# Questions tagged [quantum-computation]

Quantum Computation deals with considering computation as fundamentally physical, as well as replacing the classical binary digit (bit) with the quantum binary digit (qubit). While the classical bit is either 0 or 1, the qubit can be in a superposition of these states. Computation systems that use quantum phenomena, such as superposition and entanglement, can solve certain complex problems very quickly.

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### Suppose $tr(A(P\otimes Q))\geq0$ for all semi-definite positive matrices P and Q, does it implies that A is semi-definite positive? [closed]

Suppose $\text{tr}(A(P\otimes Q))\geq0$ for all semi-definite positive matrices P and Q, does it imply that A is semi-definite positive? If it is not true, please provide some ideas on restricting $A$ ...
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1 vote
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Consider the following circuit : where $|\psi\rangle$ is a qubit in $\mathbb{C}^2$, $|0\rangle= \begin{pmatrix}1 \\ 0 \end{pmatrix}$, $T= \begin{pmatrix}1 & 0\\ 0 & e^{i\pi/4} \end{pmatrix}$ ...
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### Justification of bra ket notation for operators

Let ${\lvert e_1 \rangle, \ldots, \lvert e_m \rangle}$ be a basis for the first arbitary space $V$, and write $A = \sum_{i,j=1}^{m} a_{ij} \lvert e_i \rangle\langle e_j \rvert$ (the $(i,j)$-th element ...
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### What is the result of $\ell^1 \otimes \ell^1$?

I recently stumbled upon the concept of a tensor product while studying quantum computing, and felt that my understanding of it from preliminary readings was incomplete. I challenged myself to ...
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1 vote
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### Are Convex Linear Functions of Density Matrices Actually Linear Functions?

I feel like I often see passing comments about this without proof in quantum info literature, but I wasn't able to show it myself! For instance p17/p18 of this classic Lets say that we have some ...
1 vote
86 views

### Usefulness of a Hermitian Matrix

A Hermitian Matrix is defined as a self-adjoint matrix (so the complex conjugate transpose of M is equal to M). But the usefulness of Hermitian Matrices seems to be that they necessarily have ...
117 views

### Derivative of $A^x$ w.r.t $x$ inside Trace Operation

In my reference Page 520, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that The relative entropy $S(ρ||σ)$ is jointly convex in its ...
• 7,594
1 vote
52 views

### Mystery constant in an approximation of the number of stabilizer states

The number of n-qubit stabilizer states is: $$|\text{StabilizerStates(n)}| = 2^n \prod_{k=0}^{n-1} (2^{n-k}+1)$$ This is a bit cumbersome, so it's nicer to use this approximation that you get by ...
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### Validity of Commutation relations on Superoperators

Lieb's theorem states that, let $X$ be a matrix and $0\le t\le 1$ then the function $f(A,B)\equiv tr(X^\dagger A^tXB^{1-t})$ is jointly concave in positive matrices $A$ and $B$, ie.,  tr\Big[X^\...
• 7,594
1 vote
249 views

### Show that if $\mu,\eta\in I\implies \dfrac{\mu+\eta}{2}\in I$ and $\{0,1,1/2\}\subset I$ implies $[0,1]\subset I$

Let $R_1,R_2,S_1,S_2,T_1,T_2$ be positive operators such that $[R_1,R_2]=[S_1,S_2]=[T_1,T_2]=0$, and $R_1\ge S_1+T_1$ $,R_2\ge S_2+T_2$ then for all $0\le t\le 1$, \begin{align} R_1^tR_2^{1-t}\ge S_1^...
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### How to shift the eigenvalues of a quantum Hermitian operator G to ±r?

Consider a gate $\mathcal{G}(\mu)=e^{-i \mu G}$ generated by a Hermitian operator G. If G has just two distinct eigenvalues(which can be repeated) we can, without loss of generality, shift the ...
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### Measuring in Different Bases - Deterministic versus Random outcomes [closed]

The Qiskit Textbook on https://qiskit.org/textbook/ch-states/single-qubit-gates.html in section 4: Digression: Measuring in Different Bases, says – Z-basis is not intrinsically special, and that ...
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1 vote