2003 - Quantum Computation Lecture 4 - The Schrödinger Picture

Duration: 00:48:23

Transcript

David Deutsch

00:01:00 - 00:03:25

So far, whenever I’ve described the dynamics of quantum systems to you, and they’ve all been quantum gates, I’ve always just told you the laws of motion of each gate by fiat. A set of equations for how the representative observables of the qubits are changed by passage through that particular gate. Now I’ll go to the other extreme and show you the most general framework for quantum dynamics and tell you in principle which sets of such equations describe processes that occur in nature and which don’t. Laws of motion are about how things change with time. But as I said in the first lecture, the notion of a system changing itself implies that some things about it are invariant. That’s what I called the constitution of the system. The laws of motion, the dynamics, are part of the constitution. The other part is the algebra of the observables at any one time, which in a system of $n$ qubits is always the same as the algebra of all Hermitian matrices of dimension $2^n$ . So for instance, you’ll recall that this Pauli algebra is one way of summarizing the algebra of 2 by 2 Hermitian matrices representing the observables of a single qubit at any given time. Now since the algebra is invariant, the laws of motion have to be such that even though the observables change, their algebra does not. And that’s obviously a constraint on what the laws of motion can be. For instance, take this equation for the commutator of the observables $X$ and $Y$ of a qubit.

00:03:25 - 00:05:54

Rearrange it and you get a quantity which the laws of motion have to prohibit from ever changing at all. It has to stay fixed at zero for all time. Differentiate it with respect to time. That too has to be zero. So there we have a relationship between the time derivatives of observables. And we know that this relationship must be not just compatible but deducible from the law of motion of any qubit that exists anywhere in nature. And we can get similar constraints from any other equation holding among the observables at any given time. It may sound as though it’s going to be tricky to find any law of motion that enforces all of these intricate constraints on how observables can change with time. But no, it’s easy to find one. Here’s one. Pick any observable $H$ of a quantum system at a given time. If the system is a qubit, then like all observables, $H$ can be expressed as a linear combination with real coefficients of the representative observables $X$ , $Y$ , and $Z$ at that time and the unit observable. Do the same for every other time. In other words, pick a whole one parameter family of observables, $H_t$ , one for each time. The reason why I’m not calling that $H(t)$ with the $t$ in parentheses is that we’re already using that notation to refer to the time evolution of a single observable, whereas in this construction we’re allowed to choose a different observable at each time. Now given that one parameter family of observables, consider the following law of motion. For each observable $A(t)$ of the system, $\frac{dA}{dt}=i[H_t,A(t)]$ .

00:05:54 - 00:08:42

We can express that law in a form that’s more like the specific laws of motion I’ve told you in previous lectures by just integrating it over an infinitesimal period dt. So we get $A(t+dt)=A(t)+i\,dt[H_t,A(t)]$ . In that form it expresses each observable at one time, t plus dt, in terms of observables at another time t. $H_t$ is by construction some observable at time t, and so is $A(t)$ . For gates we were only interested in the relationship between observables before and after the gate acted. So we used laws of motion referring to a unit time instead of an infinitesimal time. Such laws could be obtained by integrating this differential equation of motion between t and t plus 1. Okay, now does our new law of motion preserve all the algebraic relationships that may exist between observables? Well consider any such relationship at time t. We can always write it in the form $f(A(t),B(t),\ldots)=0$ , where $A$ and $B$ and so on are observables at time t, and the function f uses any of the operations of matrix algebra, namely matrix addition, matrix multiplication and multiplication by a constant. It’s sufficient to prove that f itself obeys the law of motion. Well what can f be? Suppose f is a real multiple of something that does obey the law of motion. Then $F$ obeys it too. Suppose f is the sum of two things that each obey the law of motion. Then again f also obeys it.

00:08:48 - 00:11:13

What if f is the matrix product of two things that obey the law of motion? Well then $\frac{d}{dt}(PQ)=\frac{dP}{dt}Q+P\frac{dQ}{dt}$ , and since p and q obey the laws of motion, multiply that out and you’ll see that it’s $i$ times the commutator of $H$ with the product $PQ$ , which is $i$ times the commutator of $H$ with $F$ again. So the upshot is, however $F$ is composed out of observables obeying our equation of motion, $F$ obeys it too, and hence it remains zero over time. So any equation of motion of this form leaves any algebra of observables invariant. And actually we’re done, because as you can prove in the worked examples, the converse is also true. Any law of motion that leaves the algebra of observables of some quantum system invariant takes this form. $H$ is called the Hamiltonian of the system. This is called the Heisenberg equation of motion, and the way of representing observables that I’ve been using is called the Heisenberg picture of quantum theory. Consider the special case where the Hamiltonian is actually the same observable at all times. So then $H_t$ does equal some $H(t)$ , where $H(t)$ is a bona fide observable. In such cases we say that the Hamiltonian has no explicit time dependence. How does this observable change with time? Well, as you can see, it doesn’t. When the Hamiltonian has no explicit time dependence, it has no time dependence at all. So under those circumstances, it’s not only an observable, it’s a conserved quantity.

