Finite-Dimensional Postulates

Finite-dimensional quantum mechanics is the matrix realization of the standard postulates. After an orthonormal basis is chosen, a $d$-level system has state space $\mathbb C^d$; pure states are normalized columns up to phase, density operators are positive unit-trace matrices, observables are Hermitian matrices, closed evolutions are unitary matrices, and composite-system operations are Kronecker products.

This is the natural first language for qubits, fixed-spin systems, finite-level atoms, lattice sites, quantum circuits, and numerically truncated models. It is not a different theory from the Minimal Postulates. It is what those postulates become once every relevant Hilbert space has finite dimension and coordinates have been chosen.

Scope

A finite-dimensional model may be either:

1. intrinsically finite, as for an ideal spin-$j$ degree of freedom with dimension $2j+1$; or
2. an effective truncation, as when only the lowest $d$ oscillator levels are retained.

The algebra is exact in both cases, but the physical claim is different. An intrinsically finite model can be exact within its declared domain. A truncation is an approximation whose leakage, energy range, and observable errors must be checked.

Finite dimension removes several analytic complications:

• every linear operator is bounded and defined on the whole space;
• Hermitian matrices are self-adjoint;
• spectra consist of finitely many eigenvalues;
• every matrix admits a singular-value decomposition;
• traces and matrix products are finite sums;
• positivity can be checked from eigenvalues or principal minors.

It does not remove quantum structure. Noncommutativity, interference, entanglement, incompatible measurements, and mixed reduced states already occur for qubits.

Coordinate Dictionary

Choose an orthonormal basis

$$\mathcal B = \left\{ \lvert0\rangle,\ldots,\lvert d-1\rangle \right\}.$$

Then a ket is represented by a column

$$\lvert\psi\rangle \longleftrightarrow \boldsymbol\psi = \begin{pmatrix} \psi_0\\ \vdots\\ \psi_{d-1} \end{pmatrix},$$

and its bra is the conjugate transpose

$$\langle\psi\rvert \longleftrightarrow \boldsymbol\psi^\dagger = \begin{pmatrix} \psi_0^*&\cdots&\psi_{d-1}^* \end{pmatrix}.$$

The inner product and norm become

$$\begin{aligned} \langle\phi\mid\psi\rangle &= \boldsymbol\phi^\dagger\boldsymbol\psi,\\ \lVert\psi\rVert^2 &= \boldsymbol\psi^\dagger\boldsymbol\psi. \end{aligned}$$

An operator $A$ is represented by the matrix with entries

$$A_{jk} = \langle j\rvert A\lvert k\rangle,$$

so its action is ordinary matrix multiplication:

$$A\lvert\psi\rangle \longleftrightarrow \boldsymbol A\boldsymbol\psi.$$

Coordinates are not physical by themselves. A different orthonormal basis changes the columns and matrices but leaves probabilities unchanged. This distinction between an abstract state and its coordinate representation is easy to overlook precisely because finite-dimensional notation is so convenient.

Postulate 1: States as Vectors and Matrices

For a $d$-level system, choose

$$\mathcal H \cong \mathbb C^d.$$

Pure states

A normalized pure-state representative is

$$\lvert\psi\rangle = \sum_{j=0}^{d-1} \psi_j\lvert j\rangle, \qquad \sum_{j=0}^{d-1} \lvert\psi_j\rvert^2 =1.$$

The vectors $\lvert\psi\rangle$ and $e^{i\alpha}\lvert\psi\rangle$ represent the same ray. In matrix language the phase disappears in the rank-one projector

$$\rho_\psi = \lvert\psi\rangle\langle\psi\rvert \longleftrightarrow \boldsymbol\psi\boldsymbol\psi^\dagger.$$

