Case Study 1: The Teleportation Demo That Proved Nothing

"It matched in the computational basis" is a sentence that should always be followed by "and in which others?"

Executive Summary

A teaching demo implements quantum teleportation, runs it, and shows that the teleported qubit's measurement distribution matches the original's. The audience is satisfied. The demo is wrong — not in the sense that the circuit is broken, but in the sense that the evidence presented does not support the claim made.

This case study builds a circuit that fakes teleportation, shows it passing the demo's test perfectly, and then constructs the verification that separates them. It is short, and it is the third time this book has made the same argument — Chapter 4 for entanglement, Chapter 5 for statistical power, and now here — because the pattern is the single most common evidentiary failure in quantum computing.

Skills applied: teleportation (§9.3); basis rotations (Ch. 5 §5.6); the two-basis discipline (Ch. 4 Case Study 2); expectation values and sampling error (Ch. 5 §5.4).

The Demo

theta, phi = 0.7, 1.3          # the state to teleport

original = QuantumCircuit(1)
original.ry(theta, 0)
original.rz(phi, 0)
print("original: ", Statevector(original).probabilities().round(4))

# ... full teleportation circuit ...
print("teleported:", bob_marginal(run(teleport(theta, phi, "Z"))))

original:   [0.8824 0.1176]
teleported: {'0': 0.8854, '1': 0.1146}

"The state arrived." And the circuit is genuinely correct — that is what makes this instructive. The problem is not the implementation; it is that the demonstration does not distinguish the correct implementation from several incorrect ones.

The Impostor

Here is a circuit that requires no entanglement, no Bell measurement, no classical feedforward, and no teleportation of any kind:

def fake_teleport(theta, phi):
    """Prepare the right PROBABILITIES on q2. No teleportation whatsoever."""
    qc = QuantumCircuit(3, 1)
    qc.ry(theta, 2)                # just prepare a state with the same P(0)
    qc.measure(2, 0)               # ... and no phase information at all
    return qc

ideal probs    [0.8824 0.1176]
impostor probs [0.8824 0.1176]

Not merely close — identical, exactly. The impostor prepares $R_y(\theta)|0\rangle$, which has precisely the same $Z$ distribution as $R_z(\phi)R_y(\theta)|0\rangle$, because $R_z$ changes only the phase and a computational-basis measurement is blind to phase. Chapter 3 §3.5 established exactly this: two states can have identical measurement probabilities and be completely different states.

So the demo's evidence is consistent with:

  • correct teleportation,
  • a circuit that ignores the phase entirely,
  • a circuit that never entangles anything,
  • and a circuit in which qubits 0 and 1 are not even connected to qubit 2.

The Verification

Add the other two Bloch axes — $X$ (h before measuring) and $Y$ (sdg then h), from Chapter 5 §5.6 — and compare expectation values:

   basis |  ideal <P> |  teleported |  difference
  -----------------------------------------------
       Z |    +0.7648 |     +0.7708 |      0.0059
       X |    +0.1723 |     +0.1768 |      0.0044
       Y |    +0.6207 |     +0.6240 |      0.0033

  worst disagreement 0.0059, sampling error ~0.0110  ->  consistent

And the impostor, on the same test:

   basis |  ideal <P> |   impostor  |  difference
  -----------------------------------------------
       Z |    +0.7648 |     +0.7678 |      0.0030    passes
       X |    +0.1723 |     +0.6536 |      0.4812    FAILS
       Y |    +0.6207 |     +0.0007 |      0.6200    FAILS

The $Y$ measurement is decisive — the impostor has no phase, so its $Y$ component is $+0.0007$, statistically indistinguishable from zero, while the real state's is $+0.62$. That is a discrepancy of 56 standard errors. One extra basis and the two circuits separate beyond any doubt.

Three Bloch components completely determine a single-qubit pure state, so passing all three is not merely better evidence; it is a complete characterization. That is single-qubit state tomography, and it is the honest standard for a teleportation claim.

⚠️ Common Pitfall — Choosing a test state that hides the failure.

