On Quantum Entanglement
In this post, I aim to outline the theoretical foundations for one of the most canonical quantum phenomena — entanglement. I aim to do this in a way that can be fully understood by anyone with a basic understanding of arithmetic and (very basic) linear algebra. Consequently, the majority of this post will be comprised of (hopefully understandable) descriptions of the background knowledge needed to fully understand quantum entanglement from a mathematical/scientific perspective. These tools come from both quantum mechanical (e.g., quantum states, superposition, observables) and mathematical (e.g., complex numbers, tensor products, eigenvectors/values) worlds. I will begin with the mathematical concepts and follow with quantum mechanical concepts, concluding with a description of entanglement that combines everything together.
The Math!
In this section, I will introduce several mathematical concepts that are necessary for a basic understanding of quantum entanglement. It should be noted that these concepts are not exhaustive. In other words, there is a lot of other math that is important in quantum mechanics, but will not be covered in this post. For simplicity, I focus on the bare minimum of required mathematical concepts, but attempt to cover them in depth.
Complex Numbers
The first mathematical concept relevant to entanglement is that of a complex number. A complex number can be denoted as follows.
In this equation, z is the complex number, while a and b are real number coefficients. On the right side of the equality, a is referred to as the “real component” of z , while ib is referred to as the “imaginary component”.
Complex Conjugation and Magnitude
Once you understand complex numbers, it is also quite easy to understand two important operations on them: complex conjugation and finding the magnitude. To find the complex conjugate of a complex number z , one simply flips the sign of the imaginary component. For example, given the expression for z above, the complex conjugate of z can be formulated as follows (i.e., the complex conjugation operation is represented by the overline on z).
If one understands complex conjugation, finding the magnitude of a complex number is not much more complicated. In particular, finding the magnitude involves multiplying the complex number by its complex conjugate, then taking the square root of this product. This expression takes the following form.
Interestingly, as shown above, multiplying a complex number by its complex conjugate always yields a real-valued result. From this basic definition and formulation of major operations on complex numbers, one can gain a basic grasp on the behavior of complex numbers. For example, arithmetic on complex numbers is performed using the normal arithmetic rules (e.g., i² = sqrt(-1)² = -1). For more details on operations for complex numbers, please read this blog post.
Polar Representation of Complex Numbers
Although the basic form of a complex number is not difficult to grasp, the confusion (at least for me) stems from the different possible forms of complex numbers that one may encounter. In some cases, one may encounter complex numbers in their polar form shown below, where the equality (i) is a consequence of Euler’s formula.
Within the form above, r encodes the magnitude of the complex number (i.e., the magnitude of zis just r²!). Interestingly, multiplying z by any complex number with r = 1 (i.e., these are typically referred to as “phase factors”) will not alter the magnitude of z, as the the value of r will remain unchanged when z is multiplied by this complex number. An understanding of phase factors is important for understanding quantum mechanical concepts, as they may arise in numerous scenarios (e.g., phase factors can be used to explain why qubits are represented with only two free variables).
Vectors of Complex Numbers (Bras and Kets)
Once you understand complex numbers, vectors of complex numbers are not much different. In fact, the main peculiarity of understanding such vectors within quantum mechanics is usually learning how to denote them. First, we can start by constructing an n-dimensional, complex column vector as follows.
To anyone who is unfamiliar with quantum mechanics, this denotation (i.e., denoting vectors with brackets and angle brackets) may seem peculiar. However, the vector notation shown above is simply a common way of denoting column vectors within quantum mechanics, which was invented/popularized by Paul Dirac. We refer to the bracketed vector shown above as a “ket” vector. Ket notation is just a common way of denoting an n-dimension column vector, where each entry within the vector is a complex number. But, why do we call it a ket? Well, this naming convention becomes much more clear if we define the dual of a “ket” vector— the “bra vector”.
As can be seen, each ket vector (i.e., a column vector in the n-dimensional complex vector space) has a corresponding “bra” vector, which is a row vector of equal dimension. Given an arbitrary ket vector, its corresponding bra vector can be produced by taking the conjugate transpose of the ket vector (i.e., transpose the vector and take the complex conjugate of every entry in the vector), denoted by the dagger superscript in the equation above. As a result, each column vector, or ket, has a corresponding row vector within the dual space of bra vectors. In quantum mechanics, we always use the construct of “bras” and “kets” (i.e., notice that “bra” + “ket” → “bracket”) to denote complex vectors. For example, the state of a quantum mechanical system is typically represented as an n-dimensional ket vector.
