Bertrand Einstein IV
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Why is the transpose of a density matrix positive and trace preserving?
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8 votes

Transposing a matrix is trace preserving since for $\rho = \sum_{a,b} \rho_{a,b} | a \rangle \langle b |$: $$\text{Tr}(\rho)= \sum_c \langle c| \big( \sum_{a,b} \rho_{a,b} | a \rangle \langle b | \...

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How is a Toffoli gate built without using T gates?
6 votes

The $T$ gate as well as all possible single qubit rotations are non-entangling operations. That means if we have a circuit composed of single bit rotations, any non-entangled $n$-bit input, it will ...

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Why are all the eigenvalues of a "Hermitian block-encoding" equal to $\pm1$?
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5 votes

$U_A$ is defined to be a unitary matrix in the paper and your question. Consider the eigenvalue $\lambda$ of the general unitary matrix $U$ given by $U|\lambda\rangle=\lambda|\lambda\rangle$. $$U|\...

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Showing that $e^{i \sigma_z \otimes \sigma_z t} = \text{CNOT}(I \otimes e^{i \sigma_zt})\text{CNOT}$
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5 votes

If your question is only regarding why $| 0 \rangle \langle0 | \otimes \sigma_z - | 1 \rangle \langle 1 | \otimes \sigma_z$ ; you can simply factor it given that trivially: $\sigma_z = | 0 \rangle \...

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Aren't qubits just ternary?
5 votes

Simple answer: no. Qubits are the same as regular bits in almost every way; except two fundamental differences, superposition and entanglement (I will only address superposition since it is the focus ...

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What's the 'physical consistency' in the partial trace scenario?
5 votes

The point of physical consistency is about how we can define the operator $M$ as acting on system $\rho^A$, or we can define the operator $M \otimes \mathbb{1}_B$ as acting on the system $\rho^{AB}$, ...

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Minimum number of 2 qubit gates to build any unitary
4 votes

Let me try to reformulate your question: Given a Universal Set of Quantum Gates $\mathcal{G}$; and some $n$-bit Unitary $U$. Can we find some $q$ such that $q$ is the minimum number of gates selected ...

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What do the off-diagonal elements of a density matrix physically represent?
4 votes

A good way to think about density matrices is to think about them as Bloch Vectors (I assume you are familiar with the Bloch Sphere). This won't tackle your question head on; but I hope will give some ...

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How to combine/calculate for interference using density matrices?
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3 votes

By definition a density matrix is given by: $$\rho= \sum p_i |\psi_i \rangle \langle \psi_i|$$ Where there is a probability $p_i$ of finding the state to be $|\psi_i\rangle$. Now if we are "...

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Gate SWAP vs Physical SWAP in Trapped Ions for chain reordering
3 votes

Prelim: I am no expert on implementation techniques or the frontier of what gate technology is being used in current renditions of Trapped Ion QC. The Molmer-Sorensen gate is generally what is used in ...

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How to theoretically compare the complexity of quantum and classical algorithms?
2 votes

To analyze the complexity of Quantum Algorithms we use what is known as Query Complexity. The Query Complexity of an algorithm is the number of times it must Query the solution associated with the ...

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Confusion regarding the tensor product usage in book
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2 votes

In general given two matrices $A \in \mathbb{R}^{m_A \times n_A}$ and $B \in \mathbb{R}^{m_B \times n_B}$ for the matrix $C=A \otimes B$, we have its dimensions given by $C \in \mathbb{R}^{m_Am_B \...

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How do I determine if a given pure two-qubit state is separable?
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2 votes

Your approach is correct, but you are taking the partial trace wrong: $$\rho_A=\text{Tr}_B\big( \rho \big) = \sum_{i} \langle i_B | \rho |i_B \rangle = \langle 0_B | \rho |0_B \rangle + \langle 1_B | \...

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Is there some nice physical intuition to get for quantum teleportation
1 votes

I will try to give a high level interpretation of what is going on during Quantum Teleportation. We first begin with Alice and Bob each holding 1 bit from a Bell State, otherwise known as a maximally ...

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Is there any possibility to draw graphs for a maximally entangled state?
1 votes

In the specific $|\psi \rangle$ there is symmetry on the order of bits, so we must only show that one bit is entangled and it then follows that they all are. Notice we may write: $$|\psi \rangle = \...

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Find expectation value of the observable $X_1\otimes Z_2$ for a maximally entangled two-qubit system
1 votes

An intuitive way to think about it is that $E[M]=E[X_1 \otimes Z_2]=E[X_1 \otimes \mathbb{1}]E[\mathbb{1} \otimes Z_2]$ If we only think about $E[\mathbb{1} \otimes Z_2]$, it is just the expectation ...

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Why isn't FANIN required to be able to simulate all other elements in a classical circuit
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0 votes

A NAND gate is a universal classical gate. All boolean logic can be implemented using only NAND gates. This is clear because SOP/POS implementation needs only AND/OR/NOT and NAND can make all three of ...

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