Bits, Qubits, Qudits, and Modes

An information carrier is a physical system together with a declared set of states used to represent information. A classical bit has two distinguishable values. A qubit has a two-dimensional quantum state space. A qudit has a $d$-dimensional quantum state space. A continuous-variable mode has an infinite-dimensional state space generated by a canonical pair of observables, often represented by bosonic creation and annihilation operators.

Helpful background. Quantum states explains pure and mixed states; tensor products explains how subsystem dimensions combine.

Carriers and their physical encodings

Those statements describe abstract carriers. They do not yet say how a laboratory device stores them. Three layers must be kept separate:

1. Abstract carrier: a bit alphabet, a Hilbert space, or an oscillator mode.
2. Encoding: a chosen basis or code subspace representing the information.
3. Physical realization: voltages, atomic levels, spins, photons, circuit excitations, mechanical motion, or another controlled system.

For a qubit, the distinction is especially important:

• A pure qubit state is a normalized ray in a two-dimensional Hilbert space; a general qubit state is a density operator on that space.
• A physical qubit is a controlled physical system used to store one qubit.
• A logical qubit is an encoded qubit manipulated at the code level, ideally with error detection or correction that suppresses physical faults.

A transmon, trapped ion, photon, spin, and oscillator code can all carry a qubit, but their unused levels, dominant errors, controls, measurements, connectivity, and loss mechanisms are different. The carrier label alone does not determine the engineering problem.

Carrier, Encoding, and Device

The abstract-to-physical map can be expressed mathematically. Let $\mathcal H_{\mathrm{car}}$ be the carrier Hilbert space and $\mathcal H_{\mathrm{phys}}$ the physical Hilbert space. An ideal encoding is an isometry

$$V:\mathcal H_{\mathrm{car}}\longrightarrow\mathcal H_{\mathrm{phys}}, \qquad V^\dagger V=I_{\mathrm{car}}.$$

Its image

$$\mathcal C=V\mathcal H_{\mathrm{car}}$$

is the code subspace, with projector

$$P_{\mathcal C}=VV^\dagger.$$

The same abstract qubit can therefore be represented by very different subspaces. Conversely, one physical system may support several encodings. A photon can encode a qubit in polarization, path, time bin, frequency bin, or a code spread over multiple modes. An oscillator can be used directly as a continuous-variable carrier, truncated to a finite set of number states, or used to host an encoded qubit such as a cat or Gottesman–Kitaev–Preskill code.

Leaving a chosen qubit subspace and leaving an error-correcting code space are different events. In (a), excitation to a device’s third level is physical leakage. In (b), a bit flip leaves the repetition-code space but remains within the three-qubit Hilbert space, where a syndrome can identify it. Both diagrams are schematic subspace pictures, not energy spectra.

This layered description prevents two common category errors. First, a device with many energy levels is not literally a two-level system merely because two levels are selected for control. Second, an abstract two-dimensional code space is not automatically protected: protection depends on the noise, syndrome information, recovery operations, and measured logical performance.

Classical Bits

A symbol, a random variable, and a storage element

The abstract classical bit alphabet is

$$\mathcal X=\{0,1\}.$$

A deterministic bit is one selected symbol $x\in\mathcal X$. A random bit is instead a probability distribution

$$p(0)=p, \qquad p(1)=1-p.$$

Its Shannon entropy is

$$H(X) = -p\log_2p -(1-p)\log_2(1-p),$$

which ranges from $0$ bits for a known value to $1$ bit for a uniform random bit. The word bit thus appears in at least three related senses:

• a two-symbol alphabet;
• a unit of classical information measured with a base-$2$ logarithm;
• a physical storage element designed to realize two reliably distinguishable states.

These senses should not be silently interchanged. A biased random bit has a two-element alphabet but entropy below one bit. A memory cell may have more microscopic states than its two logical macrostates. Error-correcting storage may use many physical cells for one protected logical bit.

Physical distinguishability

An ideal classical memory uses states that can be read without ambiguity at the chosen level of description. Real devices use separated regions of a noisy state space: two voltage ranges, two magnetization directions, two charge configurations, or two optical intensities. A decoder maps the analog readout to $0$ or $1$.

