3.2.1 Quantum States

States embody all information about the preparation of a quantum system that has potential consequences for later statistical measurements. States correspond to density operators ρ satisfying (2,3)

Expectation values of measurements of observables A are given by 〈A〉 _ρ  = tr[Aρ]. So density operators can be thought of defining the linear positive normalized map mapping observables onto their expectation values. Finite‐dimensional quantum systems such as two‐level or spin systems are equipped with a finite‐dimensional Hilbert space ℋ. In a basis{|0〉, …, |d〉}, any state ρ can be represented as

The set of all density operators is typically referred to as state space.

The state space for a single qubit is particularly transparent: it can be represented as the unit ball in ℝ³, the Bloch ball. The Hilbert space of a qubit is spanned by {|0〉, |1〉}. In terms of this basis, a state can be written as

where X, Y, and Z denote the Pauli matrices

So states of single qubits are characterized by vectors (x ₁ , x ₂ , x ₃) ∈ ℝ³ taken from the unit ball, so by Bloch vectors.

In general, the state space of a d‐dimensional quantum system is a (d ² − 1)‐dimensional convex set: if ρ ₁ and ρ ₂ are legitimate quantum states, then the convex combination λρ ₁ + (1 − λ)ρ ₂ with λ ∈ [0, 1] is also a quantum state. Such a procedure reflects mixing of two quantum states. Convex sets have extreme points. The extreme points of state space are the pure quantum states. This can be represented as vectors in the Hilbert space,

c ₁, …, c_d  ∈  . State space is convex, but not a simplex: so there are typically infinitely many different representations of states

in terms of pure states, where (p ₁, …, p_K ) is a probability distribution. This innocent‐looking fact is at the root of the technicalities in mixed‐state entanglement theory: even the very definition of separability or classical correlations refers to the notion of a convex combination of products. Meaning, there must exist a decomposition in terms of extreme points such that each of the terms corresponds to a product, or – in other words – that a state is contained in the convex hull of product states.

Let us end this subsection with a remark on the composition of quantum systems, which is of key relevance when talking about entanglement. The composition of quantum systems is incorporated in the state concept via the tensor product: the Hilbert space of a composite system consisting of parts with Hilbert spaces ℋ₁ and ℋ₂ is defined to be ℋ = ℋ₁ ⊗ ℋ₂. The basis of ℋ can then identified to be

where {|1〉, …, |d ₁〉} and {|1〉, …, |d ₂〉} are bases of ℋ₁ and ℋ₂, respectively.

3.2.2 Quantum Operations

A quantum operation or a quantum channel reflects any processing of quantum information, or any way a state can be manipulated by an actual physical device. When grasping the notion of a quantum operation, two approaches appear to be particularly natural: on the one hand, one may list the elementary operations that are known from any textbook on quantum mechanics and conceive a general quantum operation as a concatenation of these ingredients. On the other hand, in an axiomatic approach one may formulate certain minimal requirements any meaningful quantum operation has to fulfill in order to fit into the framework of the statistical interpretation of quantum mechanics. Fortunately, the two approaches coincide in the sense that they give rise to the same concept of a quantum operation. We only touch upon this issue, as this will be discussed in great detail in Chapter 5 on quantum channels. To start with the former approach, any quantum operation

can be thought of being a consequence of the application of the following elementary operations:

• Unitary dynamics: Time evolution according to Schrödinger dynamics gives rise to a unitary operation
  
• Composition of systems: For states ω, this is
  
  This is the composition with an uncorrelated additional system.
• Partial traces: This amounts to
  
  in a composite quantum system.
• von Neumann measurements : This is a measurement associated with a set of orthogonal projections, π ₁, …, π_K .

Now, to mention the latter approach, any quantum operation T consistent with the statistical interpretation of quantum mechanics must certainly be linear and positive: density operators must be mapped onto density operators. Trace preservation of the map incorporates that the trace of the density operator remains to be given by unity.