Inner Products
The idea of bras and kets make the notation for inner products of complex vectors very simple and intuitive. See the example below for a demonstration of the inner product of arbitrary bra and ket vectors. Notice that an inner product must be computed between a row vector and a column vector (i.e., between a bra and a ket!). The way we denote bras and kets makes this easy to remember.
In a similar vein, the magnitude of a complex vector can be easily computed using the inner product operation of bras and kets; see below (i.e., I use the squared magnitude to avoid writing a bunch of square roots throughout the equation).
Eigenvectors, Eigenvalues, and Bases
In addition to the simple explanation of complex vector spaces provided above, there are several concepts within linear algebra that will be useful for understanding entanglement and quantum mechanics in general. The first concept is that of a basis. Given an n-dimension complex vector space, a basis is formed by a set of n linearly independent vectors within the space. We call this set of vectors a basis because any vector within the space can be written as a linear combination of vectors in the basis. In other words, the basis spans the entire n-dimensional complex vector space. For more information on bases, I recommend watching this video.
Another useful concept within linear algebra is the idea of eigenvalues and eigenvectors of a linear operator (e.g., a matrix). Given some linear operator, the eigenvalues and eigenvectors of this operator can be defined with the following identity.
As can be seen above, an eigenvector is simply a vector for which the application of the linear operator simplifies to multiplication by a constant. The constant within this multiplication is the eigenvector’s associated eigenvalue. Many of such eigenvectors can exist for a linear operator, but no more than z orthogonal eigenvectors may exist, where z is the minimum of the number of columns and the number of rows for the linear operator. There is a lot more to eigenvalues and eigenvectors than this simple definition. However, there are many resources online that explain this concept much better than I can within this condensed post. For example, I recommend watching this video or reading this blog post.
Kronecker Products
A Kronecker product is an operation applied on two linear operators of arbitrary size. Informally, it is the generalization of the outer product to the space of matrices. The Kronecker product can be formalized as shown below.
In words, the Krocker product takes as input linear operators of size (m x n) and (p x t), then outputs a block matrix of dimension (mp x nt). A visualization of the Kronecker product is shown below.
Kronecker products are not horribly difficult to understand. Furthermore, it should be noted that Kronecker products can also be applied to vectors (i.e., the inputs have abitrary dimension), as shown in visualization below.
Kronecker products — and similarly tensor products — arise practically in numerous different scenarios. For example, given two quantum states (e.g., spin states or qubits), these quantum states can be combined into a single system by taking the Kronecker of their state vectors. Similarly, given two observables (i.e., hermitian operators), these observables can be combined into a single observable on a multi-particle system using the Kronecker product. However, because we have not introduced any notions of quantum states, observables, or quantum systems, it is more than likely that neither of those sentences make any sense. So, let’s learn some basic quantum mechanics!
The Science!
In this section, I will provide an (extremely) brief introduction to basic ideas in quantum mechanics. Again, I cover only the minimal concepts that are required to understand entanglement at a high level. Therefore, this introduction to quantum mechanical concepts is in no way exhaustive.
A Quick Disclaimer…
For those unfamiliar with quantum mechanics, the concepts outlined in this section will be somewhat puzzling. In general, quantum mechanics studies the behavior of very small objects, which is drastically different from what we, as humans, perceive in the world around us. As a result, this behavior is oftentimes counterintuitive. The most confusing aspect of the behavior of quantum systems (in my opinion) is that they are non-deterministic (i.e., governed by probabilities). If we prepare a particle within a certain quantum state then make a measurement on this particle, we may get differing results each time this procedure is repeated. Such non-deterministic behavior does not make sense in classical mechanics. For example, if we measure the mass of an object multiple times in a row, we expect to obtain the same result every time. The relationship between the state of a system and a measurement of that system is fundamentally different in quantum mechanics. So, one must embrace the peculiarity of the subject, and do one’s best to develop novel intuition for such peculiar concepts.
Quantum States
Quantum states are fundamentally different from classical states because knowing a quantum state does not imply that we know everything about it. Namely, the behavior of this state is non-deterministic — we can only know the probabilities associated with different possibilities for the state. Generally, a quantum state, in its simplest form, is just a ket. With this in mind, a quantum state can be written with respect to some basis as follows.