The physical variable need not itself be discrete. What makes the encoding a bit is the declared pair of codewords and the readout rule. Reliability is set by overlap of readout distributions, thermal activation, drift, retention time, and correction overhead, not by the symbol names.

For $n$ classical bits, the register alphabet has size

$$\lvert\mathcal X_n\rvert=2^n.$$

At any instant an ideal deterministic register occupies one of those $2^n$ configurations. A probability distribution over configurations represents uncertainty or deliberate randomization. This contrasts with a pure $n$-qubit state, which may contain coherent amplitudes over the same labels. Coherent amplitudes do not make all labels simultaneously readable.

Qubits

The abstract two-dimensional carrier

A qubit is a quantum system whose information-bearing Hilbert space is isomorphic to

$$\mathcal H_2\cong\mathbb C^2.$$

After choosing an orthonormal computational basis $\{|0\rangle,|1\rangle\}$, a pure state can be written

$$|\psi\rangle = \alpha|0\rangle+\beta|1\rangle, \qquad |\alpha|^2+|\beta|^2=1.$$

Vectors that differ by an overall phase represent the same pure state:

$$|\psi\rangle \sim e^{i\chi}|\psi\rangle.$$

The computational basis is part of an encoding convention, not a preferred basis imposed by the definition of a qubit. Another orthonormal pair can serve as the logical basis. Quantum states develops rays and density operators, while the Bloch sphere gives the geometry of one qubit.

The coefficients $\alpha$ and $\beta$ are probability amplitudes. A computational-basis measurement has probabilities

$$p(0)=|\alpha|^2, \qquad p(1)=|\beta|^2,$$

but the state contains more operational structure than this one probability distribution. Relative phase affects measurements in other bases and interference under later transformations.

A qubit is not two classical bits

The pair of complex amplitudes does not permit arbitrary classical data to be extracted from one qubit. A measurement returns an outcome from its declared outcome set, and incompatible measurements generally disturb later statistics. Unknown nonorthogonal states cannot be perfectly distinguished in one shot, and No-Cloning and No-Signaling forbids a universal device that copies an arbitrary unknown qubit.

A qubit can nevertheless do things a classical bit cannot. It can preserve relative phase, become entangled with another system, and be transformed in bases that have no joint classical refinement. The operational consequences depend on a protocol; “being in superposition” alone is not an advantage claim.

Registers and tensor products

An $n$-qubit register has Hilbert space

$$\mathcal H_{2}^{\otimes n} \cong \mathbb C^{2^n}.$$

The computational basis is indexed by bit strings:

$$|x\rangle = |x_1\rangle\otimes\cdots\otimes|x_n\rangle, \qquad x\in\{0,1\}^n.$$

A general pure state is

$$|\Psi\rangle = \sum_{x\in\{0,1\}^n} c_x|x\rangle, \qquad \sum_x|c_x|^2=1.$$

The exponential dimension is a statement about state-space geometry and simulation cost. It is not permission to read out $2^n$ arbitrary classical numbers from one preparation. Algorithms gain value only when controlled dynamics and measurements make a task-relevant property accessible. Tensor Products develops the composition rules.

Qudits

A qudit is a $d$-dimensional quantum carrier,

$$\mathcal H_d\cong\mathbb C^d, \qquad d\geq2,$$

with a chosen computational basis

$$\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}.$$

A pure state has the form

$$|\psi\rangle = \sum_{j=0}^{d-1}c_j|j\rangle, \qquad \sum_{j=0}^{d-1}|c_j|^2=1.$$

A qubit is the case $d=2$; a qutrit is the case $d=3$. Qudits arise naturally from atomic angular-momentum manifolds, multilevel ions, molecular rotational states, photon paths or time bins, orbital angular momentum, and controlled levels of nonlinear circuits.

Dimension accounting

Dimensions multiply under composition:

$$\dim(\mathcal H_A\otimes\mathcal H_B) = d_A d_B.$$

It is often useful to express a finite dimension in qubit-equivalent log-dimension

$$n_{\mathrm{dim}}=\log_2 d.$$

This expresses dimension on a logarithmic scale; it is not automatically a communication capacity or an experimentally accessible number of bits. A single $d$-level system supports at most $d$ mutually orthogonal codewords. What can actually be communicated depends on the state ensemble, prior probabilities, channel, receiver, error criterion, and resource accounting.