The demo above used $\theta = 0.7$, $\phi = 1.3$ — a generic state with all three Bloch components nonzero. That was lucky.

Had it teleported $|0\rangle$, or $|1\rangle$, or any state on the $Z$ axis, then $\langle X\rangle$ and $\langle Y\rangle$ would both be zero for the real state and for the impostor, and even the three-basis test would pass.

Test with a state that has all three Bloch components nonzero. This is Chapter 5 §5.3's asymmetric-test rule in a new costume, and it is the third distinct place it has appeared: palindromic bitstrings hide ordering bugs, symmetric entangled states hide endianness bugs, and axis-aligned states hide phase bugs.

A test case that is special in any way is a test case that cannot detect a failure that respects that speciality.

Analysis

Three properties made the demo's evidence insufficient, and all three generalize.

The measurement was blind to the property most at risk. Teleportation's hard part is preserving phase; the computational basis measures only populations. The demo tested the easy part.

The impostor is not adversarial — it is the natural bug. A teleportation implementation with the corrections omitted, or with $Z$ and $X$ swapped, or with the Bell measurement in the wrong order, produces something with the right populations and the wrong phase. The impostor is what a broken teleportation circuit looks like, which is why passing this test is so weak.

The fix was cheap. Two extra circuits — a h and a sdg+h — and about a second of runtime. There is no version of this where the strong evidence was expensive.

🔬 Honest Assessment — This is the third time.

Chapter 4's Case Study 2 built an impostor with zero entanglement that matched a Bell state's counts. Chapter 5's Case Study 2 showed a comparison whose error bars were never computed. This chapter builds an impostor with no phase that matches a teleported state's counts.

The pattern is the same each time: a measurement in one basis, presented as evidence about a property that basis cannot see. It recurs because computational-basis measurement is the default, the cheapest, and the one every tutorial shows.

The general rule, and it is worth carrying: ask what property your claim depends on, and whether your measurement can see it. Phase claims need a phase-sensitive basis. Entanglement claims need a second basis or a witness. Precision claims need an error bar.

None of these are expensive. They are simply not the default, and the default is what gets published.

Lessons

  1. A computational-basis distribution is blind to phase. If your claim is about phase, it is not evidence.
  2. Verify in all three Bloch bases for a single-qubit claim. That is tomography, and it is complete.
  3. Build the impostor. Handing someone a circuit that passes their test while doing none of what they claimed is more persuasive than any argument.
  4. The natural bug looks like the impostor. Omitted corrections and swapped Paulis all produce right-populations/wrong-phase.
  5. Choose a test state with all three Bloch components nonzero. Axis-aligned states hide phase errors exactly as palindromes hide ordering errors.
  6. A special test case cannot detect a failure that respects its speciality. This is the same rule as Chapter 5 §5.3's, in a third costume.
  7. Ask what property the claim depends on, and whether the measurement can see it.

Questions

  1. Build both the real and the fake teleportation circuits and reproduce all six numbers. Which basis separates them most strongly, and why that one?

  2. Teleport $|0\rangle$ instead. Does the three-basis test still separate the impostor? Explain, and state the rule this illustrates.

  3. Construct a better impostor: one that passes the $Z$ and $X$ tests and fails only $Y$. What would it have to do? Is there an impostor that passes all three without teleporting?

  4. Break the real circuit in three ways — omit the $X$ correction, omit the $Z$ correction, swap them — and run the three-basis test on each. Which bases catch which bug? Build a diagnosis table.

  5. The verification used expectation values rather than raw distributions. Why is that the better comparison? (Consider what a distribution gives you that an expectation value does not, and whether you need it here.)

  6. How many shots do you need for the three-basis test to reliably detect a phase error of size $\delta$? Use Chapter 5 §5.4's arithmetic, and check your answer empirically.

  7. Hardest. Three Bloch components determine a single-qubit pure state. Teleporting a two-qubit state would require how many measurement settings for a complete verification? At what qubit count does full tomography stop being feasible, and what do people do instead? (Chapter 26 §26.8.)