There are a few important things to understand within this equation. First of all, the kets that form our basis (i.e., vectors on the right-hand side of the quation above) are all orthogonal to each other (i.e., this is part of the definition of a basis) and there exists n kets within our basis. Furthermore, all of our coefficients (i.e., scalar lambdas in the equation above) are simply complex numbers. We typically refer to these coefficients as “probability amplitudes” (i.e., this will be explained shortly). Therefore, all we have done in this equation is write our quantum state as a linear combination of kets that form an arbitrary basis within our complex vector space. Because our quantum state is just a ket vector, there is nothing peculiar about this. From the previous discussion, you should know that any vector within a vector space can be expressed as a linear combination of vectors that form a basis in that space.
Measurements
So, when does this get interesting? Well, the interesting part of a quantum state is how we choose the basis. In particular, we construct this basis such that each of its vectors represent a possible state for our system. Therefore, a quantum state is simply a linear combination of its possible states. Although this statement may seem completely absurd, remember that there is a huge difference between our quantum state and what we get as a result when this quantum state is measured. Namely, the result of a measurement will not be a linear combination of possible states (i.e., the quantum state as shown in the equation above). Rather, it will be one of the vectors in our basis. When we measure the quantum state, this measurement will cause the state to “collapse” to one of the states in its basis (i.e., the measurement perturbs the state and makes it something else!). Which one? The answer to this question is non-deterministic. But, we can get the probabilities that the quantum state will collapse to any one of the states in its basis as follows.
As shown above, the probability that our quantum state will collapse to a certain state in its basis when measured is given by the squared magnitude of the probability amplitude associated with that state (i.e., highlighted in blue in the equation above). So, while probability amplitudes are not probabilities, their magnitude is used to express probabilities associated with measurements of the quantum state, thus revealing why we call them probability amplitudes. Because probabilities within our quantum system are defined as shown above, we generally assume the quantum state is a unit vector (i.e., so that the probabilities sum to one), yielding the identities shown below.
Superposition
If you paid close attention to the section above, you will have noticed a very important detail in the definition of quantum states. When we measure our quantum state, it has a certain probability of existing within any of the basis states (though some of these probabilities may be 0). In other words, if our quantum state is an n-dimensional complex vector, this single state can be used to simultaneously represent n different states! This idea, which is fundamental to quantum mechanics, is called superposition. While in classical systems our state must exist in one possible state (e.g., a bit within a computer cannot be both 1 and 0 simultaneously), in quantum systems, the state is allowed to be in multiple states at once with a certain probability. When we measure this quantum state, however, it must collapse to one of the possible states in its basis.
Consecutive Measurements
Once we measure a quantum state, this state collapses to a new state corresponding to one of the basis vectors. So, what happens if we make this same measurement on our state a second time? We will get the same result 100% of the time. Why? After we measure our state the first time, assume (without loss of generality) that this state collapses to the i-th vector in the basis. Then, our new quantum state is represented as follows.
Clearly, making the same measurement on the quantum state shown above will yield a deterministic result (i.e., all of the probabilities are 0 except for one). This highlights a very important point in quantum mechanics. As soon as we make a measurement on our quantum state, the state has collapsed to something different (i.e., the state is modified as a result of the measurement). Therefore, the order and manner in which we perform measurements in quantum mechanics is very important. If we wanted to perform repeated measurements on our original quantum state, we would have to “prepare” this quantum state (i.e., construct a quantum state that somehow exists within this same superposition) each time before making a measurement.
Example of a Quantum State: Qubits
To solidify the concept of quantum states, I think it is useful to present a concrete example of a quantum system that is (somewhat) simple, but very useful in modern research — the quantum bit, or “qubit”. In a classical computer, we have the notion of bits, which correspond to values of 0 or 1. Each bit must exist in one of these two possible states, and many bits can be combined together to form complex computer systems. The possible states of a bit can easily be represented as follows.
A qubit is similar to a bit, as it shares the same basis states. However, for qubits we consider complex vector spaces (as opposed to real vector spaces). A single qubit can be represented as follows.
In this equation, the basis vectors are defined identically as for bits. Additionally, if we measure a qubit, the result will be either 0 or 1— same as classical bits. However, a qubit can exist in a superposition of these possible states, allowing it to encode significantly more information than a classical bit. Such a system can be visualized within something called the Bloch Sphere, shown below.
