No-Cloning Theorem
Photon Polarized Along a Direction
Photon polarized along a direction
Quantum cryptography leverages the fundamental principle that the quantum state of an individual quantum system cannot be precisely measured.
Consider a photon polarized at an angle \theta impinging upon a beam splitter aligned along the x-axis.
A single measurement yields either +1 or -1, yet this outcome alone does not reveal the initial polarization angle \theta:
|\mathbf 1_\theta\rangle = \cos \left(\theta\right) |\mathbf 1_x\rangle + \sin \left(\theta\right) |\mathbf 1_y\rangle
This limitation is a basic of quantum security.
However, when dealing with a collection of identically prepared quantum systems, such as numerous photons all polarized at the same angle \theta, the situation is different:
|\mathbf 1_\theta\rangle \otimes |\mathbf 1_\theta\rangle \dots \otimes |\mathbf 1_\theta\rangle
By performing repeated measurements, we can statistically infer the initial quantum state.
A basic approach involves conducting a sufficiently large number of measurements with a polarizer oriented along the x-axis. This allows for an estimation of the probabilities of obtaining +1 or -1 outcomes:
\begin{aligned} & \mathcal P (+1) = \left|\langle \mathbf 1_x | \mathbf 1_\theta \rangle\right|^2 = \cos^2 \left(\theta\right) \\ & \mathcal P (-1) = \left|\langle \mathbf 1_y | \mathbf 1_\theta \rangle\right|^2 = \sin^2 \left(\theta\right) = 1 - \mathcal P (+1) \end{aligned}
Assuming a prior knowledge that the polarization is linear, these probabilities enable the determination of \pm \theta, modulo \pi:
\theta = \pm \arccos \left( \sqrt{\mathcal P (+1)} \right) + n\pi
This \pi ambiguity is inconsequential because linear polarization is defined by a direction, and therefore \theta and \theta + \pi are indistinguishable.
To resolve the remaining ambiguity between positive and negative angles, a supplementary measurement with the polarizer in a non-collinear orientation is required.
For example, we can orient the polarizer along the direction \theta for which |\mathbf 1_\theta\rangle would be transmitted (+1).
If the actual polarization is |\mathbf 1_{+\theta}\rangle and we measure with a polarizer at +\theta, the probability of transmission (+1) is |\langle \mathbf 1_{+\theta} | \mathbf 1_{+\theta} \rangle|^2 = 1 we will consistently obtain +1 outcomes.
If the actual polarization is |\mathbf 1_{-\theta}\rangle and we measure with a polarizer at +\theta, the probability of transmission (+1) is |\langle \mathbf 1_{+\theta} | \mathbf 1_{-\theta} \rangle|^2 = \cos^2((+\theta) - (-\theta)) = \cos^2(2\theta).
If \theta \neq 0 and \theta \neq \pi/2, then \cos^2(2\theta) < 1. In this case, we will observe a mix of +1 and -1 outcomes. This is because there is a non-zero probability of the photon being projected onto the orthogonal state |\mathbf 1_{-\theta}\rangle when the initial polarization is |\mathbf 1_{-\theta}\rangle and the measurement is along |\mathbf 1_{+\theta}\rangle.
If \theta = 0 or \theta = \pi/2, then \cos^2(2\theta) = 1. However, in these cases, \theta = -\theta (modulo \pi), so the sign ambiguity is already trivial.
To confirm that in the second case the polarization was -\theta, we can change the polarizer direction and orient it along -\theta and then we consistently obtain +1 outcomes, and it confirms that the initial polarization angle was -\theta, as |\langle \mathbf 1_{-\theta} | \mathbf 1_{-\theta} \rangle|^2 = 1.
If the polarization of photons is unknown and could be linear, circular, or elliptical, we can still derive its nature through multiple measurements, assuming a sufficiently large collection of identically polarized photons.
Quantum cloning
Quantum cloners, hypothetical devices capable of perfectly duplicating quantum states, would undermine quantum cryptography.
The no-cloning theorem, proven by Wootters and Zurek in 1982, states that perfect cloning of an arbitrary quantum state is fundamentally impossible, ensuring the security of many quantum cryptography protocols. This theorem, applicable to any qubit, will be demonstrated for polarized photons.
While laser amplifiers appear to create copies of input photons, preserving frequency, direction, and polarization, we must examine if they function as perfect photon cloners for arbitrary input polarizations.
Consider a hypothetical perfect cloner that produces two identical copies of an input photon at outputs \text{out}_1 and \text{out}_2:
| \boldsymbol \Psi_{\text{in}} \rangle \Longrightarrow | \boldsymbol \Psi_{\text{out}} \rangle = | \boldsymbol \Psi_{\text{out}} \rangle_1 \otimes | \boldsymbol \Psi_{\text{out}} \rangle_2
A perfect cloner must function for all input polarizations, including a general polarization like \theta, not just specific x or y polarizations:
\begin{aligned} & | \boldsymbol \Psi_{\text{in}} \rangle = |\mathbf 1_x\rangle \\ & | \boldsymbol \Psi_{\text{in}} \rangle = |\mathbf 1_y\rangle \\ & | \boldsymbol \Psi_{\text{in}} \rangle = |\mathbf 1_\theta\rangle = \cos \left(\theta\right) |\mathbf 1_x\rangle + \sin \left(\theta\right) |\mathbf 1_y\rangle \end{aligned}
where we also express the polarization along \theta on basis along the directions x and y.
