# Questions tagged [entanglement]

For questions about the principle and application of quantum entanglement. It is a physical phenomenon which occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the other(s), even when the particles are separated by a large distance—instead, a quantum state must be described for the system as a whole. (Wikipedia)

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### How do I show that a two-qubit state is an entangled state?

The Bell state $|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle )$ is an entangled state. But why is that the case? How do I mathematically prove that?
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### Is entanglement necessary for quantum computation?

Entanglement is often discussed as being one of the essential components that makes quantum different from classical. But is entanglement really necessary to achieve a speed-up in quantum computation?
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Two of the most well known entangled states are the GHZ-state $|\psi\rangle = 1/\sqrt{2}\left( |0\rangle^{\otimes n} + |1\rangle^{\otimes n}\right)$ and the $W_n$-state, with $W_3 = 1/\sqrt{3}\left(|... • 1,517 2 votes 2 answers 3k views ### What is the result of measuring$\sigma_x$on the state$|01\rangle+|10\rangle$? I confused about how to calculate the probabilities and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be ... 10 votes 2 answers 1k views ### How to show whether a bipartite high-dimensional system is entangled? "How do I show that a two-qubit state is an entangled state?" includes an answer which references the Peres–Horodecki criterion. This works for$2\times 2$and$2\times3$dimensional cases; however, ... • 3,312 6 votes 1 answer 388 views ### What separable$\rho$only admit separable pure decompositions with more than$\mathrm{rank}(\rho)$terms? As shown e.g. in Watrous' book (Proposition 6.6, page 314), a separable state$\rho$can always be written as a convex combination of at most$\mathrm{rank}(\rho)^2$pure, separable states. More ... • 25.5k 6 votes 2 answers 525 views ### Is it true that for a quantum algorithm to be efficient it must feature a highly entangled state at some point? I'm wrapping my head around how and why quantum computers can provide advantage over classical. A basic and naive argument is that the dimension of the Hilbert space of$n$qubits grows as$2^n$. ... • 1,655 5 votes 1 answer 509 views ### How do you build a circuit to make an equal superposition of$n$outcomes? Suppose we start with$|00...0\rangle$. We want to build an equal superposition over$|0\rangle + ... + |n-1\rangle$. When$n=2^m$for some$m$, I know I can do this using$H^{\otimes m}$. What is the ... 14 votes 2 answers 2k views ### Why is an entangled qubit shown at the origin of a Bloch sphere? I'm unclear why the Bloch sphere representation of a maximally entangled qubit shows the state of the bit as being at the origin of the sphere. For example, this illustration shows the effect of ... • 241 2 votes 2 answers 2k views ### Qiskit CNOT-gate matrix mixup? In the qiskit textbook chapter 1.3.1 "The CNOT-Gate" it says that the matrix representation on the right is the own corresponding to the circuit shown above, with q_0 being the control and ... • 23 -3 votes 1 answer 825 views ### Quantum entanglement for faster-than-light (FTL) network communication? Quantum entanglement is 2 atoms that are paired together and when you stop one from spinning the other also stops with the same spin. Can you use these pairs to have faster-than-light (FTL) ... • 161 19 votes 3 answers 11k views ### What does it mean for two qubits to be entangled? I have done some sort of online research on qubits and the factors making them infamous i.e allowing qubits to hold 1 and 0 at the same time and another is that qubits can be entangled somehow so that ... 12 votes 3 answers 1k views ### When was the first use of the word Entanglement? Schrödinger wrote a letter to Einstein after the 1935 EPR paper, and in that letter Schrödinger used the German word "Verschränkung" which translates into "entanglement", but when was the word first ... 8 votes 1 answer 574 views ### How does the conditional min-entropy$H_{\rm min}(A|B)_\rho$relate to the conditional entropy$H(X|Y)_\rho$? Suppose we have a classical quantum state$\sum_x |x\rangle \langle x|\otimes \rho_x$, one can define the smooth-min entropy$H_\min(A|B)_\rho$as the best probability of guessing outcome$x$given$\...
Consider the two-qubit Werner state, defined as $$\rho_z = z |\Psi_-\rangle\!\langle \Psi_-| + \frac{1-z}{4}I, \quad |\Psi_-\rangle\equiv\frac{1}{\sqrt2}(|00\rangle-|11\rangle),$$ for $z\ge0$. Using ...