Observables (Measureables)
Based on the definition of quantum states and measurements given so far, you are probably thinking that our possible states (i.e., the basis we use to write our quantum state) must have come out of nowhere. How do we know what the possible states are when we take a measurement? Well, this question is answered in quantum mechanics by the concept of an observable (i.e., sometimes this is called a measureable). The name pretty much explains exactly what this is — measurables (or observables) represent the quantities within a quantum system that can be measured. Concretely, measurables are hermitian linear operators, as shown by the equation below.
As can be seen in the equation above, a hermitian matrix is simply a matrix that is equal to its conjugate transpose. Matrices of this form have a few useful properties. Firstly, all of the eigenvalues of a hermitian matrix must be real-valued. Additionally, the eigenvectors of a hermitian matrix are a complete set (i.e., they form an orthonormal basis). In the equation above, these properties imply that the operator H has a set of normalized eigenvectors that form a basis for the n-dimensional space of complex vectors. Therefore, any vector in this space can be written as a linear combination of the eigenvectors for this hermitian operator. Therefore, given an observable (i.e., a hermitian operator), we can then expand any quantum state that is given to us in a familiar fashion.
The equation above is the same exact equation presented in the explanation of quantum states. Now, however, this representation should be seen from a slightly different perspective. Given some arbitrary quantum state, we can expand it as a linear combination of the eigenvectors associated with some observable. Then, these eigenvectors correspond to the possible states to which our quantum state could collapse after performing a measurement with said observable. The possible outputs of this measurement (i.e., the values that we get as a result) are the eigenvalues of the observable. In particular, whichever state we collapse to after making a measurement, its corresponding eigenvalue will be measured as a result. Since the eigenvalues of a hermitian operator are known to be real-valued, the result of the measurement will be a real value.
How do we know what the result of our measurement will be?
Just like before, the answer to this question is non-deterministic. However, as we know from the previous discussion on measurements, we can easily derive the probability that our quantum state will collapse into a certain basis state. Given some observable with associated eigenvectors, we can first expand the quantum state as a linear combination of eigenvectors for this observable. Then, the probability of collapse to any given state, just as before, is given by the squared magnitude of the probability amplitude associated with this state; see below.
Creating a Multi-Qubit System
Before moving on to the explanation of entanglement, it is important to understand how simple systems can be combined to form complex systems in quantum mechanics. Generally, multiple systems can be combined together by taking their Kronecker product. Because this statement is quite vague, I will provide a better explanation with the use of a concrete example.
From the previous discussion, we know how to represent a single qubit. However, what if we have two qubits together within a single system? Let us first consider the possible results of a measurement on this two qubit system. If we know the possible results of a measurement (i.e., the basis states for our combined system), then we can write any state of the two qubit system as a linear combination of these basis states. Because each qubit may provide a measurement of either 0 or 1, we have the following possibilities for measurements: 00, 01, 10, and 11. We can construct the vector representation of these combined states by taking the Kronecker product of their components. This is shown in the equation below, where all of the possible states for a two qubit system are constructed.
Notice that these states form a basis for the four-dimensional space of complex vectors. The states shown above reveal a more general pattern — if we form a system of n qubits, then this system is capable of representing 2^n states (i.e., in the case above there are 2² = 4 possible states). Given these possible states for our two qubit system, a quantum state within this combined system can be easily expressed as follows.
Once we have constructed the quantum state as shown above, everything we have learned so far applies (i.e., it is not any different than the general quantum systems we have been discussing so far). Therefore, we now know how smaller systems can be combined to form more complex systems! An understanding of this concept is pivotal for truly grapsing the concept of entanglement.
Observables of Complex Systems
Similarly to the concept outlined in the example above, if we want to combine observables for smaller systems into single observables of a combined system, we would take their Kronecker product. For example, an identity operator (i.e., this is a hermitian matrix and, therefore, an observable) for a single qubit system is a 2x2 matrix. To form an identity operator for a two qubit system, we would take the Kronecker product of two identity operators, forming a 4x4 identity matrix. This 4x4 identity matrix is an identity operator for the combined, two qubit system. Similar logic applies for different types of observables.
Entanglement (finally…)
Now, we finally understand enough about quantum mechanics to gain a basic grasp of quantum entanglement. Quantum entanglement assumes that there exists a complex system, composed of several smaller components. Luckily, we just outlined how quantum systems can be combined together, and I will continue using the same running example — a multi-qubit system — to explain entanglement. I will begin by introducing the notion of a product state. In words, a product state is the simplest state in which a combined system can exist. It is formed by taking the product of individual component states within the system. For example, see the equation below, which combines two individual qubit states together to form a resulting product state for a two qubit system.