This matrix satisfies

$$\rho_\psi^2=\rho_\psi, \qquad \operatorname{rank}\rho_\psi=1.$$

General states

A density matrix is a $d\times d$ matrix satisfying

$$\rho^\dagger=\rho, \qquad \rho\geq0, \qquad \operatorname{Tr}\rho=1.$$

Its spectral decomposition is

$$\rho = \sum_{r=1}^{d} \lambda_r\lvert r\rangle\langle r\rvert,$$

where

$$\lambda_r\geq0, \qquad \sum_r\lambda_r=1.$$

Thus density matrices are exactly the positive matrices whose eigenvalues form a probability distribution. A state is pure exactly when one eigenvalue is one and the rest are zero. Equivalent finite-dimensional tests are

$$\begin{aligned} \rho\text{ pure} &\Longleftrightarrow \rho^2=\rho\\ &\Longleftrightarrow \operatorname{Tr}\rho^2=1. \end{aligned}$$

For a mixed state,

$$\frac1d \leq \operatorname{Tr}\rho^2 <1.$$

The lower bound is attained by the maximally mixed state $I/d$.

Convex mixtures are computed entry by entry:

$$\rho = \sum_k q_k\rho_k, \qquad q_k\geq0, \qquad \sum_kq_k=1.$$

Different ensembles can produce the same matrix. Once $\rho$ is fixed, every measurement probability is fixed, regardless of which decomposition was used to prepare it. See Density Operators for the canonical treatment.

Postulate 2: Observables as Hermitian Matrices

A sharp real-valued observable is represented by a Hermitian matrix

$$A=A^\dagger.$$

The finite-dimensional spectral theorem gives a unitary matrix $V$ and real eigenvalues $a_j$ such that

$$A = V \begin{pmatrix} a_0&&0\\ &\ddots&\\ 0&&a_{d-1} \end{pmatrix} V^\dagger.$$

Grouping equal eigenvalues yields the basis-independent form

$$A = \sum_a aP_a,$$

with

$$\begin{aligned} P_a^\dagger&=P_a,\\ P_aP_b&=\delta_{ab}P_a,\\ \sum_aP_a&=I. \end{aligned}$$

The projector $P_a$ can have rank greater than one. Diagonalizing $A$ therefore finds both the possible sharp outcomes and the subspace associated with each outcome.

The expectation and variance in state $\rho$ are

$$\langle A\rangle_\rho = \operatorname{Tr}(\rho A)$$

and

$$(\Delta_\rho A)^2 = \operatorname{Tr}(\rho A^2) - \left[ \operatorname{Tr}(\rho A) \right]^2.$$

Two Hermitian matrices need not commute. If

$$[A,B]\neq0,$$

they cannot be simultaneously diagonalized by one unitary matrix. The noncommutativity is not a coordinate accident: under a basis change, the commutator transforms by unitary conjugation.

The Observables and Spectral Decomposition pages own the general conceptual and operator-theoretic development.

Postulates 2 and 3: Measurements and Probabilities

Finite dimension turns measurement rules into matrix constraints.

Projective measurements

A projective measurement is a set of orthogonal projectors $\{P_a\}$ satisfying

$$P_aP_b=\delta_{ab}P_a, \qquad \sum_aP_a=I.$$

The Born probability is

$$p(a) = \operatorname{Tr}(\rho P_a).$$

For a normalized pure state,

$$p(a) = \langle\psi\rvert P_a\lvert\psi\rangle.$$

If $P_a=\lvert a\rangle\langle a\rvert$, then

$$p(a) = \lvert\langle a\mid\psi\rangle\rvert^2.$$

Generalized measurements

A POVM is a set of positive matrices $\{E_a\}$ with

$$E_a\geq0, \qquad \sum_aE_a=I.$$

The same Born rule becomes

$$p(a) = \operatorname{Tr}(\rho E_a).$$

Unlike projectors, effects need not be mutually orthogonal or idempotent. POVMs describe noisy readout, coarse graining, nonorthogonal discrimination, and indirect measurements without requiring an artificial sharp observable on the original system.

