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physics
Interactive Course
Interactive chapters from intuition to mastery
Structured Lessons
Eight modules with formulas and self-checks
Quantum Brain
Navigate lessons, laws, gates, devices, and tools
Guided Experiments
Hands-on circuits that teach one idea each
simulator
Circuit Lab
Build circuits, run them, and see the results
Gate Reference
Quick reference for all quantum gates
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Cryostat Studio
3D cryostat design and simulation
Component Catalog
Browse all cryostat components
System Checks
Check your design for errors
Quantum Principles, Theorems, and LawsNo-Cloning and Information Limits0/6 mastered
Module overview
Quantum Principles, Theorems, and Laws0/6 mastered
How to Read Quantum Results
Superposition, Born Rule, and Measurement
Schrodinger Dynamics and Commutator Laws
Bell, Contextuality, and No-Signalling
No-Cloning and Information Limits
Symmetry, Statistics, and Classical Limits
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Entanglement and Multi-Qubit Thinking
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Quantum Hardware Foundations
Module 4
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Quantum Principles, Theorems, and Laws
How to Read Quantum Results
Superposition, Born Rule, and Measurement
Schrodinger Dynamics and Commutator Laws
Bell, Contextuality, and No-Signalling
No-Cloning and Information Limits
Symmetry, Statistics, and Classical Limits
Previous module
Entanglement and Multi-Qubit Thinking
Next module
Quantum Hardware Foundations
Home/Lessons/Quantum Principles, Theorems, and Laws/No-cloning and quantum information limits
5Lesson 5 of 6inQuantum Principles, Theorems, and Laws

No-cloning and quantum information limits

The no-cloning family of results explains why unknown quantum information cannot be copied, broadcast, deleted like a classical file, or read out arbitrarily well.

These limits are not side notes; they are design constraints for quantum cryptography, teleportation, error correction, and quantum networking. They also explain why a simulator can display the state vector for learning, while real hardware must infer states indirectly through repeated preparation and measurement.

Quantum Brain

Follow this lesson into the surrounding principles, theorems, tools, and modules.

No-CloningTheoremNo-BroadcastingTheoremHolevo BoundTheoremTeleportationProtocol
1
The intuition
Start with the plain-language idea

Classical information can be copied almost without thought. Quantum information cannot. The no-cloning theorem says there is no universal physical operation that perfectly copies an unknown arbitrary state. The reason is not poor engineering; it is linearity. A device that copied ∣0⟩ and ∣1⟩ would not also copy their unknown superposition correctly.

2
See it concretely
A real example before the abstraction

Teleportation is the clean example. The protocol transfers an unknown state using entanglement and two classical bits, but Alice's Bell measurement destroys the original. The result is movement, not duplication. Superdense coding shows the complementary resource accounting: entanglement changes communication capacity without allowing arbitrary state readout.

3
Tempting but wrong
The mistake most people make
Tempting but wrong
It is tempting to think no-cloning means copying any quantum state is impossible. That is too broad. You can copy known computational-basis states. What is impossible is a universal copier for arbitrary unknown states.
Also watch out for
✕Confusing state transfer with state copying.
✕Assuming a full state vector can be read from one physical qubit.
4
The precise version
Now with the formal detail
U∣ψ⟩∣0⟩=∣ψ⟩∣ψ⟩ for all ∣ψ⟩ is impossible

The no-cloning theorem applies to arbitrary unknown states. Known orthogonal states can be copied because a measurement can distinguish them without ambiguity. No-broadcasting extends the idea to mixed states: noncommuting quantum states cannot be broadcast into two systems with the same local marginals. The no-deleting theorem gives the time-reversed-looking companion: unknown quantum information cannot simply be erased from one copy while preserving another by a universal operation. The Holevo bound adds a measurement-side limit: quantum systems may be described by continuous amplitudes, but the accessible classical information from an ensemble of quantum states is bounded.

Check your understanding
Why can known orthogonal states be copied but arbitrary unknown states cannot?
Think about this against what you just read.
How does teleportation respect no-cloning?
Think about this against what you just read.
Try it yourself

Open the simulator and see this concept in action. Watch how the state changes and compare it to what you just learned.

▶ Inspect teleportation↗ MIT OCW 6.845: no-cloning theorem lecture notes↗ Harvard Physics 160: quantum information lecture notes↗ Wootters and Zurek, A single quantum cannot be cloned↗ IBM Quantum Learning: limitations on quantum information↗ Barnum et al., no-broadcasting theorem↗ Pati and Braunstein, no-deleting theorem↗ MIT OCW MAS.865J: quantum channels and Holevo theorem
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What does the no-cloning theorem forbid?
5 of 6 in Quantum Principles, Theorems, and Laws
Previous
Bell, Contextuality, and No-Signalling
Why entanglement defeats local hidden-variable shortcuts without sending messages
~10 min
Next
Symmetry, Statistics, and Classical Limits
How Pauli exclusion, symmetry, and correspondence connect particles to devices
~9 min