There is an exact Hilbert-space equivalence between one $d$-level carrier and an $n$-qubit register only when

$$d=2^n.$$

For example,

$$\mathbb C^4\cong\mathbb C^2\otimes\mathbb C^2,$$

but a qutrit is not exactly one and a half qubits. A qutrit can be embedded into two qubits with one unused basis state, or several qutrits can be encoded into qubit registers at an asymptotic rate approaching $\log_2 3$ qubits per qutrit. For $m$ arbitrary qutrits, an exact isometric embedding needs $\lceil m\log_2 3\rceil$ qubits. Even when dimensions match, the chosen isomorphism does not preserve physical locality or make single-carrier controls into inexpensive gates between separate qubits.

Why use qudits?

Higher-dimensional carriers can match a system’s natural level structure, reduce the number of physical subsystems, or make some gates and codes more economical. They can also make calibration, selective control, readout, and leakage more difficult. Compare performance at the same accuracy, duration, and available resources rather than by dimension per component alone.

Continuous-Variable Modes

One mode, infinitely many number states

A bosonic mode is commonly described by annihilation and creation operators $a$ and $a^\dagger$ satisfying

$$[a,a^\dagger]=1.$$

Its Fock basis is

$$\{|0\rangle,|1\rangle,|2\rangle,\ldots\},$$

with number operator

$$\hat n=a^\dagger a, \qquad \hat n|n\rangle=n|n\rangle.$$

The ideal mode Hilbert space is infinite dimensional. The label continuous variable refers to canonical observables such as dimensionless quadratures

$$\begin{aligned} \hat q &= \frac{a+a^\dagger}{\sqrt2}, \\ \hat p &= \frac{a-a^\dagger}{i\sqrt2}, \\ [\hat q,\hat p] &=i. \end{aligned}$$

whose measurement outcomes are continuous. The same mode has a discrete number basis. “Continuous-variable system” and “continuous energy spectrum” are therefore not synonyms.

Optical, microwave, acoustic, and mechanical normal modes are examples. A mode is not the same object as a particle: the state $|n\rangle$ describes $n$ excitations of one mode, while a single excitation can be distributed across several modes. Mode Occupations develops this distinction, and Continuous-Variable Systems develops quadratures and phase-space descriptions.

Infinite dimension does not mean infinite usable information

The ideal mode contains arbitrarily high number states, but physical tasks impose constraints. A common energy constraint is finite mean excitation number,

$$\langle\hat n\rangle_\rho = \operatorname{Tr}(\rho\hat n) \leq \bar n.$$

Bandwidth, preparation time, detector range, nonlinearity, loss, and available squeezing impose further limits. Without an energy or amplitude constraint, statements about capacities and approximation errors can become physically misleading.

Numerical calculations often truncate the Fock basis at $N$:

$$\mathcal H_{\leq N} = \operatorname{span} \{|0\rangle,\ldots,|N\rangle\}.$$

This is an approximation, not a change in the physical definition of the mode. The truncated ladder operators also cease to obey the exact oscillator commutator: their highest retained level introduces a boundary term. Use the cutoff to test convergence, not as an exact finite-dimensional realization of $[a,a^\dagger]=I$. A trustworthy calculation reports $N$ and verifies convergence of observables or bounds the omitted probability

$$\epsilon_N = \operatorname{Tr} \left[ (I-P_{\leq N})\rho \right].$$

Modes can host discrete encodings

An oscillator need not carry continuous-variable information directly. One may choose a two-dimensional subspace and encode a qubit. Examples include:

• Fock encodings using selected number states;
• cat codes using superpositions of coherent states;
• binomial codes using finite superpositions of number states;
• Gottesman–Kitaev–Preskill encodings using grid-like states in phase space.

The physical Hilbert space remains infinite dimensional while the logical carrier is finite dimensional. Ideal GKP grid states are nonnormalizable limits; physical implementations use finite-energy approximations, as explained in GKP codes.

Physical, Encoded, and Logical Qubits

Physical qubits are controlled subsystems

A physical qubit is a physical degree of freedom used as the elementary qubit in an architecture. It may be:

• two spin projections in a fixed orbital manifold;
• two hyperfine or electronic states of an atom or ion;
• the lowest two levels of an anharmonic superconducting circuit;
• two photon polarizations, paths, or time bins;
• a pair of charge configurations;
• a two-state subspace of a larger oscillator or collective mode.