We can use the x, y basis representation of a \theta-polarized photon to describe the output of this hypothetical cloner:
\begin{aligned} | \boldsymbol \Psi_{\text{out}} \rangle = & | \boldsymbol \Psi_{\text{out}_1} \rangle \otimes | \boldsymbol \Psi_{\text{out}_2} \rangle = |\mathbf 1_\theta\rangle_1 \otimes |\mathbf 1_\theta\rangle_2 \\ = & \left(\cos \left(\theta\right) |\mathbf 1_x\rangle_1 + \sin \left(\theta\right) |\mathbf 1_y\rangle_1\right) \otimes \left(\cos \left(\theta\right) |\mathbf 1_x\rangle_2 + \sin \left(\theta\right) |\mathbf 1_y\rangle_2\right) \\ = & \cos^2\left(\theta\right) |\mathbf 1_x\rangle_1 \otimes |\mathbf 1_x\rangle_2 + \cos\left(\theta\right)\sin\left(\theta\right)|\mathbf 1_x\rangle_1\otimes | \mathbf 1_y\rangle_2 \\ & +\sin\left(\theta\right) \cos\left(\theta\right)|\mathbf 1_y\rangle_1 \otimes | \mathbf 1_x\rangle_2 + \sin^2\left(\theta\right)|\mathbf 1_y\rangle_1\otimes |\mathbf 1_y\rangle_2 \\ = & \cos^2\left(\theta\right) |\mathbf 1_x,\mathbf 1_x\rangle + \cos\left(\theta\right)\sin\left(\theta\right)|\mathbf 1_x, \mathbf 1_y\rangle \\ & +\sin\left(\theta\right) \cos\left(\theta\right)|\mathbf 1_y, \mathbf 1_x\rangle + \sin^2\left(\theta\right)|\mathbf 1_y,\mathbf 1_y\rangle \end{aligned}
A realistic quantum device acting as a perfect cloner must adhere to quantum mechanics principles. Specifically, its operation is governed by a Hamiltonian, and the relationship between input and output states is described by a linear evolution operator \mathbf U.
This linearity dictates that the output corresponding to a superposition of inputs must be the superposition of individual outputs:
\begin{array}{|l|l|} \boldsymbol \Psi_{\text{in}} \rangle & \boldsymbol \Psi_{\text{out}} \rangle \\ |\mathbf 1_x\rangle & |\mathbf 1_x\rangle_1 \otimes |\mathbf 1_x\rangle_2 \\ |\mathbf 1_y\rangle & |\mathbf 1_y\rangle_1 \otimes |\mathbf 1_y\rangle_2 \\ | \alpha |\mathbf 1_x\rangle + \beta |\mathbf 1_y\rangle & \alpha \left(|\mathbf 1_x\rangle_1 \otimes |\mathbf 1_x\rangle_2\right) + \beta \left(|\mathbf 1_y\rangle_1 \otimes |\mathbf 1_y\rangle_2\right) \end{array}
This principle applies when considering a \theta-polarized photon as a superposition of x and y polarized photons, \alpha = \cos\left(\theta\right), \beta = \sin\left(\theta\right):
\begin{aligned} \boldsymbol \Psi_{\text{out}} \rangle = & \cos\left(\theta\right) \left(|\mathbf 1_x\rangle_1 \otimes |\mathbf 1_x\rangle_2\right) + \sin\left(\theta\right) \left(|\mathbf 1_y\rangle_1 \otimes |\mathbf 1_y\rangle_2\right) \\ = & \cos\left(\theta\right) |\mathbf 1_x, \mathbf 1_x\rangle + \sin\left(\theta\right) |\mathbf 1_y, \mathbf 1_y\rangle \end{aligned}
The contradiction arising from the two perspectives, combined with the established principle of linear quantum evolution, leads to the conclusion that perfect quantum cloners, capable of duplicating any state within a two-dimensional input space, are impossible.
This no-cloning theorem is a fundamental principle in quantum physics. It prevents the complete determination of a single quantum system’s state and is essential for the security of quantum cryptography.
Furthermore, it prevents exploiting quantum non-locality for faster-than-light communication.
The no-cloning theorem’s scope is important to understand. While it prohibits cloning every state within a two-dimensional space, it does not preclude perfectly cloning specific, predetermined states. Therefore, if a scheme requires cloning only a subset of quantum states, the no-cloning theorem does not immediately invalidate it.
For example, we can consider a scenario where you are certain that the input photon polarization is always horizontal, denoted as |\mathbf H\rangle. In this specific case, a device that consistently outputs two horizontally polarized photons, regardless of the input (or even without any input), acts as a perfect cloner for this predetermined input state |\mathbf H\rangle.
This does not contradict the no-cloning theorem because the theorem prohibits cloning of arbitrary quantum states, not specific, predefined states like |\mathbf H\rangle in this restricted scenario.
References
WOOTTERS, William K., ZUREK, Wojciech H., 1982. A single quantum cannot be cloned. Nature, 299 (5886), 802–803.