00:11:13 - 00:13:59

This is a conservation law, and in fact the Hamiltonian is up to a constant of proportionality the energy observable. You can also see that any other observable that commutes with the Hamiltonian is a conserved quantity too. And if an observable commutes with a conserved Hamiltonian at any one instant, even one instant, then because the algebra is invariant, it will commute at all instants and will be conserved too. And again, the converse is also true. In a system with a conserved Hamiltonian, every conserved observable commutes with that Hamiltonian. Now, in the case where the Hamiltonian is not only the same observable at all times, but is one of the system’s own observables, the system is said to be isolated. The idea of an isolated system is an idealisation. There are no physical systems that remain unaffected by the outside world for all possible states of the outside world. Remember that the idea of a physical system itself is not perfectly well defined because there is no physical system that even remains in existence in all states. And for both these reasons, the idea of the Hamiltonian for a given physical system is always an idealisation or an approximation. Except perhaps for the multiverse as a whole, that’s not interacting with anything outside itself. So the multiverse is the only truly isolated system. Nevertheless, in real physical phenomena, it’s often a very good approximation to say that a system is isolated for a period. For instance, I said that the qubit in our interference experiment in Lecture 2 was isolated during periods when the photon was travelling between gates. In fact, it was well described as having a Hamiltonian of zero during those periods, even though its real Hamiltonian, ultimately the Hamiltonian of the multiverse, would have contained terms describing how the photon would interact with a passing bumblebee that just happened to fly through the apparatus at the same time, and indeed how it did interact with one in some universes.

00:14:00 - 00:17:16

Now, even when a system isn’t isolated, it is sometimes possible to encode the whole influence of the outside world on that system into a law of motion that involves only that system’s observables, but with a time-dependent Hamiltonian. Such systems are said to be coherent. Another way of stating the definition of coherent is that a coherent quantum system is one in which the observables at one time are functions only of the observables of the same system at another time. That would be in a given state or in a given class of states. In summary, in states where the dynamics of a system can be well described by a Hamiltonian that’s simply one of its own observables, it’s said to be isolated. When it can be well described by a Hamiltonian built out of its own observables, but possibly with explicit time dependence, it is said to be coherent. So an isolated system is also coherent. When it can only be described by a Hamiltonian involving other systems, it is said to be decoherent. But ultimately, all these terms are approximative. Short of the multiverse as a whole, there are no isolated systems and every system is decoherent to some extent. And there are no time-dependent Hamiltonians. Every time-dependent Hamiltonian is just an approximate way of taking into account the effects of other systems in states where the decoherence that they cause in the system of interest is negligible. In our single photon interference experiment, for example, the qubit was effectively isolated during its motion between the beam splitters and the mirrors, but as it interacted with those, its Hamiltonian changed briefly, as I described, but only to some function of the qubit’s own observables. And then it changed back to zero. So the qubit remained coherent throughout the whole experiment until the moment when it interacted with the detector at the end. At that point, the qubit’s Hamiltonian would have depended on observables of the detector and vice versa, thus causing decoherence. And also, as we saw, if the qubit interacted with another qubit during the experiment, then its Hamiltonian would depend on observables of both qubits, and we saw that the interference was then reduced or eliminated. And it’s true in general that for an interference phenomenon to occur, or for a quantum computation to be performed, the system doesn’t have to be isolated, but it does have to be coherent.

00:17:25 - 00:19:58

Okay, now, for an isolated system, we can solve the equation of motion in closed form. It’s just $A(t)=e^{iHt}A(0)e^{-iHt}$ . So that gives A at any time in terms of A at time zero, and $H$ . We can also express the constant Hamiltonian $H$ in terms of observables at time zero. And so this equation would express A at an arbitrary time t in terms of observables of the system at time zero. For a general coherent system, the solution will be a slight generalization of that. It’ll be $A(t)=U_t^\dagger A(0)U_t$ , where $U_t^\dagger$ is the Hermitian adjoint of $U_t$ . $U_t$ is a one parameter family of unitary matrices. Unitary means $U^\dagger U=I$ , whose own equation of motion is $\frac{dU}{dt}=-iUH$ . U is called the evolution matrix between time zero and time t. The state of a system is defined by giving its expectation value function, which is a linear function mapping its observables to real numbers. I’ve used this notation to denote the expectation value of an observable $A(t)$ . But because it’s a linear function, it must also be possible to write it in any given matrix representation like this, where $\alpha$ and $\beta$ are matrix indices and $\rho$ is some matrix that doesn’t change with time. In other words, the expectation value is $\operatorname{tr}(A(t)\rho)$ . Since the expectation value of the unit observable is always one, $\operatorname{tr}(\rho)$ has got to be one. $\rho$ is called the density matrix of the system.

