Why It's Impossible to Perform Kage Bunshin no Jutsu on a Quantum Bit

I’m quite sure a significant number of the audience reading this have watched an episode from the Japanese anime series called Naruto, or is at least familiar with the characters, terms, and jutsu (skills/techniques) involved in the series. Perhaps one of the most eye-catching skills that the main character Naruto used earlier on in the series is Kage Bunshin no Jutsu, or the Shadow Clone Technique.

In case you’re unfamiliar with Naruto, the series follows a young boy named Naruto Uzumaki who started out as a loner due to his inherent terrible experience as a little baby. The story really began when Naruto reached the age when he needs to train as a ninja. From there on, he met friends, teachers, and life-long mentors who are going to accompany him for the rest of the series. The entire Naruto timeline itself is divided into multiple series, following the development of Naruto.

Shadow Clone Technique

That said, as a ninja, Naruto has to master some special techniques, one of which is the Shadow Clone Technique. With the technique, Naruto essentially creates tons of clones/copies of himself, which can be used as an offense method to raid and overwhelm enemies, or alternatively a form of defense by acting as a decoy of the real Naruto.

Unlike the ordinary Clone Technique, the Shadow Clone Technique creates physically real copies of its users. Moreover, because they are visually identical, shadow clones are indistinguishable from their real user.

Tobirama Senju using the Shadow Clone Technique. Naruto: Shippūden Episode #375.

More importantly, shadow clones are great for gathering intelligence, as once their chakra (energy) returns to their original user, the original user will also gain whatever information they have gathered, hence simply transferring what they’ve learned back to the user without having the need to place them in danger.

Recalling the existence of this technique after not watching Naruto for so long reminded me of two things: how useful this is going to be, if ever possible, in real life and a computer’s ability to copy and paste information. In the process of copy-pasting, the information is cloned to a particular destination. Like the shadow clones, they are very much identical and are hard to distinguish from one another.

Copy and Paste

Though we are not going to go deep into technical details, the copy-paste mechanism is not a difficult feature to implement in many of our day-to-day computers, hence why most, if not all, computers have this inherent feature we all take for granted.

Behind the scene, the information we have on our computers is stored as bits, a fundamental unit which all of our computers rely on. A bit can take only one out of two possible values, a zero (0) or a one (1). In reality, these bits are implemented using electrical voltage or current pulse.

In the case of electrical voltage, a one (1) is determined when the voltage is about 5V (volts), or commonly called HIGH. On the flip side, a zero (0) is represented with near 0V or called LOW. Once the physical implementation is set, everything else is built upon it – including even the world’s most advanced programs.

The language that our computers speak is thus encoded as binary digits, although we don’t necessarily view them that way. When you read this text, a standard set of binary digits are structured to be understood as an “A” or “B” or the symbol π, and so on.

Quantum Computers

Enter the realm of Quantum Computers, arguably the bread and butter of 21st-century Physics, Engineering, and Computation. With the existence of Quantum Computers, the computers we use on a daily basis are dubbed “Classical Computers.” The word “classical” stems from the habit/tradition of calling non-Quantum Mechanical Physics, Classical Physics.

As much as Classical Physics suffice to comprehend the world around us, tracing its roots way back to figures like Isaac Newton, Classical Physics fail to foster Quantum Mechanical phenomena found in the realm of the very small, like subatomic molecules such as electrons, photons, etc.

The study of Quantum Mechanics exploded in the 20th-century with world-renown Physicists like Neils Bohr, Albert Einstein, Werner Heisenberg, Max Planck, Max Born, Wolfgang Pauli, Erwin Schrödinger, and many others. Their discoveries and ideas provide the foundation of Quantum Physics, an alien realm outside of Classical Physics.

Their findings are still continued by the world’s latest Scientists and Researches, who, in this 21st-century begin to exploit the Quantum-Mechanical properties exhibited by these particles. We’ve discussed this more thoroughly in a separate article, where you can also read about Quantum Computer’s impact on modern encryption schemes.

Quantum Bits

Quantum Computer’s fundamental units are called Quantum Bits, or qubits for short. Unlike classical bits, qubits are not implemented merely using transistors or electrical voltage. Instead, they’re implemented using subatomic particles whose properties accord to the laws of Quantum Mechanics that govern them.

D-Wave TwoX 1000 Qubit quantum annealing processor chip. Mwjohnson0, CC BY-SA 4.0, via Wikimedia Commons.

A vital property to that of a qubit is its ability to superpose, or the phenomenon of Superposition. While a classical bit can be in either a zero (0) state or a one (1) state, qubits can be in both states at once. This clearly seems unintuitive, as Quantum Mechanics can be at times.

