number of 0 eigenvalues of the following operator:
$$ T = PHP$$
where $P$ is a ...
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why Hamiltonian can be expressed by sum of outer product in two level systems?
I can not figure out why Hamiltonian can be like this. Does H should be kinetic energy puls potential energy? Your help would be highly appreciated.
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Distinguishing $n$ pure states in an $n$ dimensional Hilbert space
Suppose we have $n$ pure states in an $n$ dimensional Hilbert space, and we would like to distinguish them using POVM or PVM. We get any one of the pure states with equal probability, and we may set ...
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Are the eigenvalues of projectors always zero and/or one?
Nielsen and Chuang, page 87, defining projective measurements, refers to projectors with "eigenvalue m." However, exercise 2.16 on page 70 seems to imply that the eigenvalue is always one or ...
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Are projections determined by their action on a full-rank density matrix?
Consider (self-adjoint) projections $P$ and $Q$ defined on a finite-dimensional Hilbert space. If $\rho$ is the maximally-mixed state, then we have that $P \rho P = Q \rho Q$ implies $P = Q$, since $\...
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What is the state after a projective measurement?
Given an observable $M = \sum_m \lambda_m P_m$ and assuming that $P_m = |v_m\rangle \langle v_m|$, the state after measurement after getting result $\lambda_m$ is given as $$ \frac{P_m |\psi\rangle}{||...
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Proof that the projector onto the symmetric subspace of the Swap $F$, with $n=2$, equals $\frac{1}{2}(I+F)$
I've seen in some papers and notes that we can write the projector onto the symmetric subspace as
$$
\Pi^{d, 2}_{sym} = \frac{1}{2}(I+F)
$$
but I can't really figure out how specifically this follows ...
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What does the expression $\langle y^{(n)}|\otimes𝟙 \,|\Psi_b\rangle$ mean?
I'm trying to understand the following paper, https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.013167, but I'm new to quantum computing. In it they use this expression:
$$\langle ...
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When discussing error correction, what are the objects in the expression $PE_i^\dagger E_j P=\alpha_{ij} P$?
I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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What is meant by a "projection operator" in the book "Quantum Computation and Quantum Information"?
I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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Does ${\rm tr}(\Pi_z\rho\Pi_z)\le p$ imply $\cal E(\rho)$ and $\cal E(\Pi_{-z}\rho\Pi_{-z})$ are close in trace distance?
Suppose I have a quantum operation $\mathcal{E}$ and a state $\rho$ such that:
$$ \operatorname{tr}(\Pi_z \rho \Pi_z) \le p $$
for some probability $p$ and some projection $\Pi_z$ onto some subspace ...
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Relation between symmetric subspaces and $n$-exchangeable density matrices
Let us consider $n$ elements, each taken from the set $\{1, 2, \ldots, d\}$ and let $S_n$ be the set of all permutations on these $n$ elements.
Define a permutation operator on the set of $n$ qudits ...
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What are the eigenvectors of the superoperator $[H,\cdot]$ with $H$ the Hamiltonian?
Let $\{A_\alpha\}$ be a set of hermitian operators and $\{\Pi(\varepsilon)\}$ a set of projectors on the (finite-dimensional) $\varepsilon$ subspace. Define $$A_\alpha(\Delta\varepsilon)=\sum_{\...
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Applying projectors with mid-circuit measurements
I am trying to apply a non-unitary projector (see image) to my two-qubit quantum circuit using mid-circuit measurements.
$$
\begin{pmatrix}
0 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 &...
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Increasing entropy for projective LCPT mapping
Given a set of projectors $\{P_i\}$ acting on a space $\mathcal H_S$, let $\Phi$ be the LCPT map defined by $$\Phi(\rho)=\sum_i P_i\rho P_i.$$ The goal is to show that $S(\Phi(\rho))\ge S(\rho)$. The ...
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Implement a projection operator as a quantum circuit
Let the state $|\Psi\rangle\equiv a|0\rangle\otimes|\psi_0\rangle + b|1\rangle\otimes|\psi_1\rangle$, where $|\psi_0\rangle$ and $\psi_1\rangle$ belong to a multi-qubit register $R$ and the ...
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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 $P_-=\...
user13341
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How to project a composite system down into a smaller subspace in Python?
If we have a composite system over five qubits ($|\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 ...
glS♦
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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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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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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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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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