Kraus matrices and conditional states

When state disturbance matters, introduce matrices $M_{a\mu}$ for each recorded outcome $a$ and unresolved internal branch $\mu$. They obey

$$\sum_{a,\mu} M_{a\mu}^\dagger M_{a\mu} = I.$$

The unnormalized conditional branch is

$$\widetilde\rho_a = \sum_\mu M_{a\mu}\rho M_{a\mu}^\dagger,$$

with probability

$$\begin{aligned} p(a) &= \operatorname{Tr}\widetilde\rho_a\\ &= \operatorname{Tr}(\rho E_a), \end{aligned}$$

where

$$E_a = \sum_\mu M_{a\mu}^\dagger M_{a\mu}.$$

For $p(a)>0$, the conditional output state is

$$\rho_a = \frac{\widetilde\rho_a}{p(a)}.$$

Many Kraus families can produce the same effects $\{E_a\}$. Consequently, the POVM fixes the outcome statistics but not the state disturbance. The special ideal projective choice $M_a=P_a$ gives the Lüders update.

See Generalized Measurements Overview for the conceptual separation of effects and instruments.

Postulate 4: Unitary Matrices and Hamiltonians

Closed finite-dimensional systems evolve by unitary matrices. A pure state and density matrix transform as

$$\begin{aligned} \lvert\psi'\rangle &= U\lvert\psi\rangle,\\ \rho' &= U\rho U^\dagger, \end{aligned}$$

with

$$U^\dagger U = UU^\dagger =I.$$

Unitarity preserves inner products:

$$\langle U\phi\mid U\psi\rangle = \langle\phi\mid\psi\rangle.$$

It therefore preserves norms, transition probabilities, eigenvalues of $\rho$, and purity.

For a time-independent Hermitian Hamiltonian,

$$U(t) = \exp\!\left( -\frac{i}{\hbar}Ht \right).$$

If

$$H = \sum_nE_n\lvert n\rangle\langle n\rvert,$$

then the matrix exponential is

$$U(t) = \sum_n e^{-iE_nt/\hbar} \lvert n\rangle\langle n\rvert.$$

Diagonalizing $H$ therefore converts time evolution into phase multiplication in the energy basis. Direct diagonalization is not always the best numerical method for large sparse matrices, but the spectral formula is exact in finite dimension.

For a time-dependent Hamiltonian, the propagator solves

$$i\hbar \frac{dU(t,t_0)}{dt} = H(t)U(t,t_0),$$

with

$$U(t_0,t_0)=I.$$

When Hamiltonians at different times do not commute, one cannot replace the propagator by the exponential of the ordinary time integral without time ordering or an appropriate approximation.

Open-system matrices

A general finite-dimensional quantum channel can be written

$$\mathcal E(\rho) = \sum_\mu K_\mu\rho K_\mu^\dagger,$$

where trace preservation requires

$$\sum_\mu K_\mu^\dagger K_\mu = I.$$

Channels are the appropriate maps for open systems, uncontrolled noise, and outcomes that have been ignored. They extend the closed-system postulate; they are not unitary matrices on the subsystem in general. The Unitary Time Evolution page develops the closed case in full.

Postulate 5: Tensor and Kronecker Products

For distinguishable systems with dimensions $d_A$ and $d_B$,

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B \cong \mathbb C^{d_Ad_B}.$$

Choose the lexicographically ordered product basis

$$\left\{ \lvert i\rangle_A\otimes\lvert j\rangle_B \right\}_{i,j}.$$

For two qubits, the convention

$$\lvert00\rangle, \quad \lvert01\rangle, \quad \lvert10\rangle, \quad \lvert11\rangle$$

identifies a joint ket with a four-component column. If

$$\lvert\psi\rangle_A \longleftrightarrow \begin{pmatrix}a\\b\end{pmatrix}, \qquad \lvert\phi\rangle_B \longleftrightarrow \begin{pmatrix}c\\d\end{pmatrix},$$

then

$$\lvert\psi\rangle_A \otimes \lvert\phi\rangle_B \longleftrightarrow \begin{pmatrix} ac\\ ad\\ bc\\ bd \end{pmatrix}.$$

Operators become Kronecker products. In block form,

$$A\otimes B = \left[ A_{ij}B \right]_{i,j=0}^{d_A-1}.$$

The ordering convention matters. With $A$ listed first, an operator local to $A$ is $A\otimes I_B$, while one local to $B$ is $I_A\otimes B$.

