Don't believe in superficial analogies or risk being eaten alive
Most explanations of qubits are based on misleading analogies that obscure more than they reveal. Let's clear up this confusion and show how qubits really work by comparing them to classical probability and then expanding on amplitudes, entanglement, and interference. If you're looking for a clear introduction to quantum computing without the usual myths, start here.
by Frank ZickertSeptember 30, 2025
If any one unwarily draws into close and hears the singing of the Sirens, his wife and children will never welcome him home again, for they sit in a green field and warble him to death with the sweetness of their song.
Superficial analogies in Quantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. Learn more about Quantum Computing sound appealing. But they are often useless (e.g., the A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit has wavelike properties), misleading (e.g., when one entangled particle changes, the other instantly changes too. Does this mean it is faster than the speed of light?), or simply incorrect (e.g., a A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit is in multiple states at once). Such analogies create the false impression that learning quantum computing is easy.
Not Faster Than The Speed Of Light
This doesn't mean analogies are a bad thing. I love analogies. Rather, it means analogies must offer a genuine conceptual overlap that is useful.
Because if you skip analogies entirely, you'll be exposed to the rocks of pure mathematical definitions, against which the deep blue waves of Amphitrite beat with Homer, 2022, FORGOTTEN BOOKS, ;
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It is unreasonable to teach a new concept based solely on unfamiliar mathematical definitions. Equations require thorough analysis, which is exactly the opposite of what is helpful when familiarizing oneself with a topic for the first time.
Fortunately, there is a safe way. Even more fortunately, you don't have to be tied to a mast to follow it.
Figure 1 Tossing a coin, hoping for it to be fair
This safe way is the conceptual analogy of a A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit as a A probabilistic system is one where outcomes are not fixed but occur with certain probabilities due to inherent randomness or uncertainty. Instead of producing a single deterministic result, it describes possible states and their likelihoods. Such systems are modeled using probability theory to predict behavior and quantify uncertainty. Learn more about Probabilistic System that is grounded in simple and comprehensible mathematics. A A probabilistic system is one where outcomes are not fixed but occur with certain probabilities due to inherent randomness or uncertainty. Instead of producing a single deterministic result, it describes possible states and their likelihoods. Such systems are modeled using probability theory to predict behavior and quantify uncertainty. Learn more about Probabilistic System is a system in which results cannot be predicted with certainty, but only with likelihoods.
For example, when you toss a coin, you cannot predict whether it will land on heads or tails. You can only say that each outcome has a probability, assuming the coin is fair. The system is not deterministic but is defined by a distribution of possible outcomes.
A probability distribution assigns a number between and to each outcome, whereby the probabilities must add up to . With a fair coin, this means
If we define a random variable that takes value for heads and for tails, then the probability distribution for is as depicted in ?.
Figure 2 A binary random variable (e.g. fair coin)
This compactly represents the core components of a A probabilistic system is one where outcomes are not fixed but occur with certain probabilities due to inherent randomness or uncertainty. Instead of producing a single deterministic result, it describes possible states and their likelihoods. Such systems are modeled using probability theory to predict behavior and quantify uncertainty. Learn more about Probabilistic System: the possible outcomes, their probabilities, and the applicable rule.
A A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit works on a similar principle, but instead of ordinary probabilities, it uses something that is literally more complex: In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude See ?. Mathematically, this is written as .
Here, is a basis state. Learn more about and is a basis state. Learn more about are the possible outcomes, and and are complex numbers called In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitudes. But, You never directly see or .
The rule that links In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude to probability is simple: square the magnitude. The probability of measuring is a basis state. Learn more about is , and the probability of measuring is a basis state. Learn more about is . That is what makes A quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results. Learn more about Quantum System probabilistic. It is The act of In quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition. Learn more about Measurement that forces a random outcome according to the invisible A quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin. Learn more about Quantum State.
Figure 3 The qubit as a random variable
Of course, A probabilistic system is one where outcomes are not fixed but occur with certain probabilities due to inherent randomness or uncertainty. Instead of producing a single deterministic result, it describes possible states and their likelihoods. Such systems are modeled using probability theory to predict behavior and quantify uncertainty. Learn more about Probabilistic Systems do not consist of just a single random variable. They only become really interesting when we consider more than one random variable. Let's assume, for example, that we toss two coins, and . Normally, we assume that the coins are independent of each other, i.e., the result of one coin has no influence on the result of the other.
So, knowing the result of coin tells you nothing about coin . There are four different outcomes and they all occur with the same probability.
Now suppose the two coins are glued together so they always show the same face. The sample space is the same, but the probability distribution changes to
Figure 4 How to create correlation classically
The outcomes of these two coins are perfectly correlated. Classically, correlation means that two random variables are dependent on each other. For example, if coin lands on heads, coin also lands on heads; if lands on tails, also lands on tails. Each coin taken individually still appears random ( heads, tails), but thejoint probability distribution shows a perfect correlation:.