00:19:58 - 00:21:57

It has to satisfy certain conditions to make sure that the expectation value can never be higher than the highest eigenvalue of A nor lower than the lowest. You can find out what those conditions are in the worked examples. Now look at the expression for the expectation value of an arbitrary observable. It’s often more convenient to write this in the form $\operatorname{tr}(U_t^\dagger A(0)U_t\rho)$ . Why? Because in this expression, the only quantity that changes with time is the unitary evolution matrix U, which is the same in the corresponding expression for any observable. Therefore, to track everything that the system is doing over time, we don’t have to solve the equations of motion for all the observables in terms of their values at their previous times. We need only solve this one equation for the evolution matrix $U_t$ , given the Hamiltonian. And with that, we can read off the time evolution of any observable of the system evolving under that Hamiltonian. Actually, we can do better than that. The way the dynamics of quantum systems are encoded in these unitary matrices allows for a whole alternative way of describing quantum systems that’s often much more efficient. What I’ve described so far has been the Heisenberg picture, and the alternative way I’ll show you now is called the Schrödinger picture. First, look again at this expression for the expectation value of a general observable at an arbitrary time.

00:21:57 - 00:24:22

We can rewrite it using the cyclic invariance of the trace, like this. $\langle A(t)\rangle=\operatorname{tr}(U_t\rho U_t^\dagger A(0))$ . Let’s call this quantity $\rho_t$ . $\rho_t$ is called the density matrix in the Schrödinger picture. So to evaluate an arbitrary expectation value, we need only know this density matrix in the Schrödinger picture and all the observables at any one time, say t equals 0. So for each observable $A$ , we only need to know one matrix, $\hat{A}(0)$ , which in the Schrödinger picture we just call $\hat{A}$ . To summarise the Schrödinger picture then, each observable is represented by a constant matrix, while the density matrix changes with time. This is to be compared with the Heisenberg picture, where the observables are functions of time and the density matrix is a constant. From this expression, we can read off the law of motion for the Schrödinger picture density matrix. It’s $\frac{d\rho}{dt}=-i[H,\rho]$ . This has almost the same form as the law of motion for an observable in the Heisenberg picture, but with the opposite sign. I’ll come back to that sign in a moment. Now, consider the eigenvectors of the Schrödinger density matrix at time t. The standard notation for vectors in quantum theory is called the Dirac notation. It uses a symbol called a ket, which looks like this, to denote vectors. We typically write inside the ket symbol the information specifying which vector it is. For instance, we can call the nth eigenvector of $\rho(t)$ the ket with label n and t.

00:24:25 - 00:27:00

The bra vector, that is the Hermitian transpose or dual of a given ket, say with label $\alpha$ , is written like this. This symbol is called a bra. The origin of this terminology is that the scalar product of a bra with a ket is written like this, $\langle\beta|\alpha\rangle$ , the whole thing being a bracket. So the left half of it is a bra and the right half a ket. The scalar product of a bra with the corresponding ket, say $\langle\alpha|\alpha\rangle$ , is by definition the squared magnitude or norm of $|\alpha\rangle$ . The vector space of kets is therefore a Hilbert space, basically a vector space with a norm, and it’s usually called the Hilbert space of the given system. Observables represented by matrices are linear operators on the Hilbert space. The set of all eigenkets of the density matrix at any given time, or of any observable at any time, constitutes a basis for the system’s Hilbert space. So we can expand the density matrix in terms of its eigenvalues and eigenvectors like this. $\rho(t)=\sum_n p_n |n(t)\rangle\langle n(t)|$ . And you can prove in the worked examples that the eigenvalues $p_n$ remain constant and that the equation of motion for the eigenvectors is this. $\frac{d}{dt}|n\rangle=-iH|n\rangle$ . Or more generally, if $\psi$ is any eigenvector of the density matrix, $\frac{d\psi}{dt}=-iH\psi$ . This is called the Schrödinger equation. Over a period t, its general solution will have this form where the $U_t$ are the same unitary matrices as in the solution of the Heisenberg equation of motion.