Not every vector in $\mathbb C^{d_Ad_B}$ is a Kronecker product. For example,

$$\lvert\Phi^+\rangle = \frac{ \lvert00\rangle+\lvert11\rangle }{\sqrt2}$$

cannot be factored into one two-component column for $A$ and another for $B$. Its density matrix is

$$\rho_\Phi = \frac12 \begin{pmatrix} 1&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&1 \end{pmatrix},$$

while either reduced density matrix is

$$\rho_A=\rho_B=\frac{I_2}{2}.$$

This is the smallest example showing that a pure joint state can have mixed subsystem states. See Tensor Products and Reduced Density Matrices for the canonical developments.

Basis Changes Do Not Change Predictions

Let the columns of a unitary matrix $V$ define a new orthonormal basis. The same abstract state and operators have new coordinate matrices

$$\begin{aligned} \boldsymbol\psi' &= V^\dagger\boldsymbol\psi,\\ \rho' &= V^\dagger\rho V,\\ A' &= V^\dagger A V,\\ E_a' &= V^\dagger E_a V. \end{aligned}$$

The Born probability is invariant:

$$\begin{aligned} \operatorname{Tr}(\rho'E_a') &= \operatorname{Tr} \left( V^\dagger\rho VV^\dagger E_aV \right)\\ &= \operatorname{Tr}(\rho E_a). \end{aligned}$$

Likewise,

$$(\boldsymbol\phi')^\dagger \boldsymbol\psi' = \boldsymbol\phi^\dagger \boldsymbol\psi.$$

This is a passive coordinate change: the physical preparation and measurement are unchanged. By contrast, actively applying a unitary gate changes the state relative to a fixed laboratory basis. Confusing active transformations with basis changes is a common source of sign and dagger errors.

Worked Example: Rotation with Noisy Readout

Consider a qubit prepared in

$$\rho_0 = \lvert0\rangle\langle0\rvert = \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}.$$

Apply a rotation about the $y$ axis:

$$\begin{aligned} U_y(\theta) &= \exp\!\left( -\frac{i\theta}{2}\sigma_y \right)\\ &= \begin{pmatrix} \cos(\theta/2)&-\sin(\theta/2)\\ \sin(\theta/2)&\cos(\theta/2) \end{pmatrix}. \end{aligned}$$

The output vector is

$$\lvert\psi_\theta\rangle = \begin{pmatrix} \cos(\theta/2)\\ \sin(\theta/2) \end{pmatrix},$$

and the density matrix is

$$\rho_\theta = \begin{pmatrix} \cos^2(\theta/2)& \tfrac12\sin\theta\\ \tfrac12\sin\theta& \sin^2(\theta/2) \end{pmatrix}.$$

Model a symmetric noisy computational-basis detector with visibility $0\leq\eta\leq1$:

$$\begin{aligned} E_0 &= \frac12(I+\eta\sigma_z),\\ E_1 &= \frac12(I-\eta\sigma_z). \end{aligned}$$

In matrix form,

$$E_0 = \begin{pmatrix} \tfrac{1+\eta}{2}&0\\ 0&\tfrac{1-\eta}{2} \end{pmatrix},$$

and $E_1=I-E_0$. Their eigenvalues are nonnegative for $0\leq\eta\leq1$, so they form a valid POVM.

The click probability is

$$\begin{aligned} p(0) &= \operatorname{Tr}(\rho_\theta E_0)\\ &= \frac12 \left( 1+\eta\cos\theta \right), \end{aligned}$$

and

$$p(1) = \frac12 \left( 1-\eta\cos\theta \right).$$

The limits check the model:

• $\eta=1$ gives an ideal projective measurement;
• $\eta=0$ gives a completely uninformative fair coin;
• $\theta=0$ gives $p(0)=(1+\eta)/2$;
• $\theta=\pi$ exchanges the two probabilities.

This one calculation uses a vector, density matrix, unitary matrix, Pauli observable, POVM, and trace rule. It also illustrates why a realistic detector need not be represented by projectors.

