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Definition A two-qubit state $|\psi\rangle \in \mathbb{C}^4$ is an entangled state if and only if there not exist two one-qubit states $|a\rangle = \alpha |0\rangle + \beta |1\rangle \in \mathbb{C}^2$ and $|b\rangle = \gamma |0\rangle + \lambda |1\rangle \in \mathbb{C}^2$ such that $|a\rangle \otimes |b\rangle = |\psi\rangle$, where $\otimes$ ...
A two qudit pure state is separable if and only if it can be written in the form $$|\Psi\rangle=|\psi\rangle|\phi\rangle$$ for arbitrary single qudit states $|\psi\rangle$ and $|\phi\rangle$. Otherwise, it is entangled. To determine if the pure state is entangled, one could try a brute force method of attempting to find satisfying states $|\psi\rangle$ and $... 12 The whole point is that CNOT cannot be written in the form$A\otimes B$. This is absolutely essential because if we only ever had operators of the form$A\otimes B$, states of the form$|\psi\rangle\otimes|\phi\rangle$would always remain separable. There's be no entanglement, and all quantum circuits would be easy to simulate on a classical computer. As ... 11 For many questions that appear on this site, and about quantum information and computation in general, it is possible to ask a completely classical version of the question, and often the (sometimes obvious) answer that one finds in the more familiar classical setting translates directly to the quantum setting. In this case, a reasonable classical version of ... 8 Marsl is correct, and his "hint" is really more a sketch of a solution than a hint. Rather than viewing the question or its solution as just formal algebra, you can also approach his same solution more conceptually. The conceptual reasoning is really identical to the algebra, just phrased differently. You can rely on the following two facts: 1) Trace ... 7 Here the important fact is that the maximally mixed state is in fact an identity matrix. Let me rewrite the expression on the left in index notation (the summation sign is omitted according to the Einstein convention): $$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = [\rho^{AB}]_{ijkl} [\sigma^A]_{ji} [I/d]_{lk}$$ But$[I/d]_{lk} = \frac1d \delta_{lk}$, ... 7 Does$|0\rangle\langle0|$represent a tensor product or is it just matrix multiplication? You can think of$|0\rangle\langle0|$as tensor product of$|0\rangle$and$\langle0|$, or equivalently as the matrix multiplication (more precisely, Kronecker product) of the vectors representing them. Also, I thought that we must always be able to write a ... 7 Imagine you have a vector that can be written in the form $$|\psi\rangle=\sum_{i=0}^{d_A-1}\sum_{j=0}^{d_B-1}c_{ij}|i\rangle|j\rangle.$$ The coefficients can be arranged as a$d_A\times d_B$matrix$C$, with the elements$c_{ij}$(in your special case, you're talking about setting$d_A=d_B=\sqrt{m}$). Now, if you calculate$\rho_A=CC^\dagger, this is ... 7 Hint: To make your induction work, write \eqalign{p^{\otimes n} - q^{\otimes n} & = & \left(p^{\otimes(n-1)}\otimes p \right)-\left(q^{\otimes (n-1)} \otimes q\right)\\ & = & \left(p^{\otimes(n-1)}-q^{\otimes (n-1)} \right)\otimes p+\left(q^{\otimes (n-1)} \right) \otimes (p-q)} Then, use triangle inequality and finally the fact that ... 7 No. The minimal size of the environment is just the rank of the Choi matrix of\mathcal E$, call it$J(\mathcal E)$. Since$J(\mathcal E^{\otimes n}) = \big(J(\mathcal E)\big)^{\otimes n}$and$\text{rank}(A \otimes B) = \text{rank}(A)\text{rank}(B)$, the minimal size of the environment is just$\text{rank}\big(J(\mathcal E)\big)^n$. 7 As is the case with ordinary multiplication, tensor product distributes over addition, so we can pull$|0\rangleon the first qubit out in front \begin{align} |\Psi⟩ &= \frac{1}{\sqrt{2}}|\color{red}{0}0\rangle+\frac{i}{\sqrt{2}}|\color{red}{0}1\rangle \\ &= \frac{1}{\sqrt{2}}\color{red}{|0\rangle}\otimes|0\rangle+\frac{i}{\sqrt{2}}\color{red}{|0\... 7 No, that doesn't work. It's fine to use an arbitrary pure state because the unitary U can always be used to take it to any pure state you want. This argument doesn't work for a mixed state, as unitaries cannot take mixed states to pure states. As a concrete example, consider the CPTP map \Lambda(\rho) = |0\rangle\langle 0| \operatorname{tr}(\rho), $$... 6 The tensor product is not commutative, i.e. in the computational basis,$$X\otimes Z=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{array} \right)$$while$$Z\otimes X=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\... 6 I think that you misunderstood the concept of tensor product here. There is no need of a Clifford gate in order to have a multi-qubit system. The fact that a multi-qubit system is described by the tensor produt of their state vectors comes from the fourth postulate of quantum mechanics, refer to the94^{th}$page of Nielsen and Chuang to see the formulation ... 6 The equation at the top of the question is not correct: there is a missing factor of$1/d$on the right-hand side. Let's eliminate this factor from the left-hand side to make it simpler, so that the equation we want is this: $$\text{Tr}\bigl(\rho^{AB} \bigl(\sigma^A \otimes I\bigr)\bigr) = \text{Tr}\bigl(\rho^A \sigma^A\bigr).