Wiesner State

Quantum states used in Wiesner’s quantum money scheme, randomly chosen from conjugate bases to prevent counterfeiting.

Wiesner states are the quantum states used in Stephen Wiesner’s pioneering 1983 quantum money proposal. Each state is randomly chosen from two conjugate bases, making them impossible to copy without knowing the basis.

The States

For each qubit in a quantum “bill,” the bank randomly chooses one of four states:

State	Basis	Bit Value
$∥0 ⟩$	Computational (Z)	0
$∥1 ⟩$	Computational (Z)	1
$∥ + ⟩ = \frac{1}{2} (∥0 ⟩ + ∥1 ⟩)$	Hadamard (X)	0
$∥ - ⟩ = \frac{1}{2} (∥0 ⟩ - ∥1 ⟩)$	Hadamard (X)	1

Why They’re Unforgeable

A counterfeiter trying to copy a Wiesner state faces a dilemma:

Unknown basis: They don’t know if the state is in the Z or X basis
Measurement disturbs: Measuring in the wrong basis destroys information
No-cloning: Can’t copy without knowing the state

If the counterfeiter guesses the wrong basis:

Measuring $∣0 ⟩$ or $∣1 ⟩$ in X basis → random result
Measuring $∣ + ⟩$ or $∣ - ⟩$ in Z basis → random result

Any attempt to learn the state changes it, making counterfeiting detectable.

The Protocol

Bank Creates Money

Generate random serial number $s$
For each of $n$ positions, randomly choose basis (Z or X) and bit (0 or 1)
Prepare the corresponding quantum state
Store $(s, bases, bits)$ in secret database
Bill = serial number + $n$ qubits in Wiesner states

Verification

Customer presents bill with serial number $s$
Bank looks up the bases for $s$
Measures each qubit in the recorded basis
If all measurements match recorded bits → valid
If errors → counterfeit detected

Security Analysis

If a counterfeiter tries to copy by measuring and re-preparing:

Probability of guessing one basis correctly: 1/2
Probability of all $n$ bases correct: $(1/2)^{n}$
For $n = 20$ : Success probability $\approx 1 0^{- 6}$

Historical Significance

Wiesner’s 1983 paper (written in 1970, published later) was foundational:

First application of quantum mechanics to cryptography
Introduced conjugate coding
Inspired BB84 quantum key distribution
Showed quantum information has unique security properties

Limitations

Limitation	Issue
Bank verification	Must return to bank to verify
Quantum memory	States must be preserved
Single use	Verification reveals basis information

Connection to BB84

BB84 directly uses Wiesner’s conjugate coding idea:

Same four states
Random basis choice
Security from measurement disturbance

BB84 is essentially “Wiesner states for key distribution.”

See also: Quantum Money, BB84 Protocol, No-Cloning Theorem, Conjugate Coding