Bertrand Einstein IV
• Member for 11 months
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• Cambridge, UK

Transposing a matrix is trace preserving since for $\rho = \sum_{a,b} \rho_{a,b} | a \rangle \langle b |$: $$\text{Tr}(\rho)= \sum_c \langle c| \big( \sum_{a,b} \rho_{a,b} | a \rangle \langle b | \... View answer 6 votes The T gate as well as all possible single qubit rotations are non-entangling operations. That means if we have a circuit composed of single bit rotations, any non-entangled n-bit input, it will ... View answer Accepted answer 5 votes U_A is defined to be a unitary matrix in the paper and your question. Consider the eigenvalue \lambda of the general unitary matrix U given by U|\lambda\rangle=\lambda|\lambda\rangle.$$U|\...

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: $\sigma_z = | 0 \rangle \... View answer 5 votes Simple answer: no. Qubits are the same as regular bits in almost every way; except two fundamental differences, superposition and entanglement (I will only address superposition since it is the focus ... View answer 5 votes The point of physical consistency is about how we can define the operator$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}$, ... View answer 4 votes Let me try to reformulate your question: Given a Universal Set of Quantum Gates$\mathcal{G}$; and some$n$-bit Unitary$U$. Can we find some$q$such that$q$is the minimum number of gates selected ... View answer 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 ... View answer Accepted answer 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 "... View answer 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 ... View answer 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 ... View answer Accepted answer 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 \...

Your approach is correct, but you are taking the partial trace wrong: $$\rho_A=\text{Tr}_B\big( \rho \big) = \sum_{i} \langle i_B | \rho |i_B \rangle = \langle 0_B | \rho |0_B \rangle + \langle 1_B | \... View answer 1 votes I will try to give a high level interpretation of what is going on during Quantum Teleportation. We first begin with Alice and Bob each holding 1 bit from a Bell State, otherwise known as a maximally ... View answer 1 votes In the specific |\psi \rangle there is symmetry on the order of bits, so we must only show that one bit is entangled and it then follows that they all are. Notice we may write:$$|\psi \rangle = \...

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 ...