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Newest 'projection-operator' Questions - Quantum Computing Stack Exchange
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Learn more about TeamsQuestions 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.
79 questions
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3
votes
2
answers
104
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Logical coset probabilities
Suppose we have an $n$-qubit stabilizer code with the logical state $\rho_L$, which goes through a quantum channel $\mathcal{E}(\rho_L)$. Here, I assume that the quantum channel is arbitrary, for ...
0
votes
0
answers
45
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Equivalence of projective measurement with general measurement
Nielsen and Chuang in page 91 say a very important thing: "projective measurements augmented by unitary operations turn out to be completely equivalent to general measurements, as shown in ...
6
votes
1
answer
268
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Trace of projected Gaussian operator product: puzzling difference between projection onto filled or empty fermionic mode
This question was motivated by a Mathoverflow posting.
I am comparing two traces of fermionic creation/annihilation operators $a_n,a_n^\dagger$ ($n=1,2,\ldots N$):
$$T_{\rm empty}= \operatorname{tr}\...
2
votes
1
answer
55
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If $g \in S$, does this necessitate that $g^{\dagger} \in S$?
In Lecture 5 of Aram Harrow's lecture series "Quantum Information Science II", he sets about proving that $\Pi_{S} = \frac{1}{|S|} \sum_{g \in S} g$ can be written as $\Pi_{S}= \prod_{i=1}^{...
3
votes
1
answer
57
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Showing that $V_{S} \subseteq \Pi_{S}$?
From Aaaron Harrow's Lecture series: Quantum Information Science II, Lecture on stabilizers, I am a little confused with one of the details of the following proof.
Claim 1. $$\Pi_{S} = \frac{1}{|S|} \...
2
votes
1
answer
283
views
Why are only projective measurements considered in nonlocal games to get optimal strategy?
In the CHSH game where Alice and Bob share one EPR the optimal strategy is to measure the following observables $A_0=Z,A_1=X,B_0=1/\sqrt{2}(X+Z),B_1=1/\sqrt{2}(X-Z)$ depending on their question. We ...
0
votes
1
answer
114
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Deriving the number operator in a Cooper Pair Box (CPB)
The details of my question follows closely from this source (pg 10 - 12): https://clerkgroup.uchicago.edu/PDFfiles/0210247.pdf
The standard CPB hamiltonian in charge basis is written as
$$
H = 4E_{c}\...
0
votes
0
answers
56
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Is there any well defined notion of "the state of a qubit X" if $X$ is a member of a larger qubit system
I'm trying to understand the effect of "measurement" as a gate. But my question is more elementary than that. Consider the following simple quantum circuit:
$(X,Y)$ begins in the state $1 |...
1
vote
1
answer
71
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relationship between helstrom operators acting on different pairs of quantum states
Let $\rho_1, \rho_2, \rho_3, \rho_4$ be arbitrary single-qubit density matrices.
Let $A$ be an Hermitian operator and its spectral decomposition as $A = \sum_i \lambda_i \lvert i \rangle \langle i \...
0
votes
2
answers
63
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Does $\frac{I + K}{2}\otimes\frac{I+L}{2} = \frac{I+K\otimes L}{2}$ hold for operators $K,L$ acting on different subsystems?
Let's index $i= 1, 2$ be the index representing different systems. A $Z_i$ projective measurement has projectors $P_i = \{ \frac{I+Z_i}{2}, \frac{I-Z_i}{2} \}$. One can verify that the measurement $...
1
vote
1
answer
67
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Projective measurement notation
I don't understand how projective measurements work; I think my confusion comes from the notation.
How would this be written in matrix notation?
Is it just [w row vector] [Ma matrix] [w column vector]^...
4
votes
1
answer
248
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confusion on the LCU method regarding the normalization
Let $A = \sum_{k} a_k U_k$ where $a_k$ are real, positive coefficients $U_k$ are unitary matrices. I have realized that $\sigma = A \rho A$ can be implemented on a quantum computer by using the LCU ...
2
votes
1
answer
99
views
Explicit calculation for multiplying two projection operators
Can someone explain the explicit calculations for:
$$(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
160
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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
125
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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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