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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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1 answer
191 views

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 ...
0 votes
0 answers
12 views

relationship between helstrom operators acting on different pairs of quantum states

Let $\rho_1, \rho_2, \rho_3, \rho_4$ be arbitrary single-qubit density matrices. Let $A$ be an Hermitian operator and its spectral decomposition as $A = \sum_i \lambda_i \lvert i \rangle \langle i \...
0 votes
2 answers
27 views

Does $\frac{I + K}{2}\otimes\frac{I+L}{2} = \frac{I+K\otimes L}{2}$ hold for operators $K,L$ acting on different subsystems?

Let's index $i= 1, 2$ be the index representing different systems. A $Z_i$ projective measurement has projectors $P_i = \{ \frac{I+Z_i}{2}, \frac{I-Z_i}{2} \}$. One can verify that the measurement $...
2 votes
1 answer
209 views

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} (\...
1 vote
1 answer
41 views

Projective measurement notation

I don't understand how projective measurements work; I think my confusion comes from the notation. How would this be written in matrix notation? Is it just [w row vector] [Ma matrix] [w column vector]^...
4 votes
1 answer
63 views

confusion on the LCU method regarding the normalization

Let $A = \sum_{k} a_k U_k$ where $a_k$ are real, positive coefficients $U_k$ are unitary matrices. I have realized that $\sigma = A \rho A$ can be implemented on a quantum computer by using the LCU ...
2 votes
1 answer
56 views

Explicit calculation for multiplying two projection operators

Can someone explain the explicit calculations for: $$(I \otimes ( |00\rangle \langle 00| + |11 \rangle \langle 11| ) ) \times ( (|00 \rangle \langle 00| + |11 \rangle \langle 11|) \otimes I) = |000 \...
1 vote
1 answer
63 views

How to find projection operators for spectral decomposition

I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information". The idea ...
2 votes
0 answers
101 views

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 ...
3 votes
1 answer
106 views

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:...
2 votes
1 answer
44 views

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, ...
1 vote
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56 views

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 ...
1 vote
0 answers
32 views

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 ...
1 vote
0 answers
50 views

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 ...
3 votes
0 answers
155 views

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) = \...
1 vote
0 answers
83 views

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 ...
4 votes
1 answer
137 views

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 ...
1 vote
1 answer
70 views

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 ...
0 votes
1 answer
95 views

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: $M_{m}^{T*} = M_{m}$ ...
0 votes
1 answer
142 views

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 ...
1 vote
2 answers
186 views

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 ...
1 vote
1 answer
1k 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 ...
1 vote
0 answers
101 views

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 ...
1 vote
1 answer
128 views

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 ...
8 votes
0 answers
194 views

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 ...
3 votes
2 answers
304 views

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 ...
3 votes
2 answers
436 views

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 ...
3 votes
1 answer
60 views

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 ...
1 vote
0 answers
116 views

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.
2 votes
0 answers
105 views

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 ...
4 votes
1 answer
648 views

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 ...
2 votes
1 answer
684 views

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 ...
4 votes
1 answer
105 views

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 $\...
3 votes
2 answers
517 views

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}{||...
1 vote
1 answer
70 views

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 ...
3 votes
1 answer
131 views

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 ...
1 vote
1 answer
281 views

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 ...
2 votes
1 answer
57 views

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 ...
4 votes
1 answer
222 views

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 ...
1 vote
1 answer
141 views

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_{\...
2 votes
0 answers
179 views

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 &...
5 votes
1 answer
1k views

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 ...
1 vote
1 answer
40 views

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 ...
7 votes
3 answers
2k 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 = \...
4 votes
2 answers
959 views

What is 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\...
4 votes
0 answers
133 views

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 \...
4 votes
0 answers
54 views

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. ...
4 votes
1 answer
34 views

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 ...
1 vote
2 answers
224 views

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!
3 votes
2 answers
179 views

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,...,...