Random physics

Alberto Verga, research notebook

\(\newcommand{\I}{\mathrm{i}} \newcommand{\E}{\mathrm{e}} \newcommand{\D}{\mathop{}\!\mathrm{d}} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\braket}[1]{\langle{#1}\rangle} \newcommand{\bbraket}[1]{\langle\!\langle{#1}\rangle\!\rangle}\) \(\newcommand{\bm}{\boldsymbol}\)

Topics

  • Quantum principles, density matrix and entanglement
  • Random quantum states and Don Page entropy formula
  • Gates and simple circuits, superdense coding, teleportation, Deutsch algorithm, Grove search algorithm
  • Quantum walks
  • Quantum chaos: the kicked rotator and dynamical localization

Problems

1

Density matrix. A quantum system consists in two parts A and B. Show that the expected value of an operator \(O\) acting on A,

$$O = O_A \otimes I_B$$

where \(I_B\) is the identity in the B hilbert spaca \(\mathcal{H}_B\), is given by the partial trace, over B, of the AB density matrix \(\rho\),

$$\rho_A = \mathrm{Tr}_B \rho\,.$$

As an application take a two spins state:

$$\ket{\psi} = \frac{1}{\sqrt{2}}\left( \ket{0} \otimes \ket{1} + \ket{1} \otimes \ket{0} \right)\,,$$

and compute the expected value of \(X\) and \(Z\) (the A pauli matrix in the \(x\) and \(z\) directions). Compare with the single spin state \((\ket{0} + \ket{1})/\sqrt{2}\).

2

Bell state entanglement. Consider a bipartite system AB, and the dichotomic variables \(a,c=\pm1\) of A and \(b,d=\pm1\) of B. Show that,

$$x = ab + ad + cb - cd = \pm 2\,.$$

Demonstrate that the expected value \(\braket{x}\) satisfies \(\braket{x} \le 2\).

Now consider a quantum equivalent system:

$$|\braket{X}| = \left| \braket{ \bm \sigma_A \cdot \bm n_a \bm \sigma_B \cdot \bm n_b + \bm \sigma_A \cdot \bm n_a \bm \sigma_B \cdot \bm n_d + \bm \sigma_A \cdot \bm n_c \bm \sigma_B \cdot \bm n_b - \bm \sigma_A \cdot \bm n_c \bm \sigma_B \cdot \bm n_d } \right|$$

Demonstrate that,

$$|\braket{X}| \le 2\sqrt{2}$$

in contradiction to the classical result. Compute the expected value of \(X\) in the state,

$$\ket{AB} = \frac{1}{\sqrt{2}} \left( \ket{01}- \ket{10} \right)\,,$$

where the measure axes are in the plane \(xz\) with angles with respect to the \(z\) axis \(\theta_a = 0\), \(\theta_b = \pi/2\), \(\theta_c = \pi/4\), and \(\theta_d = -\pi/4\).

3

The Greenbergm Horne and Zeilinger paradox. Consider the state GHZ of a three parts system ABC:

$$\ket{GHZ} = \frac{1}{\sqrt{2}} \left( \ket{000} + \ket{111} \right).$$

Show that the operators in the set \(\mathcal{S}\), have eigenvalue 1:

$$\mathcal{S} = \{111, ZZ1, 1ZZ, Z1Z, XXX, -YYX, -YXY, -XYY\},$$

where, for instance, \(YXY = Y_A \otimes X_B \otimes Y_C\) and \(ZZ1 = Z_A \otimes Z_B \otimes I_C\).

Consider now a classical equivalent system for which the value of the observables in \(\mathcal{S}\) can take the values \(m(\mathcal{S}) = \pm 1\). Show then that,

$$m(X_A) m(X_B) m(X_C) = ?$$
$$-m(Y_A) m(Y_B) m(X_C) = ?$$
$$-m(Y_A) m(X_B) m(Y_C) = ?$$
$$-m(X_A) m(Y_B) m(Y_C) = ?$$

leads to a contradiction \(-1=1\).