$$ To see why this is true, ... 5 Perhaps it helps to take a step back and start with something simpler: why do we tabulate probability amplitudes for state vectors and unitaries? For a single quantum system with$d$distinct states, labelled 0 to$d-1$, we associate a complex number$a_i$with the probability amplitude for being in state$i$. For a unitary, we associate a probability ... 5 When you say$\neq$I presume you are talking about the implied basis in usual ordering like (00, 01, 02, 10 etc). Otherwise you would have the isomorphism of Hilbert spaces vs an equality statement. That is, AB implies a certain ordered basis and BA a different one. The experiment has it's observables on the combined system in a basis independent way. If ... 5 It is worth emphasising that the stuff that you write inside a ket is completely arbitrary. It's just a label you're attaching to something, so it should have a proper definition somewhere. Now, usually, we're talking about quantum spins with a certain size of Hilbert space, say$d$. Probably here you're talking$d\geq 3$, and perhaps specifically$d=3$. ... 5 It is certainly sufficient to always calculate the full$2^n\times 2^n$unitary matrix, and then apply it to the$2^n$-entry state vector. If that's what you choose to do, that's all you have to do as all the entanglement information is contained in that vector. A quick and easy way to see if a particular qubit is entangled is to take the partial trace of ... 5 Yes,$|\mathbf{x},0\rangle$is a shorthand for$|\mathbf x\rangle\otimes |0\rangle$. Note that$|\mathbf x\rangle$itself, with$\mathbf x = x_1x_2\dots x_N$a bit string, is just a shorthand for $$|\mathbf x\rangle \equiv |x_1\rangle \otimes |x_2\rangle \otimes\cdots |x_N\rangle\ .$$ 5 Any two qubit Controlled-U gate is not decomposable as$A\otimes B$in general. The whole point is that a conditional operator has a non-local nature and it gives rise to non-trivial correlations when operated on a state. The Physics behind is that; imagine a Controlled operation which can be factored as$A\otimes B$, and the sub-systems (or the two qubits)... 5 I think there are two ways that you could denote the same thing. The first is what is done here: $$\prod_{j =1}^n e^{-i \beta \sigma_j^x}$$ The second is $$\bigotimes_{j-1}^ne^{-i \beta \sigma^x},$$ which I imagine is what you're thinking of. In the first expression, note the subscript on the Pauli matrix. This means that it's an operator over all$n$... 5 I know the question is already answered, but there was some question on my comment and I wanted to elaborate on that. First, let us consider one system only. The$\mathbb{R}$-span of all states$\rho$is the space of Hermitian operators. Indeed, by the spectral decomposition, already the set of pure states is enough. This also implies that the$\mathbb{C}$-... 5 In general, the knowledge of the marginals$\rho_A$and$\rho_B$and the operators$A$and$B$is insufficient to compute$\mathrm{tr}_A((A\otimes B)\rho_{AB})$. Indeed, we can find two different density matrices$\rho_{AB}$and$\sigma_{AB}$with the same marginals for which $$\mathrm{tr}_A((A\otimes B)\rho_{AB}) \ne \mathrm{tr}_A((A\otimes B)\sigma_{AB}).$$... 4 I'll add a bit to the other answer. State of a two qubit system is written as$|\psi_1\rangle\otimes|\psi_2\rangle$where$|\psi_1\rangle$is the state vector of the first qubit and$|\psi_2\rangle$is the state vector of the second qubit (it is upto you to decide which one will be called "first" and which one will be called "second"). The order is ... 4 The tensor product is not a gate, but rather a way for us as humans to model the behavior of a quantum system. Whenever we're using multiple qbits, we can look at them in two ways: in their product state (a complex vector of size$2^n$for$n$qbits) or their individual state ($|\psi_0\rangle \otimes \ldots \otimes |\psi_{n-1}\rangle$). We can usually switch ... 4 For step (116), the equivalence between both of them is proved by $$(\langle\psi_1|\otimes\langle0|)C^\dagger C(|\psi_2\rangle\otimes|0\rangle) = (\langle\psi_1|\otimes\langle0|)(|\psi_2\rangle\otimes|0\rangle) = \langle\psi_1|\psi_2\rangle\otimes\langle0|0\rangle=\langle\psi_1|\psi_2\rangle\langle0|0\rangle,$$ where in the ... 4 This assertion is an axiom of quantum mechanics. It appears, for example, as Postulate 4 on page 94 of Nielsen and Chuang. It is considered to be one of the Dirac-von Neuman axioms of quantum mechanics. However, when the quantum system's Hilbert space is defined as the space of$l^2$functions on some set, then when the set is a Cartesian product, the ... 4 I presume you mean a$2^n\times 2^n$quantum operator,$U$? Let's assume we can write $$U=\sum_{x,y\in\{0,1\}^n}U_{xy}|x\rangle\langle y|.$$ All we have to do is show that we can construct any$|x\rangle\langle y|$using Pauli operators. But this is just the same as $$\bigotimes_{i=1}^n|x_i\rangle\langle y_i|,$$ so provided I can create any$|x_i\rangle\...