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All of your math is correct. Both of the states that you calculated are entangled states. An entangled state is one in which there exist measurable properties of subsystems that are correlated. In (1), measurements of the two qubits in the computational basis are correlated; it just happens to be a negative correlation. In (2), measurements in the ...

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Welcome to Quantum Computing StackExchange. To see why it would require $2^n$ complex numbers instead of $2n$ to represent a general entangled $n$-qubit state, let's assume we have a 3-qubit system, $(\alpha_1, \beta_1), (\alpha_2, \beta_2), (\alpha_3, \beta_3)$ Where $(\alpha_i, \beta_i)$ denotes the representation of the $i^{th}$ qubit in a classical ...

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I might say: $$\vert\phi\rangle=\frac{1}{\sqrt 2}(\vert 0_A0_B0_C\rangle +\vert 1_A0_B1_C\rangle)= \frac{1}{\sqrt 2}(\vert 0_A0_C\rangle +\vert 1_A1_C\rangle)\otimes \vert 0_B\rangle,$$ because I think that might be intuitively understandable without much parsing. It also might help to label each qubit if I have to give the qubits to Alice, Bob, and Charlie ...

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The definition of separability does not require any restrictions on the purity of the state. However, if $\rho$ is pure then it is separable on some bipartition $AB$ iff it can be written as $$\rho = \rho_A \otimes \rho_B$$ where both $\rho_A$ and $\rho_B$ are pure states. Here's a sketch of why that's true. Note that if $\rho$ is separable we have $$\rho ... 4 Like you said, one reason why it's hard to do this in practice is simply due to the number of gates. Each gate comes with limited fidelity (never exactly 100%), so needing to do many of them will compound the total gate error. Moreover, for the GHZ state, notice that the whole state will dephase if even a single qubit dephases. Generally, the larger the ... 4 Recall that the W state may be defined as:$$\vert W\rangle=\frac{1}{\sqrt 3}(\vert 001\rangle+\vert 010\rangle+\vert 100\rangle.$$Given three qubits, initially to prepare such a state local operations (wherein at least two of the three qubits are at the same location) will need to be performed. Depending on your background, see, for example, this question ... 3 The CNOT matrix is the same regardless of the input (i.e. regardless of which vector is involved in the matrix-vector multiplication). Your question indicates some discomfort with how CNOT works when the control qubit is in the state: H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle), but it works just the same as CNOT does with any other control ... 3 I'm going to label my 3 parties by A, B, C (just to be awkward). If we consider a bipartition of A|BC, we know that a general state can be written in the form$$ U_A\otimes I_{BC}(\alpha|0\rangle_A|\psi_0\rangle_{BC}+\beta|1\rangle_A|\psi_1\rangle_{BC}) $$where |\psi_i\rangle are mutually orthogonal and may be entangled. This is just the Schmidt ... 3 One idea is to do polarimetry. By using a polarizing beam splitter, the polarization qubit can have each of its polarization components directed to a different detector for photon counting (ideally a single-photon detector, here). A polarizing beam splitter might send horizontally polarized photons in one direction and vertically polarized photons in another.... 3 Vidal proved that, pure-state quantum computation is efficiently simulable classically if the quantum computer’s state at every time step has amount of entanglement (measured by Schmidt rank) polynomially-bounded (theorem 1 in the linked paper) And if the amount of entanglement grows subexponentially, then it can be classically simulated with sub-exponential ... 3 With just single qubit gate in your circuit, you can only generate a small subset of quantum states. In fact, the states that you can generate are called separable states. These states have no entanglement in them. Here is an example to see why you need to be able to generate entangle state to have successful VQE calculation, supposed you have the ... 3 The entanglement measure that you quote from IBM is equivalent to the "linear entropy" of the reduced system. The idea is that, if you have a pure quantum state and you trace out one of the subsystems, the resulting state will be pure if the original state had no entanglement. Conversely, if the resulting state is not pure, you know the original ... 3 Yes, they do. If it were true that$$\operatorname{Tr}[(\Pi^A_a\otimes\Pi^B_b)\rho] = \operatorname{Tr}(\Pi^A_a\rho_A) \operatorname{Tr}(\Pi^B_b\rho_B)$$for all POVMs (or projective measurements, it doesn't make a difference), then we would just have$$\rho = \rho_A \otimes \rho_B,$$because you'd be in particular doing tomography of your quantum state. Now ... 2 For diagonal Hamiltonians, you might not need entanglement at all. The intuition for why a more complicated ansatz might be useful is that the optimization landscape is somehow better, in the sense that the minimum is easier to find. However, in this paper, the author applies VQE to some combinatorial optimization problems and finds that variational forms ... 2 No, we do not require \rho_A^i and \rho_B^i to be pure. The state is separable any time it can be written in this form with positive weights p_i. As a basic example, we can choose a single p_i to be nonzero; this is equivalent to saying that$$\rho=\rho_A^i\otimes \rho_B^i$$is separable. This is true regardless of the purities of \rho_A^i and \... 2 Here is an approach that requires no specific knowledge about |\psi\rangle whatsoever. In your description you implied that each H_i has the same maximum and minimum eigenvalues \lambda_m and \lambda_M respectively so I will assume this in the derivation. The process of measuring \langle h \rangle empirically can be thought of running a series of ... 2 If I may, I wanna share an algorithmic generalization for the inspired @Danylo Y answer. This considers also The Algorithmic Method in this answer here related to the CNOT gate. In order to algorithmically build a 2-qubit SWAP gate to operate swap within n-qubits, one could define an array containing n ID gates, i.e. ID=((1,0),(0,1)) matrices, named here as ... 2 Another interesting example is a single-photon state in a superposition of two different spatial (or any other type of degree of freedom really) modes:$$\frac{1}{\sqrt2}(a_1^\dagger + a_2^\dagger)|0\rangle\equiv \frac{1}{\sqrt2}(|1\rangle+|2\rangle).$$This type of states is sometimes not considered "entangled", as there is only a single particle ... 2 It absolutely depends on the subdivision of the spaces. Take the 3-qubit system (qubits A, B and C) in a state$$ |0\rangle_A(|00\rangle+|11\rangle)_{BC} $$We can partition these qubits in various different ways. Clearly the A|BC partition has no entanglement across the partition, while the AB|C partitioning is maximally entangled. Note that if you go ... 2 You may be interested in this paper Effcient quantum algorithms for GHZ and W states, and implementation on the IBM quantum computer. The paper provides general method how to prepare n-qubit W state. Concerning your question on how place the state's qubits to different location, you can prepare W state on qubits from q_{1} to q_n and then use SWAP ... 2 To understand what entanglement is, as a crude description it means if you measure one part of the system, then knowing what the measurement outcome is will change what you expect for measurement outcomes on the remainder of the system. This is true for both your examples, in both cases if you don't measure qubit 1, you expect to measure 0 or 1 with equal ... 1 From quantiki we have$$\rho=p_{\mathrm{sym}}\frac{2\hat{P}_{\mathrm{sim}}}{d^2+d}+(1-p_{\mathrm{sym}})\frac{2\hat{P}_{\mathrm{anti-sym}}}{d^2-d}, $$where p_{\mathrm{sym}} is the probability of being in the symmetric subspace, d is the dimension of the state vectors, the projectors onto the symmetric and anti-symmetric subspaces are given by$$\hat{P}_{\...