In the equation above, the first two rows represent quantum states of individual qubits, while the last row represents the resulting product state. The product state is formed by taking the product of two individual quantum states. In this case, the product of distinct qubit quantum states is taken, resulting in a product state for a two qubit system. In this equation, each of the four entries in the product state’s vector correspond to probability amplitudes associated with each basis state in a two qubit system. Interestingly, if one closely examines the product state, it will become clear that the probabilities associated with measuring either component of the two qubit system are independent. To understand this, take a look at the table below, where I use the same exact product state, but assign concrete values to the probability amplitudes shown above. All values within the tables represent measurement probabilities of a basis state (i.e., the basis state is listed in the top-left corner of each entry).
Examining the product state in the table above, the probability of measuring the first qubit in the combined system as 0 (or 1) is 0.5 (i.e., same as if you measured the qubit in isolation). The same observation applies to the second qubit. After closely examining the above product state, it becomes clear that if the first qubit is measured as 0 (without loss of generality), the probability of the second qubit being measured as 0 (or 1) is still 0.5. In other words, measuring one qubit in the combined system does not give us any extra information about the value of the other qubit. The measurement probabilities of the two qubits in the combined system do not depend on each other whatsoever — there is no entanglement. This property of product states follows from the fact that all of the probability amplitudes in a product state are expressed as a product of probability amplitudes for each its individual components. However, it is possible to construct states within a combined system that cannot be expressed as product states. For example, consider the following state of a two qubit system.
If you try for a long time, you will realize that the above two qubit quantum state cannot be expressed as the product of individual qubit states. We will call this a “non-product state”, and the vector representation of the non-product state shown above can be seen in the equation below.
If we consider the measurement probabilities for the quantum state shown above, we will notice something interesting. Assume that we make a measurement on the first qubit and it collapses to 0. If we then measure the second qubit, what will we get as a result? Interestingly, there is a 100% chance that the second qubit will collapse to 0 when measured (i.e., examine the probabilities in the table above closely!). A similar, but reversed, phenomenon is observed if the first qubit collapses to 1. In other words, as soon as we measure the first qubit, the state of the second qubit is known— the qubits in our two qubit system are entangled! Therefore, through this example, we gain a basic theoretical (and practical) understanding of the meaning of entanglement. Namely, entangled quantum states are those that cannot be expressed as products of their individual components, causing measurements of the systems’ components to depend on each other (i.e., measuring one component of the system will provide extra information about the others). Obviously, this idea extends beyond the two qubit example given above, and can be observed within increasingly complex quantum mechanical systems.
Although entanglement might seem underwhelming at first glance, it is important to realize that the entanglement of a system has no dependence on distance. Therefore, if we prepare a two qubit state as shown above, then move each of the qubits in our system extremely far apart (e.g., assume we fly the apparatus containing the information of the second qubit to Mars and keep the other one on earth), as soon as we measure the first qubit, the state of the second qubit will be immediately known/fixed. Therefore, this information (i.e., the outcome of the first qubit’s measurement) travels faster than the speed of light as soon as the first qubit is measured — something famously described by Einstein as “spooky action at a distance”. Although this may seem completely absurd, these properties of quantum entanglement have been experimentally verified, proving that the nature of reality is somewhat different than what we perceive. For example, does this mean teleportation is possible? I will leave this as something to think about — your guess is as good as mine!
Conclusion
To summarize, entanglement can be described as a quantum state — composed of multiple, smaller components — with measurement probabilities that are dependent on each other. The components of the system are entangled, causing the measurement outcome of one component to impact the measurement outcomes of other components in the system.
I hope you found this post useful, and please feel free to leave any comments or feedback you have. If you are interested, you can find more about me and my research here.
Sources and Citations
- The cover image is from here.
- I got most of my understanding of quantum mechanics from this book, which I highly recommend to anyone who is interested in learning more.
- I really enjoyed the explanations in this blog post, which inspired me to write a similar post with a more extensive background explanations.
- I read through this series of blog posts on quantum computing found here several times (i.e., these are written by my advisor), which helped me to develop a basic understanding of quantum mechanics and qubits.
