Table of Contents
$\lvert\Phi^-\rangle$ (Phi-minus state)
The phi-minus state $\lvert\Phi^-\rangle$ is one of the four Bell states — the maximally entangled two-qubit states. It is an equal superposition of $\lvert 00\rangle$ and $\lvert 11\rangle$ with a relative minus sign. Like $\lvert\Phi^+\rangle$, both qubits always give the same outcome in the computational basis; the minus sign is only visible in phase-sensitive measurements.
$$\lvert\Phi^-\rangle = \frac{1}{\sqrt{2}}(\lvert 00\rangle - \lvert 11\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\0\\0\\-1\end{pmatrix}$$
It is prepared from $\lvert\Phi^+\rangle$ by applying $Z$ to either qubit, or directly as $\text{CX}\lvert -\rangle\lvert 0\rangle = \lvert\Phi^-\rangle$. In the X basis, $\lvert\Phi^-\rangle = \tfrac{1}{\sqrt{2}}(\lvert +-\rangle + \lvert -+\rangle)$, giving anti-correlated X-basis measurements — the opposite of $\lvert\Phi^+\rangle$.
Qiskit
# Prepare |Φ−⟩ = (|00⟩ − |11⟩)/√2 — same as |Φ+⟩ preparation, then Z on qubit 0. from qiskit import QuantumCircuit from qiskit.quantum_info import Statevector qc = QuantumCircuit(2) qc.h(0) # |00⟩ → (|00⟩ + |10⟩)/√2 qc.cx(0, 1) # CX: → (|00⟩ + |11⟩)/√2 = |Φ+⟩ qc.z(0) # Z on qubit 0: |Φ+⟩ → |Φ−⟩ print(Statevector(qc).data)
Applying gates
Single-qubit gates act on qubit 1. By symmetry of $\lvert\Phi^-\rangle$, applying $X$, $Y$, or $Z$ to qubit 2 gives the same result.
| Gate | Result | Comment |
|---|---|---|
| I gate | $I_1\lvert\Phi^-\rangle = \lvert\Phi^-\rangle$ | Identity leaves the state unchanged. |
| X gate | $X_1\lvert\Phi^-\rangle = -\lvert\Psi^-\rangle$ | Flips qubit 1; $\lvert 00\rangle\to\lvert 10\rangle$ and $\lvert 11\rangle\to\lvert 01\rangle$ with the minus sign preserved. Equivalent to $\lvert\Psi^-\rangle$ up to global phase. |
| Y gate | $Y_1\lvert\Phi^-\rangle = i\lvert\Psi^+\rangle$ | Bit-flip with phase; the two $i$ factors combine constructively, removing the relative minus sign. Equivalent to $\lvert\Psi^+\rangle$ up to global phase. |
| Z gate | $Z_1\lvert\Phi^-\rangle = \lvert\Phi^+\rangle$ | Negates the $\lvert 11\rangle$ term, cancelling the relative minus sign and recovering $\lvert\Phi^+\rangle$. |
| Hadamard gate | $H_1\lvert\Phi^-\rangle$ (not a Bell state) | Bell measurement circuit: CX then $H_1$ maps $\lvert\Phi^-\rangle\to\lvert 10\rangle$. |
| S gate | $S_1\lvert\Phi^-\rangle = \tfrac{1}{\sqrt{2}}(\lvert 00\rangle - i\lvert 11\rangle)$ | Adds a phase of $i$ to the $\lvert 11\rangle$ term; combined with the minus sign gives $-i$. Not a standard Bell state. |
| T gate | $T_1\lvert\Phi^-\rangle = \tfrac{1}{\sqrt{2}}(\lvert 00\rangle - e^{i\pi/4}\lvert 11\rangle)$ | Adds a phase of $e^{i\pi/4}$ to the $\lvert 11\rangle$ term; combined with the minus sign. Not a standard Bell state. |
| Rotation-X gate | Mixes $\lvert\Phi^-\rangle$ with $\lvert\Psi^-\rangle$ | X-type rotations couple the $\lvert\Phi^-\rangle$ and $\lvert\Psi^-\rangle$ Bell states. |
| Rotation-Y gate | Mixes $\lvert\Phi^-\rangle$ with $\lvert\Psi^+\rangle$ | Y-type rotations couple the $\lvert\Phi^-\rangle$ and $\lvert\Psi^+\rangle$ Bell states. |
| Rotation-Z gate | $R_z(\theta)_1\lvert\Phi^-\rangle = \tfrac{e^{-i\theta/2}}{\sqrt{2}}(\lvert 00\rangle - e^{i\theta}\lvert 11\rangle)$ | Modifies relative phase only; at $\theta=\pi$ gives $\lvert\Phi^+\rangle$ up to global phase. |
| CX (CNOT) gate | $\text{CX}\lvert\Phi^-\rangle = \lvert -\rangle\lvert 0\rangle$ | Disentangles $\lvert\Phi^-\rangle$ back to the product state used to prepare it. |
| SWAP gate | $\text{SWAP}\lvert\Phi^-\rangle = \lvert\Phi^-\rangle$ | $\lvert 00\rangle$ and $\lvert 11\rangle$ are symmetric under qubit exchange; eigenstate of SWAP with eigenvalue $+1$. |
| iSWAP gate | $\text{iSWAP}\lvert\Phi^-\rangle = \lvert\Phi^-\rangle$ | $\lvert 00\rangle$ and $\lvert 11\rangle$ are unchanged by iSWAP; eigenstate with eigenvalue $+1$. |
Reaching other Bell states
| State | Gates | Comment |
|---|---|---|
| $\lvert\Phi^-\rangle$ | $I\lvert\Phi^-\rangle = \lvert\Phi^-\rangle$ | The identity gate leaves $\lvert\Phi^-\rangle$ unchanged. |
| $\lvert\Phi^+\rangle$ | $Z_1\lvert\Phi^-\rangle = \lvert\Phi^+\rangle$ | Z on qubit 1 cancels the relative minus sign; same-value Z correlations are preserved. |
| $\lvert\Psi^-\rangle$ | $X_1\lvert\Phi^-\rangle = -\lvert\Psi^-\rangle$ | X on qubit 1 switches from same-value to anti-correlated Z-basis correlations, reaching $\lvert\Psi^-\rangle$ up to global phase. |
| $\lvert\Psi^+\rangle$ | $Y_1\lvert\Phi^-\rangle = i\lvert\Psi^+\rangle$ | Y on qubit 1 reaches $\lvert\Psi^+\rangle$ up to global phase $i$. |