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--- quantum-gate-two-qubit [August 26, 2026 at 17:43] – Ivan Janevski
+++ quantum-gate-two-qubit [August 26, 2026 at 18:03] (current) – Ivan Janevski
@@ Line 14: / Line 14: @@
   - [[quantum-gate-swap|SWAP]]
   - [[quantum-gate-iswap|iSWAP]]
-  - [[quantum-gate-cphase|Controlled-Phase]]
+  - [[quantum-gate-cp|Controlled-Phase]]
   - [[quantum-gate-xx|XX (Ising coupling)]]
   - [[quantum-gate-yy|YY (Ising coupling)]]
   - [[quantum-gate-zz|ZZ (Ising coupling)]]
-## Entanglement and local equivalence
-A gate $U$ is **local** if $U = A \otimes B$ for $A, B \in \mathrm{SU}(2)$: it acts independently on each qubit and can't create correlation. Local gates form the subgroup $\mathrm{SU}(2) \otimes \mathrm{SU}(2) \subset \mathrm{SU}(4)$. Every other two-qubit gate has some **entangling power**.
-$$\mathrm{CX} \left( \frac{|0\rangle + |1\rangle}{\sqrt{2}} \otimes |0\rangle \right) = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$$
-Entangling power is quantified by **concurrence**, from 0 (local) to 1 (maximally entangling). CX, CZ, SWAP, and $\text{XX}(\pi/2)$ reach 1; $R_Z(\theta) \otimes R_Z(\theta)$ stays at 0 for any $\theta$.
 ## Matrix representations
@@ Line 43: / Line 36: @@
 Note: $\mathrm{SWAP} = \mathrm{CX}_{12} \, \mathrm{CX}_{21} \, \mathrm{CX}_{12}$, and $\mathrm{iSWAP} = \text{XX}(\pi/2)\,\text{YY}(\pi/2)$ up to single-qubit phases.
-## Clifford vs non-Clifford
-**Clifford gates**: CX, CY, CZ, SWAP, iSWAP. For any Pauli string $P \otimes Q$, $U (P \otimes Q) U^\dagger$ is again a Pauli string up to sign, e.g.:
-- $\mathrm{CX} (X \otimes I) \mathrm{CX} = X \otimes X$
-- $\mathrm{CY} (X \otimes I) \mathrm{CY} = X \otimes Y$
-- $\mathrm{CZ} (X \otimes I) \mathrm{CZ} = X \otimes Z$
-Together with single-qubit Cliffords, these generate the [[quantum-gate-clifford|two-qubit Clifford group]], still efficiently simulatable via the stabilizer formalism. CY is rarely native; most hardware synthesizes it from CZ ($\mathrm{CY} = (I \otimes S) \mathrm{CX} (I \otimes S^\dagger)$).
-**Non-Clifford**: $\text{XX}$/$\text{YY}$/$\text{ZZ}(\theta)$ are Clifford only at $\theta = \pi/2$, mirroring how [[quantum-gate-t|T]] is a special-angle $R_Z$. At generic angle they're what most hardware natively implements.
 ## Group structure
@@ Line 59: / Line 41: @@
 Two-qubit gates form $\mathrm{SU}(4)$ (dimension 15); the local subgroup $\mathrm{SU}(2) \otimes \mathrm{SU}(2)$ has dimension 6. The entangling content of a gate, the part that can't be absorbed into surrounding single-qubit gates, lives in the remaining 9-dimensional quotient $\mathrm{SU}(4) / (\mathrm{SU}(2) \otimes \mathrm{SU}(2))$.
-$\mathrm{SU}(4)$ has no Bloch-sphere-like picture. Instead its entangling content is captured by a small set of **local invariants**, unchanged under pre/post-composition with local gates.
+$\mathrm{SU}(4)$ has no Bloch-sphere-like picture. Instead its entangling content is captured by a small set of **local invariants**, unchanged under pre/post-composition with local gates, and reduces to just three numbers.
 ## KAK decomposition
-Any $U \in \mathrm{SU}(4)$ factors as local gates sandwiching one canonical entangling gate $A \in \mathrm{SU}(4)$, where $K_1, K_2, K_3, K_4 \in \mathrm{SU}(2)$ are single-qubit unitaries, local by construction:
+The intuition: any two-qubit gate can be sorted into "local dressing", single-qubit rotations that don't cost an entangling operation on hardware, wrapped around one irreducible entangling core. KAK decomposition makes this split explicit. Any $U \in \mathrm{SU}(4)$ factors as local gates sandwiching one canonical entangling gate $A \in \mathrm{SU}(4)$, where $K_1, K_2, K_3, K_4 \in \mathrm{SU}(2)$ are single-qubit unitaries, local by construction:
 $$U = (K_1 \otimes K_2) \; A \; (K_3 \otimes K_4), \quad A = \exp\left(i \left( a\, X{\otimes}X + b\, Y{\otimes}Y + c\, Z{\otimes}Z \right)\right)$$
@@ Line 85: / Line 67: @@
 - **Photonic**: deterministic gates need weak optical nonlinearity, so CZ is often probabilistic/measurement-induced; fidelity limited by photon loss
-CNOT is rarely the physically native gate; it's a compiler target synthesized from whatever the hardware actually implements (usually CZ or an $\text{XX}$-type coupling).
+CX is rarely the physically native gate; it's a compiler target synthesized from whatever the hardware actually implements (usually CZ or an $\text{XX}$-type coupling).
 ## Relations