Controlled Finite Truncations

Suppose the physical Hilbert space is infinite-dimensional but a calculation retains a $d$-dimensional subspace with projector

$$P_d = \sum_{n=0}^{d-1} \lvert n\rangle\langle n\rvert.$$

A common effective Hamiltonian is

$$H_d = P_dHP_d$$

acting within $P_d\mathcal H$. The corresponding matrix is finite, but this projection alone does not prove that the truncated dynamics approximates the full dynamics.

Useful diagnostics include:

1. initial support: verify $\operatorname{Tr}(P_d\rho_0)$ is close to one;

2. boundary population: monitor occupation near the highest retained levels;

3. leakage: compare against

   $$\epsilon_d(t) = 1- \operatorname{Tr} \left[ P_d\rho_{\mathrm{full}}(t) \right]$$

   when a larger reference calculation is available;

4. convergence: repeat the calculation with increasing $d$;

5. observable stability: check the quantities of interest, not only state-vector norms;

6. time-scale dependence: a truncation accurate at short times can fail after weak leakage accumulates.

Projecting products can also matter. In general,

$$P_dABP_d \neq (P_dAP_d)(P_dBP_d)$$

because intermediate components outside the retained subspace are removed on the right. Truncated operators may therefore fail to satisfy exact commutation relations obeyed in the full space.

A finite matrix calculation is trustworthy only when the finite model or truncation is connected to the physical regime being claimed.

Numerical Validation Checklist

For a finite-dimensional implementation, test the defining relations directly.

States

Check

$$\lVert\rho-\rho^\dagger\rVert \leq\varepsilon, \qquad \left\lvert \operatorname{Tr}\rho-1 \right\rvert \leq\varepsilon,$$

and verify that the smallest eigenvalue is no less than a tolerance compatible with numerical error. Blindly clipping substantial negative eigenvalues can hide a faulty algorithm.

Measurements

For a POVM, check every $E_a$ is Hermitian and positive, then test

$$\left\lVert \sum_aE_a-I \right\rVert \leq\varepsilon.$$

For an instrument or channel, check the corresponding Kraus completeness relation.

Dynamics

For a proposed unitary, check

$$\left\lVert U^\dagger U-I \right\rVert \leq\varepsilon.$$

Also monitor conserved quantities that should commute with the Hamiltonian. Norm preservation alone does not detect every modeling or time-stepping error.

Composite systems

Declare the tensor-factor order once. Test local operators on product basis vectors before trusting a larger calculation. A matrix can have the correct dimensions and still act on the wrong subsystem.

The tolerance $\varepsilon$ is not universal. It should be chosen from floating-point precision, condition numbers, matrix size, and the accuracy required by the physical conclusion.

What Finite Dimension Hides

The finite-dimensional formulation suppresses issues that become essential in wave mechanics and rigorous quantum mechanics:

• unbounded position, momentum, and Hamiltonian operators;
• dense operator domains and self-adjoint extensions;
• continuous spectra and generalized eigenvectors;
• boundary conditions for differential operators;
• convergence of infinite sums and integrals;
• inequivalent representations in systems with infinitely many degrees of freedom;
• field-theoretic locality and variable particle number.

The statement “Hermitian equals self-adjoint” is safe for finite matrices but not for arbitrary differential operators. Likewise, every finite matrix has a complete eigenbasis, while an infinite-dimensional self-adjoint operator can have continuous spectrum with no normalizable eigenvector at a given spectral value.

See Finite vs Infinite-Dimensional Quantum Mechanics and Wave-Mechanics Postulates for the transition beyond matrices.