However, perhaps surprisingly, mere positivity of the map T is not enough: it could well be that the map is applied to a part of a composite quantum system, which has previously been prepared in an entangled state. Needless to say, the image under this map must again correspond to a legitimate density operator. This means that we have to require that

is positive for all n ∈ ℕ. It may, at first, not appear very intuitive that this is a stronger requirement as mere positivity, referred to as complete positivity. The good news is that these conditions are already enough to specify the class of maps that correspond to physical quantum operations, being identical to the above‐sketched class of concatenated maps. So obviously, quantum channels are completely positive maps and can be cast into the general form

Trace preservation is reflected as . If they are unital, they satisfy . In turn, any such completely positive map can be formulated as a dilation of the form

where U is a unitary acting in ℋ and the Hilbert space ℋ _E of an “environment.” So any channel can be thought of as resulting from an interaction with an additional quantum system, a system one does not have complete access to.

3.3.1 Phase Space

The physics of N canonical (bosonic) degrees of freedom – or modes for that matter – is that of N harmonic oscillators. Such a quantum system is described in a phase space. The phase space of a system of N degrees of freedom is ℝ^2N , equipped with an antisymmetric bilinear form (3,14,15). The latter originates from the canonical commutation relations between the canonical coordinates. Writing the canonical coordinates as (R ₁, …, R _2N ) = (X ₁, P ₁, …, X_N , P_N ), the canonical commutation relations can be expressed as

where the skew‐symmetric 2N × 2N matrix σ is given by

This matrix is block diagonal, as observables of different degrees of freedom certainly commute with each other. Here, units have been chosen such that ℏ = 1. The commutation relations are those of position and momentum, although, needless to say, this should not be taken too literal: these coordinates correspond, for example, to the quadratures of field modes of light.

A convenient tool for a description of states in phase space is the displacement operator – or, depending on the scientific community, Weyl operator. Defined as

for ξ ∈ ℝ^2N , it is straightforward to see that this operator indeed generates translations in phase space. For a single degree of freedom, this displacement operator becomes

The canonical commutation relations manifest themselves for Weyl operators as .

Equivalent to referring to a state, that is, a density operator, one can specify the state of a system with canonical coordinates by a suitable function in phase space. In the literature, one finds a plethora of such phase space functions, each of which equipped with a certain physical interpretation. One of them is the characteristic function (17,19). It is defined as the expectation value of the Weyl operator, so as

This is generally a complex‐valued function in phase space. It uniquely defines the quantum state, which can be reobtained via . The characteristic function is the Fourier transform of the Wigner function, so familiar in quantum optics,

The Wigner function is a real‐valued function in phase space. It is normalized, in that for a single mode the integral over phase space delivers the value 1. Yet, it is, in general, not a probability distribution, and it can take negative values.

One of the useful properties is the so‐called overlap property (16,17). If we define the Wigner function of the operators A ₁ and A ₂ as the Fourier transforms of and , respectively, and denote them with and , we have that

This can straightforwardly be used to determine moments of canonical coordinates. For example, assume that we know the Wigner function. How can we determine from it the first moment of the position observable? This is easily found to be

Similarly, the expectation value of the momentum operator is obtained as

Similar expressions can be found for integration along any direction in phase space.

Often, it is also convenient to describe states in terms of their moments (3). The first moments are the expectation values of the canonical coordinates, so d_k  = 〈R_k 〉 _ρ  = tr[R_kρ]. The second moments, in turn, can be embodied in the real symmetric 2N × 2N matrix γ, the entries of which are given by

j, k = 1, …, N. This matrix is typically referred to as the covariance matrix of the state. Similarly, higher moments can be defined.

3.3.2 Gaussian States

As mentioned before, Gaussian states play a central role in continuous‐variable systems, so in quantum systems with canonical coordinates. Quantum states of a system consisting of N degrees of freedom are called Gaussian (or also quasifree) if its characteristic function is a Gaussian function in phase space (3,5,15,18), that is, if χ takes the form

As Gaussians are defined by their first and second moments, so are Gaussian states. The vector d and the matrix γ can then be identified as the displacement and covariance matrix in the above sense.

