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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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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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
$$
in ...
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Implications of commuting within the code space
The question:
I have a Hilbert space $\mathcal{H}=\mathcal{H}_A\otimes \mathcal{H}_B$, and a codespace $\mathcal{H}_{code}\subset \mathcal{H}$, so that $\mathcal{H}=\mathcal{H}_{code}\oplus\mathcal{...
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What does "measuring a state" mean?
I have been reading about secret sharing schemes, and they regularly come up with a line that says 'that a person upon receiving a state measures the state in one of the basis say the computational ...
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What is the relation between POVMs and projective measurements?
I'm a little confused about the terminology of measurement.
So say that we have the single qubit state $|\phi \rangle=c_0|0\rangle+c_1|1\rangle$.
If we perform the projective measurement $P_0=|0\...
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Projecting $\lvert ++ \rangle$ on Bell Basis
I understand that, projecting $\lvert 00\rangle$ on the Bell states would produce $\lvert\Phi^+\rangle$. Because,
$$
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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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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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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