Common Mistakes

1. Treating a column as basis independent. The ray is physical; its entries depend on the chosen basis.
2. Forgetting conjugation in bras. The bra is $\boldsymbol\psi^\dagger$, not the transpose $\boldsymbol\psi^{\mathsf T}$.
3. Checking only trace one for a density matrix. Hermiticity and positivity are also required.
4. Assuming every Hermitian matrix specifies a general measurement. It specifies a sharp observable; POVMs are more general.
5. Using a POVM to infer disturbance. Conditional states require Kraus matrices or an instrument.
6. Exponentiating a non-Hermitian closed-system Hamiltonian. The result is generally nonunitary and needs a different physical interpretation.
7. Changing basis on the state but not the operators. All coordinate representations must transform consistently.
8. Reversing tensor-factor order. $A\otimes I$ and $I\otimes A$ are usually different matrices.
9. Equating finite truncation with exact finiteness. Convergence and leakage must be demonstrated.
10. Repairing numerical states without diagnosing the source. Large positivity or normalization errors are evidence, not cosmetic defects.

Connections

• Minimal Postulates is the canonical concise statement independent of coordinates.
• Density-Matrix Formulation treats density operators as primary beyond the matrix-first motivation.
• Equivalent Formulations explains when different mathematical packages make identical predictions.
• Finite-Dimensional Hilbert Spaces supplies the linear-algebra background.
• Tensor-Product Ordering fixes the site-wide coordinate convention for composite systems.
• Bloch Sphere specializes qubit density matrices to Bloch-vector coordinates.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010 — matrix-first states, gates, measurements, and composite systems.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018 — precise finite-dimensional treatment of states, channels, measurements, and norms.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — spin systems, matrix mechanics, and unitary dynamics.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994 — finite-dimensional foundations and the bridge to wave mechanics.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995 — conceptual treatment of finite-state systems and measurement.
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, 1983 — operator-sum representations and generalized operations.

Exercises

1. Density-matrix validity. For

$$\rho = \begin{pmatrix} \tfrac12&c\\ c^*&\tfrac12 \end{pmatrix},$$

find the condition on $c$ for $\rho$ to be a state, and identify when it is pure.

Solution

Hermiticity and trace one are already built in. The eigenvalues are

$$\lambda_\pm = \frac12\pm\lvert c\rvert.$$

Positivity therefore requires

$$\lvert c\rvert\leq\frac12.$$

The purity is

$$\operatorname{Tr}\rho^2 = \frac12+2\lvert c\rvert^2.$$

It equals one exactly when $\lvert c\rvert=1/2$. Thus boundary points are pure and interior points are mixed.

2. Basis covariance. Let $V$ be unitary and define $\rho'=V^\dagger\rho V$ and $E'=V^\dagger EV$. Prove that $\operatorname{Tr}(\rho'E')=\operatorname{Tr}(\rho E)$.

Solution

Substitution gives

$$\begin{aligned} \operatorname{Tr}(\rho'E') &= \operatorname{Tr} \left( V^\dagger\rho VV^\dagger EV \right)\\ &= \operatorname{Tr} \left( V^\dagger\rho EV \right). \end{aligned}$$

Cyclicity of the finite-dimensional trace then gives

$$\begin{aligned} \operatorname{Tr} \left( V^\dagger\rho EV \right) &= \operatorname{Tr} \left( \rho EVV^\dagger \right)\\ &= \operatorname{Tr}(\rho E). \end{aligned}$$

The coordinate matrices change, but the probability does not.

3. A two-outcome POVM. Let

$$E_\pm = \frac12 \left( I\pm\eta\sigma_x \right).$$

For which real values of $\eta$ is this a POVM? Find $p(+)$ for $\rho=(I+r_z\sigma_z+r_x\sigma_x)/2$.

Solution

Completeness holds because $E_++E_-=I$. The eigenvalues of $E_\pm$ are

$$\frac{1+\eta}{2}, \qquad \frac{1-\eta}{2},$$

up to exchange, so positivity requires

$$-1\leq\eta\leq1.$$

Using $\operatorname{Tr}(\sigma_i\sigma_j)=2\delta_{ij}$,

$$\begin{aligned} p(+) &= \operatorname{Tr}(\rho E_+)\\ &= \frac12 \left( 1+\eta r_x \right). \end{aligned}$$

The $z$ component does not affect this $x$-oriented measurement.