00:27:03 - 00:29:25

Now the significance of that sign in the equation of motion of the density matrix. The density matrix is not an observable. Both in the Schrödinger and Heisenberg pictures, it’s equal to a different observable at each instant. So why does it have a similar equation of motion to an observable but with the opposite sign? Well, in the Schrödinger picture, the state changes and the observables are constant matrices. In the Heisenberg picture, the state is invariant and the observables change. But in both of them, the expectation value of an observable A at a given time is $\operatorname{tr}(\rho A)$ . And the Heisenberg $A(t)$ we know is $U^\dagger$ $A(0)U$ . Remember, the Schrödinger $A$ equals Heisenberg $A(0)$ and also Heisenberg $\rho$ equals Schrödinger $\rho(0)$ . What’s happening is that the unitary transformation $U_t$ that defines the motion amounts to a rigid rotation in Hilbert space. It’s rigid in that it preserves the scalar product between any two kets that it acts on. When $|\alpha\rangle$ goes to $U|\alpha\rangle$ , $\langle\alpha|$ goes to $\langle\alpha|$ $U^\dagger$ . And so any scalar product $\langle\alpha|\beta\rangle$ is unchanged and that’s rigid rotation. The density matrix represents the state of the world. In the Heisenberg picture, the state is constant and the observables rotate in the sense that their eigenvectors rotate in Hilbert space. In the Schrödinger picture, the observables are fixed.

00:29:31 - 00:31:56

The state rotates rigidly in the opposite sense, thus ensuring that both pictures make the same predictions for the physical quantities, the expectation values of observables. The Schrödinger equation is the Schrödinger picture way of defining the dynamical evolution of a quantum system. In the general case, we have to solve the equation for each eigenvector of the density matrix, from which we can reconstitute the density matrix itself. But if $\rho$ is ever sharp, well $\rho$ is not an observable, but what I mean is if it’s ever equal at some instant to a sharp observable, then the system is said to be in a pure state. And we’re often particularly interested in systems in pure states. For instance, if the $Z$ observables of a set of qubits are all sharp, so the qubits together are like a register of a classical computer holding a single integer, then the density matrix is sharp and the system is in a pure state. You can prove that in the worked examples. And you can also prove that in a pure state, in the eigenvector representation of the density matrix, all the coefficients vanish except for one, which takes the value one. And since the eigenvalues of the density matrix don’t change with time, if a system is in a pure state at any instant, then it remains in a pure state so long as the evolution remains coherent. And the density matrix in such a case just takes this form where $\psi(t)$ is a single one parameter family of kets obeying the Schrödinger equation with respect to t. $\psi(t)$ is then called the Schrödinger state vector. For a quantum system in a pure state, all the motion of the system is summed up in the motion of this state vector.

00:31:56 - 00:33:51

Okay, this equation for pure states and this one for non-pure states are in the majority of cases the most convenient ways of analysing the motion of quantum systems, and in particular the computations of quantum computers. And I’ll mostly use it from now on. But beware, as we’ve just seen, this economy of calculation comes at the expense of additional layers of abstraction. Heisenberg observables, which are the dynamical quantities in the Heisenberg picture of quantum physics, have a lot in common with the variables of classical physics. Yes, they’re matrices rather than real numbers, but at least there’s one such matrix corresponding to each quantity we can observe. And the particular number corresponding to what we do directly observe is in there somewhere, it’s one of the eigenvalues. And in the Heisenberg picture, the changing quantities, the observables, are located in particular systems at particular locations. They obey an equation of motion that expresses how they affect other observables and are affected by them, and thereby carry information from one place to another. The Schrödinger state doesn’t have any of that. There’s only one of them. It’s not located anywhere, but refers to the whole system or ultimately to the whole multiverse. What is the Hamiltonian of the whole multiverse? Fundamentally, fundamental physics has been about what types of systems exist in nature, what their observables are, and what their Hamiltonians are.

00:33:51 - 00:36:04

That’s what elementary particle physicists call the theory of everything. But there is another way of looking at what is elementary or fundamental. Instead of asking what types of Hamiltonian are found in natural systems, which means what sorts of changes can occur in elementary systems over an infinitesimal time, one could go all the way to the bottom line and ask which transformations can be realized in nature by some quantum system evolving for some time and which cannot. And the short answer is all changes in which observables of the system undergo unitary evolution with every observable undergoing the same transformation so as to preserve their algebra, every one of those can occur in nature and nothing else can. Every unitary matrix is the evolution matrix for some quantum system evolving over some time. Or, at least, we think it is, because it turns out that the vast majority of these possible evolutions can only be realized in a very special type of physical system, a universal quantum computer. They don’t occur naturally because they require a complex computer program to bring them about. So here’s a fascinating situation. In terms of Hamiltonians, the laws of physics are very finely tuned. The multiverse has its Hamiltonian and subsystems of the multiverse only have very special Hamiltonians. Most Hermitian matrices cannot be realized in nature as Hamiltonians. But when we ask a slightly different question, which unitary evolution operators can be realized in nature, the answer is all of them, if a universal quantum computer can exist.

00:36:04 - 00:36:39

This special type of object, the universal quantum computer, in a sense contains within itself all the diversity in nature. No other system does, except perhaps systems that are capable of constructing a universal quantum computer. Suddenly we find ourselves unavoidably playing a role at the deepest level of the structure of physical reality.

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