What states are now Gaussian in this sense? Coherent states with state vectors as in Eq. 3.1 constitute important examples of Gaussian states, having a covariance matrix γ =  : Coherent states are nothing but vacuum states, displaced in phase space. The covariance matrix of a squeezed vacuum state is given by γ = diag(d, 1/d) for d > 0 (and rotations thereof), −log d being its squeezing parameter. Thermal or Gibbs states are also Gaussian states, which can in the number basis be expressed as

where  = (e^β  − 1)⁻¹ is the mean photon number of the thermal state of inverse temperature β > 0. These states are mixed, with covariance matrix

3.3.3 Gaussian Unitaries

The significance of the Gaussian states, needless to say, stems in part from the significance of Gaussian operations. Gaussian unitaries are generated by Hamiltonians, which are at most quadratic in the canonical coordinates: such Hamiltonians, yet, are ubiquitous in physics. So a Gaussian unitary operation is of the form

H being real and symmetric, corresponding to a bosonic quadratic Hamiltonian. Such unitaries correspond to a representation of the symplectic group Sp(2N, ℝ). It is formed by those real matrices for which

In other words, these transformations are the familiar transformations from one legitimate set of canonical coordinates to another. In turn, the connection from S to the Hamiltonian is determined by S = e^Hσ . It is convenient to keep track of the action of Gaussian unitaries on the level of second moments (5,14,15), that is, covariance matrices, as

Those Gaussian unitaries that are energy preserving are typically called passive. In the optical context, such unitaries preserve the total photon number. Beam splitters of some transmittivity t and phase shifts, for example, have this property. They correspond – in the convention chosen in this chapter – to

Whether a transformation is passive or not can easily be read off from the matrix S: the matrices S corresponding to passive operations are exactly those that are orthogonal, S ∈ SO(N). These transformations again form a group, Sp(2N, ℝ) ∩ O(2N). This group is a representation of U(N), which is a property that can conveniently be exploited when assessing quantum information tasks that are accessible using passive optics (see, e.g., Ref. (19)).

Active transformations, in contrast, do not preserve the total photon number. Operations that induce squeezing in optical systems are such active transformations. The most prominent example is a unitary that squeezes the quantum state of a single mode,

the number x > 0 characterizing the strength of the squeezing. We find that

this matrix in turn determines the transformation on the level of covariance matrices.

It seems a right moment to get back to the constraint that any covariance matrix actually has to satisfy. Is any real symmetric 2N × 2N matrix a legitimate covariance matrix? The answer can only be “no”; the Heisenberg uncertainty principle constrains the second moments of any quantum state. The Heisenberg uncertainty principle may be expressed as the semidefinite constraint

3.2

In turn, for any real symmetric matrix, there exists a state ρ having these second moments (3).

That this is indeed nothing but the familiar Heisenberg uncertainty principle can be seen as follows: For any covariance matrix γ of a system with N degrees of freedom, there exists an S ∈ Sp(2N, ℝ) such that

3.3

The numbers s ₁, …, s_N can be identified to be given by the positive part of the spectrum of iσγ. This is the normal mode decomposition, resulting from the familiar procedure of decoupling a coupled system of harmonic oscillators. The covariance matrix of Eq. 3.3 is then the covariance matrix of a system of N uncoupled modes, each of which is in a thermal state of mean “photon number”  = (s_i  − 1)/2 (14,15). Now, having this in mind, we can reduce 3.2 to a single‐mode problem, for a covariance matrix of the form γ = diag(s, s). For the covariance matrix of one of these uncoupled modes, in turn, the Heisenberg uncertainty principle becomes

where ΔX = 〈(X − 〈X〉 _ρ )²〉 _ρ and ΔP = 〈(P − 〈P〉 _ρ )²〉 _ρ .

This normal mode decomposition is a very helpful tool when evaluating any quantity dependent on quantum states that is unitarily invariant. For example, to calculate the (Von Neumann) entropy S(ρ) = −tr[ρ log ρ] of a Gaussian state becomes a straightforward enterprise, once the problem is reduced to a single‐mode problem using this Williamson normal form.