A A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit enables a similar but ultimately deeper connection between systems. Let's consider two A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bits. If they are prepared independently of each other, each in the same superposition, the joint state is
When measured, all four outcomes are equally likely , just as with two uncorrelated coins. The outcome of the first A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit provides no information about the second.
But if the two A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bits are prepared in an entangled state, such as
the probabilities look strikingly different:
Each individual A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit still looks random, with an equal probability of being or . But together, they behave like the coins stuck together: the results are perfectly correlated. If you measure the first A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit and get is a basis state. Learn more about , the second is also is a basis state. Learn more about . If the first is is a basis state. Learn more about , the second is guaranteed to be is a basis state. Learn more about .
This is the same situation we observed with two classic coins stuck together. Yet, with classic coins, one can always imagine a hidden rule or cause that explains why both coins show the same side, for example, that they are glued together.
For A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit such a hidden instruction cannot explain the full range of correlations that appear when we choose different In quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition. Learn more about Measurement settings. Entangled A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit can violate the limits set by classical probability theory, showing that their correlation is not just ordinary dependence but a uniquely quantum phenomenon.
Bell Inequality
The most surprising and powerful feature of A quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results. Learn more about Quantum System becomes apparent when we ask ourselves how different paths combine. In classical probability theory, paths refer to the different ways in which a result can occur.
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For example, suppose you want to know the probability of getting exactly one head when tossing two coins. There are two possible paths to this outcome: (heads, tails) and (tails, heads). Since the coins are independent of each other and each pair has a probability of ( ), we add them together:
The rule is simple: when different paths lead to the same event, we add their probabilities.
Quantum mechanics changes this rule. Each path is assigned a A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex NumberIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude not just a probability. To determine the probability of an outcome, we must first add up the In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude of all paths leading to that outcome and then take the absolute square of that sum to calculate the resulting probability.
Figure 5 Binomial expansion
And if you recall high-school math, it will strike you in the eye. Think of the simple binomial expansion you already know:
That last term, , is the cross term. It comes from the fact that when you square a sum, the terms don't just square individually; you also pick up the products of the different pieces.
But that's not all. If In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude were A real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve . Learn more about Real Number, the cross term would always be positive and therefore, increases the result.
But in quantum mechanics, and are not A real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve . Learn more about Real Number They are A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex NumberIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude of . And we don't square them directly; we take the squared magnitude (the absolute value squared). The rule for A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex Number is:
where is the The complex conjugate of a A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex Number, where and , is defined as . So you flip the sign of the imaginary part and leave the real part unchanged. Learn more about Complex Conjugate of (the same number but with the sign of the imaginary part flipped). So instead of writing , we must write
Now it looks a little more familiar. The first two terms are and , just like the squares of the individual paths. However, the last two terms look a little strange at first glance. Note, however, that they are complex conjugates of each other, because complex conjugation applies to each factor:
And that means the sum is always a A real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve . Learn more about Real Number When you multiply a A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex Number by its complex The complex conjugate of a A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex Number, where and , is defined as . So you flip the sign of the imaginary part and leave the real part unchanged. Learn more about Complex Conjugate the imaginary parts cancel and you're left with a A real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve . Learn more about Real Number, non-negative value, as summarized in ?. That's why and are ordinary real probabilities, even though and themselves may be A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex Number
Figure 6 Multiplying a complex number by its conjugate
The cross term follows the same logic. Since and are conjugates of each other, their sum doesn't carry any imaginary part. Instead, it reduces neatly to
So the whole formula is
Figure 7 Born's Rule
In short, the probability of measuring a A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. Learn more about Quantum Bit is the sum of the individual contributions plus an Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage. Learn more about Interference term that depends on the relative A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves. Learn more about Quantum Phase of and , the In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude of the possible outcomes. If the A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves. Learn more about Quantum Phase match, the A real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve . Learn more about Real Number part of is positive and the probability is increased. If the A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves. Learn more about Quantum Phase are opposite, it is negative and the probability is reduced. If the A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves. Learn more about Quantum Phase differ by , the real part disappears and there is no Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage. Learn more about Interference at all.
The cross term is therefore not just an arbitrary remnant of the binomial expansion. It specifies exactly how the relative A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves. Learn more about Quantum Phase of the paths influences the final probability. This is why Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage. Learn more about Interference plays a central role in Quantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. Learn more about Quantum Computing probabilities do not simply add up, but depend on how A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers. Learn more about Complex NumberIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitude are arranged in the complex plane.
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In summary, quantum theory replaces directly specified probabilities with complex In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitudes, so that measurement results are determined by squared quantities (superposition), joint states cannot be factorized (entanglement), and probabilities arise from squared sums of In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions. Learn more about Amplitudes rather than from direct addition (interference), which means that, strictly speaking, quantum randomness is richer than classical randomness.
