Quantum superpositions and entanglement of thermal states at high temperatures and their applications to quantum-information processing

We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum-information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two single-photon detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits.

Figure 1

The probability distributions of $x$ (left) and $p$ (right) for a “superposition” of two distant thermal states. A thermal state with a large mixedness is converted to such a “thermal-state superposition” by interacting with a microscopic superposition (see text). The variance $V$ and displacement $d$ for the thermal state are chosen as (a) $V=100$ and $d=100$, and (b) $V=1000$ and $d=300$. The fringe visibility is 1 regardless of $V$ and the fringe spacing (the distance between the fringes) does not depend on the variance (i.e., mixedness) but only on the distance $d$ between the two component thermal states.

Figure 2

The probability distributions $P$ for a “superposition” of thermal states where $V=5$, $d=2000$, $\phi =\pi ∕1000$. The $x^{\prime }$ $\left(p^{\prime }\right)$ axis in this figure has been rotated by $\pi ∕2000$ from the $x$ $\left(p\right)$ axis for clarity.

Figure 3

(Color online) The time dependent Wigner functions of the thermal state of $V=100$ at the origin $(d=0)$ after an interaction with a microscopic superposition and a conditional measurement. The measurement result on the microscopic part was supposed to be $(\mid 0⟩+\mid 1⟩)∕\sqrt{2}$. The interaction times are (a) $\theta =\lambda t=0$, (b) $\theta =\lambda t=\pi ∕32$, (c) $\theta =\pi ∕16$, (d) $\theta \approx 3.102$, (e) $\theta \approx 3.122$, and (f) $\theta =\pi$.

Figure 4

(Color online) The Wigner function of the thermal state of $V=100$ at the origin $(d=0)$ after an interaction with a microscopic superposition and a conditional measurement. The measurement result on the microscopic part was supposed to be $(\mid 0⟩-\mid 1⟩)∕\sqrt{2}$ with the interaction time $\theta =\lambda t=\pi$.

Figure 5

(a) The optimized violation $B\equiv \mid B^{+}{\mid }_{\max }$ of Bell-CHSH inequality for the “thermal-state entanglement” ${\rho }_{+}$ of $V=1000$ (solid curve) and $V=100$ (dashed curve). The Bell violation of a pure entangled coherent state, i.e., $V=1$, has been plotted for comparison (dotted curve). The Bell-violation $B$ approaches its maximum bound $2\sqrt{2}$, when $d⪢\sqrt{V}$ regardless of the level of the mixedness $V$. (b) The optimized Bell-violation $B$ against $d$ for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ${\rho }^{+}$. $V=1000$ (solid curve), $V=100$ (dashed curve), and $V=1$ (dotted curve).

Figure 6

The optimized Bell-violation $B$ against $V$ for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ${\rho }^{+}$ when $d=0$.

Figure 7

(a) The Bell-CHSH function $B$ against $\theta$ $(=\lambda t)$ for $V=1$ (solid curve), $V=10$ (dashed curve), and $V=20$ (dotted curve) for $d=30$. (b) The Bell-CHSH function for $d=10$ (solid curve), $d=20$ (dashed curve), and $d=30$ (dotted curve) for $V=10$. The Bell violations are more sensitive to the interaction time as either $V$ or $d$ increases.

Figure 8

A schematic of the thermal-Bell state measurement (a) using photon number resolving detection and (b) using homodyne measurements with cross-Kerr nonlinear interactions (NL) (see text for details).

Figure 9

The average photon number $N$ for the “many-photon case” (solid line) and the “few-photon case” (dashed line) for $V=10$ against $d$ (a) when the input state is either ${\rho }^{\Phi (+)}$ or ${\rho }^{\Psi (+)}$ and (b) when the input state is either ${\rho }^{\Phi (-)}$ or ${\rho }^{\Psi (-)}$.

Figure 10

(a) The probability distributions $P_{{\Phi }^{(+)}}^{++}$ (solid curve) and $P_{{\Psi }^{(+)}}^{++}$ (dashed curve) for homodyne measurements at detector $C$. (b) The probability distributions $P_{{\Phi }^{(-)}}^{++}$ (solid curve) and $P_{{\Psi }^{(-)}}^{++}$ (dashed curve) for homodyne measurements at detector $C$.

Figure 11

The distinguishability $P_s$ between states ${\rho }^{\Psi (+)}$ and ${\rho }^{\Phi (+)}$ by a homodyne measurement for $V=10$ (solid curve) and $V=20$ (dashed curve) against distance $d$ (see text for details).