Questions & explanations
1. Show that the Bell state |Φ+> = (|00>+|11>)/√2 violates the range criterion.
For the Bell state, the density matrix has support only on the vector (|00>+|11>)/√2. The reduced density matrix on system A is I/2, so its range is the whole space—any vector in A is possible. The tensor product of the two reduced ranges is all vectors in the full space. The full range is a single line, which is inside that huge space. So the criterion says it might be separable, which is wrong. Actually, the criterion does not detect this entanglement because the condition is necessary but not sufficient for mixed states. For pure states, it works differently: the range of the full state must be one-dimensional, and the criterion becomes trivial. So the Bell state passes the range criterion, but it is entangled. That shows the criterion is not sufficient.
2. Compare the representation of canonical commutation relations in different systems.
In the standard Schrödinger representation, position is multiplication and momentum is a derivative. In the Heisenberg representation, operators evolve in time. Both are unitarily equivalent via the Stone-von Neumann theorem. For a particle in a box, the commutation relations are the same, but the representation is different due to boundary conditions. However, the theorem still applies because the system is finite-dimensional? Actually, infinite-dimensional Hilbert space, but the theorem holds. For systems with constraints, like a particle on a circle, the commutation relations need careful treatment, and the theorem may not apply directly. In quantum field theory, infinite degrees of freedom lead to inequivalent representations.
3. How does the Horodecki criterion relate to the PPT criterion?
The PPT criterion says that if a state has a positive partial transpose, it cannot be distilled (so it is bound entangled). The Horodecki criterion is the converse: if the partial transpose is negative, the state is distillable. However, this is only true for systems of size 2×2 and 2×3. For larger systems, there exist PPT entangled states that are bound entangled but also non-PPT entangled states that might not be distillable? Actually, non-PPT always distillable? The Horodecki criterion says that for any bipartite state, if it is non-PPT (NPT), it is distillable. That is a main result: any state with a negative partial transpose can be distilled to a singlet. So in bipartite systems, NPT implies distillability.
4. Compare theoretical models of entanglement in photosynthesis vs vision.
In photosynthesis, models propose that entangled states allow excitons to explore many pathways simultaneously, increasing energy transfer efficiency. These models often involve vibronic coupling to protect coherence. In vision, models of entanglement are less common; the primary process (photon absorption by retinal) is often described semiclassically. However, some theoretical work suggests that entanglement between photons and retinal could enhance detection. Both areas share the challenge of maintaining entanglement in noisy environments. Photosynthesis models are more advanced and have experimental support for quantum coherence. Vision models are more speculative.
5. In an infinite square well, what is the probability of finding the particle in the left half of the well for the n=2 state?
The probability is exactly 1/2 because the wave function is symmetric? Actually for n=2, the probability density is sin²(2πx/L). It is symmetric about the center? Actually it is symmetric about L/2? For n=2, the density has a node at x=L/2, so the left and right halves each have equal probability? Let's check: integral from 0 to L/2 of sin²(2πx/L) dx = L/4 (since average sin² is 1/2 over half period? Actually the probability in left half is exactly 1/2 because the wave function is symmetric about the center? Wait, sin²(2πx/L) is symmetric about L/4? No, better to compute: The full integral is L/2. Half of that is L/4, so probability is (L/4)/(L/2)=1/2. So yes, 1/2.
6. What is a quantum phase transition?
A quantum phase transition is a change in the ground state of a many-particle system driven by quantum fluctuations at zero temperature. It occurs when a parameter like magnetic field or pressure crosses a critical value. Unlike classical phase transitions, temperature plays no role. Near the critical point, the system's properties change dramatically, and entanglement between particles becomes important. For example, in a chain of magnetic atoms, a quantum phase transition can happen between ordered and disordered phases. The transition is marked by a sudden change in the way particles are entangled. This makes entanglement a key tool to study such transitions.
7. Compare the Rains bound with the distillable entanglement for a given state.
The distillable entanglement E_D is the maximum amount of pure entanglement you can extract per copy from many copies of ρ. The Rains bound R(ρ) is always greater than or equal to E_D. For some states, like any pure state, E_D equals the entropy of entanglement, and the Rains bound also gives that. For mixed states, the bound can be strict: for example, for a PPT entangled state, E_D=0 but the Rains bound may be positive? Actually, for PPT states, the Rains bound is also zero because the minimum over PPT states includes the state itself, giving zero relative entropy. So it is tight for PPT states. For NPT states, the Rains bound is generally larger than zero.
8. How does the relativity of simultaneity affect entanglement?
In special relativity, different observers disagree on which events happen at the same time. For entangled particles, this affects the measurement order. The entanglement is still valid, but the correlations between measurements depend on the observer's frame. For spacelike separated particles, the order of measurements is frame-dependent, but the quantum correlations remain consistent. This leads to the concept of 'relativistic entanglement' where the entanglement measure can be frame-dependent. For example, the amount of entanglement between two modes can change under Lorentz boosts. Thus, relativity forces us to reconsider how we define entanglement.
9. In the transverse Ising model, how does entanglement behave at the critical point?
The transverse Ising model describes a chain of spins in a magnetic field. At its critical field value, the system undergoes a quantum phase transition. The entanglement between two spins decays as a power law with distance, unlike the exponential decay away from criticality. The entanglement entropy of a block of spins diverges logarithmically with block size. This shows that at criticality, the system becomes highly entangled on all length scales. The entanglement pattern is universal, meaning it depends only on the type of transition, not on microscopic details. This makes the model a classic example to study entanglement in phase transitions.
10. Compare the partial wave expansion and the Born approximation for calculating the scattering amplitude.
The partial wave expansion works well for low-energy scattering from a finite-range potential, where only a few angular momentum waves contribute. It expands the scattering amplitude in terms of Legendre polynomials and phase shifts. The Born approximation, on the other hand, is a perturbative method valid for weak potentials at any energy, computing the amplitude as the Fourier transform of the potential. The partial wave method is exact but becomes cumbersome at high energies, whereas the Born approximation is simpler but fails for strong potentials. For a square well, you might use partial waves for low energy and Born for high energy.
11. How does scaling of entanglement entropy relate to critical exponents?
At a continuous quantum phase transition, the entanglement entropy of a subsystem of size L scales as S(L) ≈ c/3 log(L) + constant, where c is the central charge. The central charge is a critical exponent that characterizes the underlying quantum field theory. Other critical exponents, like the correlation length exponent ν, appear in finite-size scaling of entanglement. The area law (entropy proportional to boundary area) is modified at critical points to a logarithmic correction. This scaling unifies many physical systems into universality classes. Thus, entanglement entropy provides direct access to these fundamental exponents.
12. What is the relationship between entanglement and operator spreading?
Operator spreading describes how a local operator, when evolved in time, becomes supported on many-body terms. Entanglement is the quantum correlation between subsystems. Both are measures of information scrambling. Under chaotic dynamics, an operator spreads out, and the size of its support grows linearly. This growth is bounded by the entanglement velocity, which also limits entanglement growth. The out-of-time-order correlator (OTOC), which measures scrambling, is related to entanglement. In holographic theories, the entanglement wedge is tied to operator growth. Thus, entanglement and operator spreading are deeply connected.