Finally, in this subsection, let us note that Gaussian states can be characterized by entropic expressions: Namely, Gaussian states are those quantum states for fixed first and second moments that have the largest entropy. Quite surprisingly, it is not at all technically involved to show that this is the case. If σ is any quantum state having the same first and second moments as the Gaussian state ρ, then

the first symbol on the right‐hand side denoting the quantum relative entropy. This argument shows that in fact, Gaussians have the largest Von‐Neumann entropy. This may be regarded as a manifestation of the Jaynes minimal information principle.

3.3.4 Gaussian Channels

A more general class of Gaussian operations is given by the Gaussian channels (20–22). Such Gaussian channels play a quite central role in quantum information with continuous variables. Most prominently, they are models for optical fibers as noisy or lossy transmission lines. A Gaussian channel is again of the form

3.4

where now U is a Gaussian unitary and ρ_E is a Gaussian state of some number of degrees of freedom. Such channels arise whenever one encounters a coupling which is at most quadratic in the canonical coordinates, to some external degrees of freedom, in turn governed by some bosonic quadratic Hamiltonian. Needless to say, such a situation is quite ubiquitous. Whenever one encounters, say, a weak coupling of canonical degrees of freedom to a some bosonic heat bath, it gives rise to a Gaussian channel in this sense.

How can such channels now concisely be described? Since they map Gaussian states onto Gaussian states, they are – up to displacements – completely characterized by their action on second moments. This action can be cast into the form

3.5

where G is a real symmetric 2N × 2N matrix and F is an arbitrary real 2N × 2N matrix (20,21). On the level of Weyl or displacement operators, this can be grasped as W_ξ  ↦ W_Fξ  exp(−ξ^T Gξ/2).

In more physical terms, the matrix X may be said, roughly speaking, to determine the amplification or attenuation part of the channel. The matrix Y originates from the “quantum noise induced by the coupling with the environment.” Not every pair of matrices F and G result in a legitimate quantum channel: from complete positivity we have that

3.6

This inequality sign originates again from the Heisenberg uncertainly principle. Equations 3.5 and 3.6 specify the most general Gaussian quantum channel as given by Eq. 3.4.

An important example of such Gaussian channels in practice is the lossy channel. This channel does what the name indicates: it loses photons. It can be modeled by a beam splitter of transmittivity t ∈ [0, 1] with an empty port in which the vacuum is coupled in. In the above language, this becomes

Then, the channel that induces classical Gaussian noise is a Gaussian quantum channel (23,24). This channel can be conceived as resulting from random displacements in phase space with a Gaussian weight,

which is reflected as a map

with a positive matrix G. This classical noise channel can also be realized as a lossy channel, followed by an amplification, which is identical to the lossy channel, yet with t > 1.

In this language, one can also conveniently read off how well an impossible operation can be approximated in a way that induces minimal noise. For example, optical phase conjugation is an impossible operation, in that there is no device that perfectly performs this operation with perfect fidelity. This would correspond to a channel of the above form with

However, if we allow for G = (2, 2), then the map γ ↦ F^TγF + G corresponds to a channel, so a legitimate completely positive map. One may say – which can also be made more precise in terms of a figure of merit – that for Gaussian states far away from minimum uncertainty, this additional offset Y hardly matters. Close to minimal uncertainty, this additional noise leads to a significant deviation from actual phase conjugation.

Then, how well can Gaussian quantum cloning be implemented? The answer to this question depends, needless to say, on the figure of merit. Natural choices would be the joint fidelity of the output with respect to two specimens of the input, or the single clone fidelity. However, if we ask which symmetric Gaussian channel approximates the perfect cloner inducing minimal noise, then the answer will take us only a single line. We fix F to be identical to

then G =  is a minimal solution of 3.6. This can be conceived as an optimal cloner inducing minimal noise (25). Indeed, it turns out that this channel is identical to the optimal 1 → 2‐cloner when the joint fidelity is taken as the figure of merit (26). So when judging clones by means of their joint fidelity, a Gaussian channel amounts, indeed, to the optimal cloner for Gaussian states, which is by no means obvious. Interestingly, it turns out that when one judges single clones (by means of the single‐copy fidelity), the optimal cloner is no longer Gaussian (27).