Mathematically, a qubit $\left|\phi\right>$ is represented by the following quantity,

$\left|\phi\right> = \alpha \left|0\right> + \beta \left|1\right>$

where $\left|0\right>$ and $\left|1\right>$ represents the zero (0) and one (1) state respectively. The quantities $\alpha$ and $\beta$ are Complex Numbers that signifies the probability amplitudes. These amplitudes are what allows for Superposition as they are tweakable such that a qubit may be in part zero (0) and in part one (1) at the same time.

Nonetheless, the biggest takeaway is this: with $n$ bits, you can simulate up to $n$ states only. On the other hand, with $n$ qubits, you can represent up to $2^n$ states. A notable ability of qubits in Superposition is that they can quickly reach the desired state that contains information needed in a specific task, like prime factors of a number, the Quantum States of a chemical molecule, etc.

Despite the ability to contain the critical information needed in a task, one problem with qubits’ states is that it is very hard to extract, or at least needs a clever workaround. Extracting information from qubits naively may even cause the information to be lost, due to another Quantum phenomenon called Quantum Measurement.

No-Cloning Theorem

In some cases, the fact that the desired state is “hidden” among the entire combination of states and is hard to extract, some would think that it is easily solvable by simply copying the collection of information to another “blank” or empty qubits.

If possible, no one ever needs to dabble with the hassle of finding a workaround to smartly extract the fatal information, as you can simply pass the desired information to a separate qubit, and voila, you get infinite copies of the array of information and can thus extract separate information per copy. In theory, you wouldn’t have to worry about extracting the wrong information, since you can repeat the same process until you get the right answer.

Unfortunately, that is not the case. Physicists before the era of Quantum Computers have noticed that such maneuver is impossible even when Quantum Mechanics is within the realm of the theoretical.

No-cloning theorem’s first discovery is still unsure but dates back to a 1982 publication by Wootters and Zurek and by Dieks, a theorem proven by Giancarlo Ghirardi 18 months prior to the former, and James L. Park in 1970.

For the sake of completeness, allow me to include a short proof of the theorem provided in a Fall 2005 lecture note from a Berkeley course titled Quantum Information Science and Technology. The lecture note presented two proofs, but I prefer the second one for its intuitiveness.

Proof

Suppose there exists a unitary operator $U_{cl}$ that can indeed clone an unknown quantum state $\left|\phi\right> = \alpha \left|0\right> + \beta \left|1\right>$ . Then

$\left|\phi\right>\left|0\right> \stackrel{U_{cl}}{\longrightarrow} \left|\phi\right>\left|\phi\right> \\ \\ = (\alpha|0\rangle+\beta|1\rangle)(\alpha|0\rangle+\beta|1\rangle) \\ \\ =\alpha^{2}|00\rangle+\beta \alpha|10\rangle+\alpha \beta|01\rangle+\beta^{2}|11\rangle$

But now if we use $U_{cl}$ to clone the expansion of $\left|\phi\right>$ , we arrive at a different state:

$(\alpha|0\rangle+\beta|1\rangle)|0\rangle \stackrel{U_{cl}}{\longrightarrow} \alpha|00\rangle+\beta|11\rangle$ .

Here there are no cross terms. Thus we have a contradiction and therefore there cannot exist such a unitary operator $U_{cl}$ .

This seemingly obscure law can be broken down into a simpler analogy. Say that you’re in the heist of cheating from a genius friend of yours, but his/her answer sheet is hidden somewhere in a pile of other answer sheets. If you look through all the sheets and found your friend’s sheet, all you can do is take it out of the pile and obviously start to copy his/her answers. However, the moment you start writing, the head principal will magically appear in front of you and of course, forbid you from cheating.

In practice, the no-cloning theorem ensures that no other Quantum Mechanical laws are violated when dabbling with the Quantum States. It similarly maintains the fact that no quantum information can be transmitted faster than the speed of light, also called the no-communication theorem. Faster-than-light communication is indeed a consequence of Albert Einstein’s Special Relativity which places the speed of light as the upper limit for speeds of objects.

Closing Remarks

Given the previous discussion, we saw how the weird behaviors of Quantum Mechanics forbid a method of cloning information, contrary to that of Naruto’s Shadow Clone Technique. As a ninja like Naruto, you can gather information without the need to leave the comfort of your couch. However, if Naruto was a collection of qubits, this technique may very well be impossible in the realm of Quantum Ninja.

Perhaps the language that the Master speaks in this universe does not allow for such convenience and would rather see us find smarter solutions as modern creatures. Yet, we’re at least allowed to poke around the smallest fundamental units that make up the Master’s entire craft; at times get the chance to peek into the secret language. Hopefully, one day, we get to understand the language wholly and use it to understand the smallest legos that the Master used to build with.

Featured Image by Masashi Kishimoto (Naruto) via Naruto Fandom.