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For an arbitrary input product pure state $|\psi_1\rangle\otimes|\psi_2\rangle$, using the notation $|\psi_i\rangle\equiv\alpha_i|0\rangle+\beta_i|1\rangle$, we have \begin{align} |\psi_1\rangle\otimes|\psi_2\rangle &= \alpha_1\alpha_2 |00\rangle + \alpha_1\beta_2 |01\rangle + \beta_1\alpha_2 |10\rangle + \beta_1\beta_2 |11\rangle, \\ &= (\alpha_1|0\...

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Just few ideas, I do not pretend this to be the definitive answer. Firstly, to exploit quantum advantage, you need to employ both superposition and entanglement. If these phenomena are not employed, a quantum computer can do anything a classical one can do but with no speed-up (I supposed that e.g. Toffoli gate is implemented in the computer, so you can ...

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I believe your problem is that you're splitting up your measurement observable too much. Yes, $M\otimes I$ has four eigenvalues, $-1,-1,1,1$. BUT, when you make the measurement, you do not get 4 difference answers. You only get 2. To see why what you've done cannot work, note that the choice of eigenvectors is not unique. So how can you sub-divide the ...

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You should use an entanglement measure. An overview of these are in this paper. In your case, you could try to use entropy of entanglement which quantifies the amount of uncertainty in a quantum state (always between 0 and 1). If we know a state is classical then there's no uncertainty because we know the state of the system exactly i.e. entropy of ...

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If we label the expansion coefficients as $c_{N_1,N_2}(t)$ as suggested in the comments, the correct expression will be $$c_{N_1,N_2}(t)= \vphantom{a}_a\langle N_1| \vphantom{a}_b\langle N_2|\psi(t)\rangle= \vphantom{a}_a\langle N_1| \vphantom{a}_b\langle N_2|e^{-i H_{NL}t}|N\rangle_a|0\rangle_b.$$ Responding to the comment: some people write their Hilbert ...

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