3.3.5 Gaussian Measurements

If we project parts of a system in a Gaussian state onto a Gaussian state of a single mode, how do we describe the resulting Gaussian state? This is nothing but a non‐trace‐preserving channel. In practice, this occurs in a dichotomic measurement associated with Kraus operators

A perfect avalanche photodiode could be described by a measurement of this type: K ₀ corresponds to the outcome that no photon has been detected, K ₁ to the one in which photons have been detected, although there is no finer resolution concerning the number of photons. Imperfect detectors may be conveniently and accurately described by means of a lossy channel, followed by a measurement of this type.

In a system consisting of N + 1 modes in a Gaussian state ρ, what would be the covariance matrix of

The covariance matrix of ρ can be written as

where A is a 2N × 2N matrix and B is a 2 × 2 matrix. It turns out that the covariance matrix of the resulting (unmeasured) N modes is given by (28)

This expression can be identified as a Schur complement of the matrix . This formula provides a very useful description of the resulting state after a vacuum projection, without the need of actually determining the resulting quantum state explicitly.

In turn, a homodyne detection leads to a covariance matrix of the form (28)

where π is a 2 × 2 matrix of rank 1. The inverse has then to be understood as the pseudoinverse. The most general Gaussian operation, including Gaussian measurements, resulting from the concatenation of the above elementary operations (28–30), gives rise to a transformation on the level of covariance matrices

Here, Γ is by itself a covariance matrix on 2N modes,

and , where

is the covariance matrix of the partial transposition of the Gaussian state described by Γ. This is the transformation law for any completely positive map that maps Gaussian states onto Gaussian states. This approach can be understood in terms of the isomorphism between completely positive maps and positive operators (29–31). If one asks a question what operations are accessible in the Gaussian setting, this is a natural starting point.

3.3.6 Non‐Gaussian Operations

It might appear illogical to think that the formalism of Gaussian states and Gaussian operations has anything to contribute once we leave the strict framework of the Gaussian setting. After all, with general quantum operations, the reduced description in terms of first and second moments becomes inappropriate.¹ However, for the probably most important Gaussian operation from the quantum optical perspective, this language is still valuable.

This measurement again corresponds to a dichotomic measurement distinguishing the absence or presence of photons, as realized with perfect avalanche photon detectors. In contrast to the case of the outcome associated with K ₀ = |0〉〈0|, the outcome of does not correspond to a Gaussian operation. Yet, it is clear how one can describe the state ρ after such a measurement in mode labeled N + 1 – corresponding to a “click” in the detector – of an entangled of N modes:

This is not a convex combination of Gaussian states, but nevertheless a sum of two Gaussians, each of which can be characterized by its moments. So in a network consisting of only Gaussian unitaries and k such yes–no detectors, the resulting state will at most be a sum of 2 ^k contributions, each of which has a description in terms of first and second moments, as can be obtained from the above Schur complements.

An important measurement of this type is the one where one “subtracts a photon.” Here, in one of the ports of a beam splitter, the input of a single mode is fed in, into the other vacuum, such that the second moments transformation becomes

Then, one postselects on the outcomes corresponding to K ₁, to a “clicking” detector. For the values of t ∈ [0, 1] close to 1, one can, to an arbitrarily good approximation (in trace‐norm), realize a transformation

at the expense that the respective outcome becomes very unlikely. Hence, this procedure amounts to essentially applying an annihilation operator to the state. Such photon subtractions have been realized experimentally to prepare non‐Gaussian states (11,32). They form, for example, the starting point of distillation procedures with continuous‐variable systems (33) or for ways to violate Bell's inequalities using homodyne detectors (34,35).