Entanglement Swapping
Line up four qubits, 1, 2, 3, 4. Qubits 1 and
2 are born entangled; separately, qubits 3 and
4 are born entangled. Now carry 1 far to the left
and 4 far to the right, leaving 2 and
3 together at a station in the middle. Qubits 1
and 4 have never interacted — they were made in different
labs and have no shared history at all. Yet a single joint measurement on the middle pair
(2,3) reaches out and leaves 1 and
4 entangled with each other. This is
entanglement swapping: you spend two short entangled links to mint one longer one.
The picture: two links in, one link out
Step through the swap below. You start with two separate entangled bonds — the arc joining
1 to 2, and the arc joining
3 to 4. A Bell measurement at
the middle station acts jointly on 2 and 3, and
the entanglement is swapped outward onto the far qubits: a fresh bond springs up between
1 and 4.
Why it works: regroup the four-qubit state
Take both pairs in the state |\Phi^+\rangle
(see the Bell states).
Writing the qubits in the order 1,2,3,4, the whole system starts as
|\Phi^+\rangle_{12} \otimes |\Phi^+\rangle_{34}
= \tfrac12\big(|00\rangle_{12} + |11\rangle_{12}\big)\big(|00\rangle_{34} + |11\rangle_{34}\big)
= \tfrac12\big(|0000\rangle + |0011\rangle + |1100\rangle + |1111\rangle\big).
Nothing quantum has happened yet — this is just two independent pairs. The trick is purely
bookkeeping: regroup each term so the middle qubits (2,3) sit
together and the outer qubits (1,4) sit together. Reading off each term,
whenever the middle pair is |00\rangle_{23} the outer pair is
|00\rangle_{14}, and so on down the list:
= \tfrac12\big(|00\rangle_{23}|00\rangle_{14} + |01\rangle_{23}|01\rangle_{14}
+ |10\rangle_{23}|10\rangle_{14} + |11\rangle_{23}|11\rangle_{14}\big).
The state is exactly the same — only the grouping changed.
Worked example: the Bell measurement projects (1,4)
Now rewrite the middle pair in the Bell basis, using
|00\rangle = \tfrac{1}{\sqrt2}(|\Phi^+\rangle + |\Phi^-\rangle),
|11\rangle = \tfrac{1}{\sqrt2}(|\Phi^+\rangle - |\Phi^-\rangle),
|01\rangle = \tfrac{1}{\sqrt2}(|\Psi^+\rangle + |\Psi^-\rangle), and
|10\rangle = \tfrac{1}{\sqrt2}(|\Psi^+\rangle - |\Psi^-\rangle). Substituting
for the (2,3) kets and collecting the outer (1,4)
terms, every cross-term lines up and the four-qubit state becomes the beautifully symmetric
|\Phi^+\rangle_{12}\otimes|\Phi^+\rangle_{34} = \tfrac12\big(
|\Phi^+\rangle_{23}|\Phi^+\rangle_{14} + |\Phi^-\rangle_{23}|\Phi^-\rangle_{14}
+ |\Psi^+\rangle_{23}|\Psi^+\rangle_{14} + |\Psi^-\rangle_{23}|\Psi^-\rangle_{14}\big).
Read this off. A Bell measurement on (2,3) returns one of the four Bell
states, each with probability \big|\tfrac12\big|^2 = \tfrac14. And whichever
one it finds, the far qubits (1,4) are left in the matching Bell
state — a genuine two-qubit entangled state, even though 1 and
4 never shared a gate. The measurement did not just read the
system; it projected (1,4) into entanglement.
Worked example: the Pauli correction
Suppose the station's Bell measurement returns the outcome |\Psi^+\rangle_{23}.
Then, from the expansion above, qubits (1,4) collapse into
|\Psi^+\rangle_{14} = \tfrac{1}{\sqrt2}\big(|01\rangle_{14} + |10\rangle_{14}\big),
which is entangled but not the "standard" |\Phi^+\rangle we may want. No
problem: the station sends its two-bit outcome over an ordinary classical channel to whoever holds
qubit 4, who applies a single Pauli correction — here a
bit-flip X — turning |\Psi^+\rangle_{14} back into
|\Phi^+\rangle_{14}. The full table mirrors
teleportation:
outcome |\Phi^+\rangle needs no correction,
|\Phi^-\rangle needs a Z,
|\Psi^+\rangle needs an X, and
|\Psi^-\rangle needs ZX. That is the whole point of
the name: swapping is teleportation of an entangled qubit — you teleport qubit
2's entanglement-with-1 across onto qubit
4.
Why it matters: quantum repeaters
Entanglement is fragile. Send a photon down an optical fibre and it is attenuated exponentially with
distance — after a few hundred kilometres almost none survive. Classically you fix this with
amplifiers that copy and boost the signal. But an unknown quantum state
cannot be copied (the no-cloning theorem) or amplified, so that trick is off the
table. Entanglement swapping is the way out: divide a long channel into short segments, generate a
shared Bell pair across each short hop (which is achievable), then swap at
every junction to fuse the little links into one end-to-end entangled pair.
A device that does exactly this — hold two short-range links, perform a Bell measurement, extend the
entanglement — is a quantum repeater. Chain enough of them and you build entanglement
across a continent. Repeaters are the missing hardware for a quantum internet: a
network where distant nodes share entanglement on demand, the raw fuel for teleportation, distributed
quantum computing, and provably secure key exchange.
- start with two independent Bell pairs, |\Phi^+\rangle_{12} and
|\Phi^+\rangle_{34}; qubits 1 and
4 have never interacted;
- a Bell measurement on the inner pair (2,3) projects
the outer pair (1,4) into a Bell state — they become
entangled;
- the four outcomes are equally likely (\tfrac14 each); the classical
outcome tells the receiver which Pauli correction fixes the final state — so it
needs shared classical information and cannot beat light;
- it consumes two pairs to make one longer-range pair — the heart of a
quantum repeater and the quantum internet.
Picture a chain of stations strung between two cities, every neighbouring pair sharing a fresh Bell
link. The first station holds its right-hand qubit and its neighbour's left-hand qubit and performs a
Bell measurement — the two little links merge into one that now spans two hops. Repeat down
the line: each swap doubles the reach, stitching short, easy-to-make links into a single strand of
entanglement running the whole way. Because you never copy a qubit — you only ever measure and
correct — the no-cloning theorem is respected at every step. This staged swapping, paired with
error correction on the stored qubits, is the leading blueprint for long-distance quantum networking:
the internet, rebuilt out of entanglement instead of amplified bits.
It is tempting to read the swap as "measuring two qubits magically entangled two far-away ones for
free." It is not free. Swapping consumes two ready-made entangled pairs to produce
one longer-range pair — it is a conversion, not a creation. It also demands a genuine
Bell measurement on the inner qubits plus the classical broadcast of its
outcome, so the receiver can apply the right Pauli correction; without that classical
message the far pair is entangled but in an unknown, unusable Bell state. And since the correction
travels on an ordinary channel, the whole protocol is firmly light-speed-limited —
no signalling faster than light, exactly as for
teleportation.