4. Hamiltonian exponential. For $H=(\hbar\Omega/2)\sigma_z$, compute $U(t)$ and evolve $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$.

Solution

Since $\sigma_z^2=I$,

$$U(t) = \cos\frac{\Omega t}{2}\,I - i\sin\frac{\Omega t}{2}\,\sigma_z.$$

Equivalently,

$$U(t) = \begin{pmatrix} e^{-i\Omega t/2}&0\\ 0&e^{i\Omega t/2} \end{pmatrix}.$$

Therefore

$$\lvert\psi(t)\rangle = \frac{ e^{-i\Omega t/2}\lvert0\rangle + e^{i\Omega t/2}\lvert1\rangle }{\sqrt2}.$$

Removing the irrelevant global phase $e^{-i\Omega t/2}$ leaves a relative phase $e^{i\Omega t}$ between the basis states.

5. Amplitude-damping channel. For $0\leq\gamma\leq1$, let

$$\begin{aligned} K_0 &= \begin{pmatrix} 1&0\\ 0&\sqrt{1-\gamma} \end{pmatrix},\\ K_1 &= \begin{pmatrix} 0&\sqrt\gamma\\ 0&0 \end{pmatrix}. \end{aligned}$$

Verify trace preservation and find the output for input $\lvert1\rangle\langle1\rvert$.

Solution

The completeness relation is

$$\begin{aligned} K_0^\dagger K_0 &= \begin{pmatrix} 1&0\\ 0&1-\gamma \end{pmatrix},\\ K_1^\dagger K_1 &= \begin{pmatrix} 0&0\\ 0&\gamma \end{pmatrix}. \end{aligned}$$

Their sum is $I$, so the map is trace preserving. Acting on the excited-state projector gives

$$\mathcal E \left( \lvert1\rangle\langle1\rvert \right) = (1-\gamma) \lvert1\rangle\langle1\rvert + \gamma \lvert0\rangle\langle0\rvert.$$

The excited-state population decays by $\gamma$ and appears in the ground state.

6. Tensor-factor ordering. In the ordered basis $\lvert00\rangle,\lvert01\rangle,\lvert10\rangle,\lvert11\rangle$, write the matrices $X\otimes I$ and $I\otimes X$. State how each acts on $\lvert01\rangle$.

Solution

The matrices are

$$X\otimes I = \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{pmatrix}$$

and

$$I\otimes X = \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}.$$

Therefore,

$$(X\otimes I)\lvert01\rangle = \lvert11\rangle,$$

whereas

$$(I\otimes X)\lvert01\rangle = \lvert00\rangle.$$

The first matrix flips subsystem $A$; the second flips subsystem $B$.

7. Purity under unitary evolution. Show that $\operatorname{Tr}[(U\rho U^\dagger)^2]=\operatorname{Tr}(\rho^2)$.

Solution

Using unitarity,

$$\begin{aligned} (U\rho U^\dagger)^2 &= U\rho U^\dagger U\rho U^\dagger\\ &= U\rho^2U^\dagger. \end{aligned}$$

Cyclicity of the trace gives

$$\operatorname{Tr} \left( U\rho^2U^\dagger \right) = \operatorname{Tr} \left( \rho^2U^\dagger U \right) = \operatorname{Tr}(\rho^2).$$

Closed-system unitary evolution cannot turn a mixed state into a pure state or conversely.

8. A truncation warning. Let $P$ project onto a retained subspace. Expand the difference

$$PABP-(PAP)(PBP)$$

and explain when it vanishes.

Solution

Insert $I=P+(I-P)$ between $A$ and $B$:

$$\begin{aligned} PABP &= PA \left[ P+(I-P) \right] BP\\ &= PAPBP + PA(I-P)BP. \end{aligned}$$

Hence

$$\begin{gathered} PABP-(PAP)(PBP)\\ = PA(I-P)BP. \end{gathered}$$

The difference vanishes when the discarded intermediate contribution is zero. Sufficient conditions include $A$ preserving the retained subspace or $B$ mapping the retained subspace into itself. In general the term need not vanish, so projecting operators and then multiplying can change their algebra.