Quantum Mechanics 101 · Unit 6: Quantum Technology & Frontiers

Quantum Academy · Chapter 24: Quantum Cryptography, Teleportation & Sensing

Unbreakable keys, teleported states, and the most precise clocks ever built

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Instructions. Work through these questions with the lesson closed first, then check yourself against it. Show every step in calculations. Useful facts: an intercept-resend attack on a fraction f of photons gives an error rate of f/4; Eve escapes n test bits with probability (3/4)ⁿ; fibre transmission is 10 to the power (−αL/10); a clock raised by Δh shifts by g·Δh/c²; h = 6.626×10⁻³⁴ J·s, e = 1.602×10⁻¹⁹ C, g = 9.81 m/s², c = 2.998×10⁸ m/s. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.No-cloning theorem
2.Quantum key distribution
3.Sifting
4.Quantum bit error rate
5.Intercept-resend attack
6.Privacy amplification
7.Quantum teleportation
8.Post-quantum cryptography
9.Quantum repeater
10.Optical clock
11.Nitrogen-vacancy centre
12.Squeezed light

A.Light whose uncertainty in one quantity, such as phase, is reduced below the vacuum level at the cost of greater uncertainty in another.
B.Classical cryptographic algorithms, such as ML-KEM, designed to resist attack by quantum computers and run on ordinary hardware.
C.A defect in diamond, a nitrogen atom beside a missing carbon, whose electron spin can be read optically at room temperature to sense magnetic fields.
D.A chain of nodes with quantum memories that extends entanglement over long distances by entanglement swapping, since signals cannot be amplified.
E.An atomic clock that counts the oscillations of an optical transition near 10¹⁵ hertz; the best now reach fractional uncertainties below 10⁻¹⁸.
F.A method for two parties to build a shared secret key from quantum states, so that eavesdropping causes detectable errors.
G.A hashing step that shortens a partly compromised key into one about which an eavesdropper knows essentially nothing.
H.The fraction of sifted bits on which Alice and Bob disagree, estimated from a sacrificed sample; abbreviated QBER.
I.An eavesdropping strategy in which Eve measures each photon in a randomly chosen basis and sends on a replacement, causing a 25 percent error rate.
J.The 1982 result that no physical process can produce a perfect copy of an arbitrary unknown quantum state.
K.The BB84 step in which Alice and Bob publicly compare bases, never results, and keep only the bits measured in matching bases.
L.Transfer of an unknown quantum state from one particle to a distant one, using a shared entangled pair, a Bell-state measurement and two classical bits.

Part B Questions

1.
Which statement best captures the no-cloning theorem?Circle one
1. ANo quantum state can ever be prepared twice
2. BClassical bits stored in qubits cannot be copied
3. CNo process can make a perfect copy of an arbitrary unknown quantum state
4. DCopying a quantum state is possible but slower than light
2.
Because of the no-cloning theorem, a qubit known to be in either |0⟩ or |1⟩ cannot be copied.Circle one
TrueFalse
3.
In BB84, discarding every bit for which Alice's and Bob's bases differ is called .Fill in the blank
4.
Eve runs intercept-resend on 40 percent of the photons in a BB84 session and leaves the rest alone. What quantum bit error rate should Alice and Bob measure, and can they still extract a secret key?Show your working
Working
Answer:
5.
Eve intercepts and resends every photon. How many sifted bits must Alice and Bob compare to make the chance of missing her smaller than one in a million?Show your working
Working
Answer:
6.
Which resources does quantum teleportation of one qubit require?Circle one
1. AOne entangled pair shared in advance and two classical bits
2. BA copy of the qubit and one classical bit
3. CA faster-than-light channel and one entangled pair
4. DTwo entangled pairs and no classical communication
7.
At the end of a successful teleportation, Alice's original qubit no longer holds the state α|0⟩ + β|1⟩.Circle one
TrueFalse
8.
Explain in two or three sentences the difference between post-quantum cryptography and quantum cryptography.Short answer
9.
Telecom fibre loses 0.2 dB per kilometre. What fraction of photons survives 250 km? If a source launches 10⁹ photons per second, how many arrive each second?Show your working
Working
Answer:
10.
Two optical clocks are 1.00 m apart in height. Find the fractional frequency difference between them and how much time the upper clock gains in one year (3.156×10⁷ s).Show your working
Working
Answer:
11.
A SQUID responds to each flux quantum Φ₀ = h/2e threading its loop. Calculate Φ₀, then find the magnetic field that puts one flux quantum through a loop of area 1.0 mm².Show your working
Working
Answer:
12.
What lets nitrogen-vacancy centres in diamond work as magnetometers at room temperature?Circle one
1. AThey are superconducting at room temperature
2. BTheir spin state changes how brightly they fluoresce under green light, so a magnetic field's effect on the spin levels can be read optically
3. CThey store magnetic flux quanta like a SQUID
4. DThey emit squeezed light whenever a field is present
13.
Why can't ordinary optical amplifiers extend a QKD link, and what are three ways around the distance limit?Short answer
14.
Since 2019, LIGO has reduced its quantum noise by injecting light, whose phase uncertainty is pushed below the vacuum level.Fill in the blank

Part C Mad scientist challenge bonus

The spy's budget. Eve runs intercept-resend on a random fraction f of the photons in a BB84 session. (a) Show that the sifted-key error rate is Q = f/4 and that Eve knows for certain a fraction f/2 of the sifted bits. (b) Alice and Bob measure Q = 2.5 percent. What f is consistent with that, and about how many of 10,000 sifted bits could Eve know for certain? (c) For ideal single-photon BB84, the Shor–Preskill analysis says that a fraction r = 1 − 2h(Q) of the sifted key survives as secret key, where h(Q) = −Q·log₂Q − (1 − Q)·log₂(1 − Q) is the binary entropy. How many secret bits remain from the 10,000? (d) At what error rate does r reach zero, and what interception fraction f does that correspond to? (e) Why does the formula subtract 2h(Q) rather than simply discarding the bits Eve knows?

Working

Memory aid

Mnemonic Five S's and a price tag. For BB84, chant Send, Shrug, Sift, Spot-check, Shrink: Alice sends random bits in random bases; Bob shrugs and measures in random bases; they sift away mismatched bases; they spot-check a sample for errors, remembering the spy's rhyme 'wrong basis half the time, wrong bit in half of those, so a quarter of the key goes bad wherever Eve goes'; and they shrink the rest with error correction and privacy amplification. For teleportation, recite the price tag one-two-zero-c: one entangled pair, two classical bits, zero copies, and nothing faster than c. Use it in the lab: say one S aloud at each stage of a Spy Hunt round, and read the price tag back to any headline that promises teleportation.

In one line Unknown quantum states can't be copied, so spies leave fingerprints, states teleport only with classical help, and fragile quantum systems make superb sensors.