The full device Hilbert space is usually larger than $\mathbb C^2$. Selecting states $|0\rangle_{\mathrm p}$ and $|1\rangle_{\mathrm p}$ defines a physical-qubit subspace

$$\mathcal C_{\mathrm p} = \operatorname{span} \{|0\rangle_{\mathrm p},|1\rangle_{\mathrm p}\}.$$

Control pulses, noise, and measurement may couple this subspace to unused levels. Treating the device as an ideal qubit is justified only on timescales and at accuracies where those couplings are negligible or explicitly modeled.

Leakage is different from an in-code error

For a physical qubit, take $P_{\mathcal C}$ to project onto its selected two-level computational subspace. For a state $\rho$ in the full device Hilbert space, the leakage probability is

$$p_{\mathrm{leak}} = 1-\operatorname{Tr} \left(P_{\mathcal C}\rho\right).$$

An error that maps $|0\rangle$ to $|1\rangle$ remains inside the qubit subspace. Leakage instead transfers population to states outside it, such as $|2\rangle$ of a weakly anharmonic circuit, vacuum in a single-photon encoding, or an unwanted atomic level. This use of leakage differs from an ordinary correctable error leaving the code space of a multiqubit error-correcting code. For example, $X_2|000\rangle=|010\rangle$ is outside the repetition code but still inside the three-qubit computational space. The distinction determines which error model and recovery to use. Erasure and Loss Channels gives the channel-level treatment when loss is flagged or represented by an orthogonal erasure state.

An ideal projection onto the computational subspace, without resolving its basis states, gives the conditional state

$$\rho_{\mathcal C} = \frac{ P_{\mathcal C}\rho P_{\mathcal C} }{ \operatorname{Tr}(P_{\mathcal C}\rho) },$$

provided the denominator is nonzero. This formula specifies a projection-based state update, not the result of every possible leakage detector. Reporting only a normalized conditional state hides rejected runs; retain its success probability as well.

Encoded qubits

An encoded qubit is any isometric embedding

$$V:\mathbb C^2\longrightarrow\mathcal H_{\mathrm{phys}}.$$

Its logical basis states are

$$|0_L\rangle=V|0\rangle, \qquad |1_L\rangle=V|1\rangle.$$

A logical operator $\bar U$ acts on the code as the desired abstract operation:

$$\bar U V=VU.$$

This definition covers both protected and unprotected embeddings. For example, the dual-rail photonic encoding uses two modes:

$$|0_L\rangle=|1,0\rangle, \qquad |1_L\rangle=|0,1\rangle.$$

The one-photon sector is the code space. Losing the photon produces $|0,0\rangle$, which is orthogonal to the code and can be interpreted as erasure when detected.

Logical qubits and protection

Usage varies across the literature: some authors call every code-space degree of freedom a logical qubit, while others reserve logical qubit for an actively error-corrected object. Here the term means an encoded qubit addressed through logical operations, with the degree of protection stated explicitly. “Logical” alone does not prove fault tolerance.

Bosonic and encoded computation models explains how physical operations act on oscillator code spaces.

For an $n$-physical-qubit code encoding $k$ qubits, the code subspace has dimension

$$\dim\mathcal C=2^k$$

inside

$$\dim\mathcal H_{\mathrm{phys}}=2^n.$$

The redundancy allows selected errors to move the state into distinguishable syndrome sectors without revealing the encoded amplitudes. A code is useful only relative to an error model and available recovery operations.

The three-qubit repetition encoding illustrates both the possibility and the limitation:

$$\begin{aligned} |0_L\rangle&=|000\rangle, \\ |1_L\rangle&=|111\rangle, \\ \alpha|0\rangle+\beta|1\rangle &\longmapsto \alpha|000\rangle+\beta|111\rangle. \end{aligned}$$

This code can diagnose and correct one bit flip when suitable syndrome measurements and recovery are available. It does not correct an arbitrary one-qubit error and does not protect against a phase flip. The map also does not clone the unknown state: the encoded superposition is generally entangled, and none of the three individual qubits has state $\alpha|0\rangle+\beta|1\rangle$.

Why quantum error correction is possible states the correctability criterion. Three-qubit codes works through the bit- and phase-flip encodings, syndrome measurements, and recoveries.