4

Teleportation. Alice and Bob share the bell state,

$$\ket{\Phi_-} = \frac{1}{\sqrt{2}} \left( \ket{01}- \ket{10} \right)\,,$$

In addition, Alice possesses an arbitrary state \(\ket{\psi} = a \ket{0} + b \ket{1}\). Show that the three qubit state can be written as,

$$\ket{\psi} \ket{\Phi_-} = \frac{1}{2} \left[ \ket{\Phi_+} (-Z) \ket{\psi} + \ket{\Phi_-} (-\ket{\psi}) + \ket{\Psi_+} (-XZ \ket{\psi}) + \ket{\Psi_-} (X \ket{\psi}) \right]$$

a decomposition in the bell states:

$$\ket{\Phi_\pm} = \frac{1}{\sqrt{2}} \left( \ket{01} \pm \ket{10} \right)\,,$$
$$\ket{\Psi_\pm} = \frac{1}{\sqrt{2}} \left( \ket{00} \pm \ket{11} \right)\,.$$

Explain how the information in possession of Alice (the bell states) may lead Bob to recover the initially Alice \(\ket{\psi}\) state.

Answer: \(\ket{\Phi_+} \rightarrow Z\), \(\ket{\Phi_-} \rightarrow I\), \(\ket{\Psi_+} \rightarrow ZX\), and \(\ket{\Psi_-} \rightarrow X\).

5

The Deutsch-Jozsa algorithm. We want to determine if certain boolean function \(f\) of \(n\) bits,

$$f(x_1,\ldots,x_n), \quad x \in \{0,1\}$$

is “constant” or “balanced”. In the first case, the constant value of \(f\) can be either 0 or 1, whatever the set of input bits. In the second case, \(f=0\) for exactly half of the possible input values of \(\{x_1,\ldots,x_n\}\), and \(f=1\) for the other half.

  • How many evaluations of \(f\) are necessary to decide classically if it is constant or balanced?

Deutsch devised a quantum algorithm which only needs one evaluation of \(f\), using the circuit:

where the unitary operator \(U_f\) is defined by

$$U_f \ket{x}\otimes \ket{b} = \ket{x}\otimes \ket{f(x) \oplus b}$$

with the notation \(\ket{x}\) for the state of the first \(n\) qubits (top wire of the circuit), and \(\ket{b}\) for the last qubit (bottom wire). Initially the \(n+1\) qubits are in the state \(\ket{0}\)

  • Compute the state \(\ket{\psi_0}\) after the application of the operator \(X\) to the last qubit.

  • Compute the state \(\ket{\psi_1}\) after the application of the Hadamard operator to all qubits.

  • Show that the state of the last qubit remains the same, up to a phase factor,

    $$ \ket{-} = \frac{\ket{0}-\ket{1}}{\sqrt{2}} $$

    after the \(U_f\) step. The phase factor depends on the value of \(f\).

    Determine the state \(\ket{\psi_2}\) after applying \(U_f\).

  • Demonstrate that after the last application of the Hadamard operator to the first \(n\) qubits, the final state is,

    $$\ket{\psi_3} = \frac{1}{2^{n}} \sum_{x,y\in\{0,1\}^n} (-1)^{\langle x,y \rangle + f(x)} \ket{y} \otimes \ket{-} $$

    Hint. As an intermediate computation you may demonstrate that,

    $$H^{\otimes n} \ket{0} = \frac{1}{\sqrt{2^n}} \sum_{x\in\{0,1\}^n} \ket{x}$$

    (which gives you \(\ket{\psi_1}\)), and

    $$H^{\otimes n} \ket{x} = \frac{1}{\sqrt{2^n}} \sum_{y\in\{0,1\}^n} (-1)^{\langle x,y \rangle} \ket{y}$$

    which will give you the \(\ket{\psi_3}\) state.

  • Explain how one may distinguish between the two cases, constant or balanced, using the properties of the final quantum state. Can we answer the question with certainty?

6

Quantum walk. Investigate the pseudo-energy spectrum of a one dimensional quantum walk defined by the step operator \(U = SC\) with coin,

$$C=R_y(\theta) = \E^{-\I \theta Y/2}$$

and motion,

$$S = \sum_x \left( \ket{x+1} \bra{x} \otimes \ket{0} \bra{0} + \ket{x-1} \bra{x} \otimes \ket{1} \bra{1} \right) \,.$$

Find the expression of the effective hamiltonian in the limit \(k \rightarrow 0\) and compare the result with the Dirac hamiltonian.