Representative Encodings

Carrier	Encoding and physical realization	Typical limitation
Classical bit	Two readout regions in voltage, magnetization, or charge	Readout distributions overlap
Qubit	Two selected atomic, spin, or circuit states	Excitation to unused levels
Qubit	One photon in two modes	Photon loss gives vacuum
Qudit	$d$ orthogonal atomic states or photon paths	Imperfect level-selective control
CV mode	Optical, microwave, acoustic, or mechanical oscillator	Loss and finite available energy
Bosonic logical qubit	A two-dimensional code in oscillator modes	Shifts or loss beyond the code’s correction ability

This table is a taxonomy, not a ranking. “More levels” does not automatically mean “more useful information,” and “logical” does not automatically mean “better.” Performance must be compared for a task under matched constraints.

Combining different carriers

An ion’s internal levels can store qubits while collective vibrational modes mediate gates. A superconducting circuit can control a cavity containing a bosonic code. A photon can carry a state between stationary matter qubits. The subsystems need not have the same logical dimension, noise, or lifetime.

An ideal interface preserves a logical state while changing its physical carrier. Real interconnects and transducers can introduce loss, noise, mode mismatch, or phase drift. A large conversion probability alone does not establish preservation of quantum coherence.

Carrier choice therefore depends on the task. Two selected levels can simplify control; a qudit can match a multilevel model; a bosonic qubit can exploit a long-lived oscillator. Compare initialization, native gates, readout, noise, connectivity, and encoding overhead for the same operation and error target.

Exercises

1. Dimensions of a mixed register

A register contains two qubits, one qutrit, and one oscillator truncated to number states $|0\rangle$ through $|5\rangle$. What is the dimension of the modeled Hilbert space? How many qubits are required for a direct binary embedding with no compression assumptions?

Solution

The subsystem dimensions are $2$, $2$, $3$, and $6$. Tensor-product dimensions multiply:

$$D=2\cdot2\cdot3\cdot6=72.$$

An $n$-qubit register has dimension $2^n$. Since

$$2^6=64<72\leq128=2^7,$$

seven qubits are required for a direct embedding. The embedding has $128-72=56$ unused basis states. This count concerns the chosen truncated model; the untruncated oscillator is infinite dimensional.

2. When is a qudit a qubit register?

For which values of $d$ can a single qudit be identified exactly with an $n$-qubit tensor-product register without unused states? Explain why $d=6$ does not qualify, even though $\log_2 6$ is well defined.

Solution

Exact equality of dimensions requires

$$d=2^n$$

for an integer $n\geq0$. In that case an isomorphism can map the qudit basis to $n$-bit strings.

For $d=6$,

$$\log_2 6\approx2.585,$$

which is not an integer. Three qubits can host a six-dimensional code subspace, but two of their eight computational-basis states remain unused. The logarithm measures dimension on a base-$2$ scale; it does not construct fractional tensor factors.

3. Leakage and conditional quality

A three-level device uses $|0\rangle$ and $|1\rangle$ as its physical-qubit subspace. After a gate,

$$\rho = 0.45|0\rangle\langle0| + 0.45|1\rangle\langle1| + 0.10|2\rangle\langle2|.$$

Find the leakage probability and the state conditioned on remaining in the qubit subspace. Why is the conditional state alone an incomplete report?

Solution

With

$$P_{\mathcal C} = |0\rangle\langle0| + |1\rangle\langle1|,$$

the survival probability is

$$\operatorname{Tr}(P_{\mathcal C}\rho)=0.90,$$

so

$$p_{\mathrm{leak}}=1-0.90=0.10.$$

Conditioning and renormalizing gives

$$\rho_{\mathcal C} = \frac1{0.90} \left( 0.45|0\rangle\langle0| +0.45|1\rangle\langle1| \right).$$

Equivalently,

$$\rho_{\mathcal C} = \frac12 \left( |0\rangle\langle0| + |1\rangle\langle1| \right).$$

This normalized state discards the ten-percent failure probability. A complete report gives both survival and conditional state quality.

4. Loss in a dual-rail qubit

A dual-rail state is

$$|\psi_L\rangle = \alpha|1,0\rangle+\beta|0,1\rangle.$$

Model each rail by an independent pure-loss channel with vacuum environment, the same single-photon transmissivity $\eta$, and matched optical phases. Show that the output is a mixture of the unchanged logical state and vacuum. What is the state conditional on a nondestructive measurement of total photon number giving one?

Solution

On the no-loss branch, both amplitudes acquire the same factor $\sqrt\eta$, and both environments remain in vacuum. The one-photon component therefore survives without relative-amplitude distortion, with probability $\eta$. On the loss branch the environment can carry rail information, but the system is vacuum. Tracing out the environments gives

$$\rho_{\mathrm{out}} = \eta|\psi_L\rangle\langle\psi_L| + (1-\eta)|0,0\rangle\langle0,0|.$$

The vacuum is orthogonal to the code space

$$\operatorname{span}\{|1,0\rangle,|0,1\rangle\}.$$

For $\eta>0$, the stated nondestructive projection preserves the conditional logical state $|\psi_L\rangle$. Resolving which rail contains the photon would instead measure the logical basis, and destructive photodetection would leave no stored qubit. Without conditioning, the pure-loss model has erasure probability $1-\eta$. Unequal transmissivities distort the surviving amplitudes; unmatched phases or additional dephasing can also spoil the state even at equal transmissivity.

5. What the repetition code protects

Encode

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$

as

$$|\psi_L\rangle = \alpha|000\rangle+\beta|111\rangle.$$

Apply a bit flip $X_2$ to the middle qubit and compute the eigenvalues of the parity checks $Z_1Z_2$ and $Z_2Z_3$. Then apply a phase flip $Z_1$ to the original codeword. What does this show?

Solution

The middle-qubit flip gives

$$X_2|\psi_L\rangle = \alpha|010\rangle+\beta|101\rangle.$$

Both parity checks change sign:

$$Z_1Z_2=-1, \qquad Z_2Z_3=-1$$

on this error sector. The syndrome identifies the middle qubit without measuring $\alpha$ or $\beta$, so applying $X_2$ recovers the codeword.

A phase flip gives

$$Z_1|\psi_L\rangle = \alpha|000\rangle-\beta|111\rangle.$$

This state still has eigenvalue $+1$ for both $Z$-parity checks. The error acts as a logical phase flip and is not corrected by the repetition code. The code protects against one selected error type, not arbitrary noise.

6. Finite energy and Fock truncation

Let a mode state have finite mean occupation

$$\langle\hat n\rangle=\bar n.$$

Show that the probability of finding $n\geq N+1$ obeys

$$\Pr(n\geq N+1) \leq \frac{\bar n}{N+1}.$$

What does the bound say about numerical truncation?

Solution

Let $p_n=\langle n|\rho|n\rangle$. For all terms with $n\geq N+1$,

$$n\geq N+1.$$

Therefore

$$\begin{aligned} \bar n &= \sum_{n=0}^{\infty}np_n \\ &\geq \sum_{n=N+1}^{\infty}np_n \\ &\geq (N+1) \sum_{n=N+1}^{\infty}p_n. \end{aligned}$$

Rearranging gives

$$\epsilon_N = \sum_{n=N+1}^{\infty}p_n \leq \frac{\bar n}{N+1}.$$

Finite mean occupation controls omitted probability, but a small probability tail alone does not bound errors in unbounded observables such as $\hat n$. For example, for fixed $\bar n$ and $N+1>\bar n$,

$$\rho_N=\left(1-\frac{\bar n}{N+1}\right)|0\rangle\langle0| +\frac{\bar n}{N+1}|N+1\rangle\langle N+1|$$

has omitted probability $\bar n/(N+1)\to0$, yet projecting onto levels through $N$ and normalizing gives vacuum and loses the entire mean occupation $\bar n$. This is a family of states, not a failure of convergence for a single fixed state with finite mean energy. Check the observables actually being calculated or use stronger tail bounds.

Continue to the Bloch sphere for one-qubit geometry and the circuit model for operations on registers.

References

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• Gottesman, D., A. Kitaev, and J. Preskill. “Encoding a qubit in an oscillator.” Physical Review A 64, 012310 (2001). doi:10.1103/PhysRevA.64.012310.
• Knill, E., and R. Laflamme. “Theory of quantum error-correcting codes.” Physical Review A 55, 900–911 (1997). doi:10.1103/PhysRevA.55.900.
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