Quantum Channel

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In the intricate field of quantum information theory, a quantum channel stands as a pivotal conduit, capable of transmitting not only the ephemeral whispers of quantum information but also the robust pronouncements of classical data. Consider, for instance, the general dynamics of a qubit – that fundamental unit of quantum information – as a prime example of what a quantum channel can convey. Conversely, the transmission of a mundane text document across the vast expanse of the Internet serves as a familiar analogy for classical information transfer.

From a terminological standpoint, quantum channels are precisely defined as completely positive (CP) trace-preserving maps that operate between spaces of operators. More specifically, a quantum channel can be understood as a quantum operation, but viewed not merely as the reduced dynamics of an isolated system, but rather as a deliberate pathway designed for the conveyance of quantum information. It's worth noting a subtle distinction in terminology: some scholars opt to use "quantum operation" to encompass all trace-decreasing maps, reserving "quantum channel" exclusively for those that strictly preserve the trace.[1]

Memoryless Quantum Channel

For the purposes of our immediate exploration, we shall assume that all state spaces, whether they pertain to classical or quantum systems, are finite-dimensional.

The term "memoryless" in this context carries the identical connotation it holds within classical information theory. It signifies that the output generated by a channel at any given moment is contingent solely upon the input provided at that precise moment, and is entirely independent of any preceding inputs.

Schrödinger Picture

Let us first consider quantum channels that are exclusively dedicated to the transmission of quantum information. These are, in essence, quantum operations, and we shall now summarize their defining characteristics.

Suppose we have two finite-dimensional Hilbert spaces, denoted as

$H_{A}$

and

$H_{B}$ ,

which represent the state spaces of the sending and receiving ends of the channel, respectively. The family of operators acting on

$H_{A}$

will be denoted as

$L(H_{A})$ .

Within the framework of the Schrödinger picture, a purely quantum channel is characterized by a map, let's call it

$\Phi$ ,

that transforms density matrices acting on

$H_{A}$

into density matrices acting on

$H_{B}$ .

This transformation must adhere to the following stringent properties:[2]

Linearity: As mandated by the fundamental postulates of quantum mechanics, the map

$\Phi$

must exhibit linearity.
Positivity Preservation: Since density matrices are inherently positive, the map

$\Phi$

is obligated to preserve the cone of positive elements. In other words,

$\Phi$

must be a positive map.
Complete Positivity: If we introduce an ancilla of any arbitrary finite dimension, say 'n', and couple it to the system, the induced map, represented as

$I_{n}\otimes \Phi$

(where $I_{n}$ is the identity map on the ancilla), must also remain positive. Consequently, it is a requirement that

$I_{n}\otimes \Phi$

is positive for every possible value of 'n'. Maps possessing this property are termed completely positive.
Trace Preservation: Density matrices are specifically defined to have a trace equal to 1. Therefore, the map

$\Phi$

must inherently preserve this trace property.

These descriptors – completely positive and trace preserving – are often abbreviated as CPTP. It’s worth noting that in some academic contexts, the fourth property (trace preservation) might be relaxed to simply require that

$\Phi$

is not trace-increasing. However, for the discussions within this article, we shall consistently assume that all channels under consideration are CPTP.

Heisenberg Picture

While density matrices acting on $H_{A}$ constitute a specific subset of operators on $H_{A}$ (and similarly for system B), a linear map $\Phi$ defined between these density matrices can, under the assumption of finite dimensionality, be uniquely extended to encompass the entire space of operators. This extension yields the adjoint map, $\Phi^{*}$ , which elegantly describes the action of $\Phi$ within the Heisenberg picture:[3]

The spaces of operators, $L(H_{A})$ and $L(H_{B})$ , are themselves Hilbert spaces, endowed with the Hilbert–Schmidt inner product. Consequently, when we consider $\Phi$ as a map between these Hilbert spaces, $\Phi : L(H_{A})\rightarrow L(H_{B})$ , its adjoint, $\Phi^{*}$ , is defined by the relationship:

$\langle A, \Phi(\rho) \rangle = \langle \Phi^{*}(A), \rho \rangle$ .

While $\Phi$ transforms states residing on system A into states on system B, its adjoint, $\Phi^{*}$ , performs the converse operation, mapping observables associated with system B to observables on system A. This duality mirrors the relationship between the Schrödinger and Heisenberg descriptions of dynamics: the statistical outcomes of measurements remain invariant whether one considers the observables to be fixed while the states evolve, or vice versa.

A direct verification reveals that if $\Phi$ is assumed to be trace preserving, then its adjoint, $\Phi^{*}$ , necessarily becomes unital. This means that $\Phi^{*}(I) = I$ , where $I$ represents the identity operator. From a physical perspective, this implies that in the Heisenberg picture, the most trivial observable, the identity, remains unchanged after the application of the channel.

Classical Information

Up to this point, our discussion has focused on quantum channels exclusively designed for the transmission of quantum information. However, as the introduction alluded to, the inputs and outputs of a channel can also encompass classical information. To incorporate this, the formulation we have developed requires a slight generalization. In the Heisenberg picture, a purely quantum channel is essentially a linear map $\Psi$ between spaces of operators:

$\Psi : L(H_{B}) \rightarrow L(H_{A})$

This map must be both unital and completely positive (CP). The operator spaces can be conceptualized as finite-dimensional C*-algebras. Therefore, we can more broadly state that a quantum channel is a unital CP map existing between C*-algebras:

$\Psi : \mathcal{B} \rightarrow \mathcal{A}$ .

Classical information can then be seamlessly integrated into this framework. The observables of a classical system can be represented by a commutative C*-algebra, which is essentially the space of continuous functions, $C(X)$ , defined over some set $X$ . Assuming $X$ to be finite, $C(X)$ can be identified with the n-dimensional Euclidean space, $\mathbb{R}^{n}$ , equipped with entry-wise multiplication.

Consequently, if classical information is to be considered part of the input in the Heisenberg picture, we would define $\mathcal{B}$ to include the relevant classical observables. An illustrative example would be a channel of the form:

$\Psi : L(H_{B}) \otimes C(X) \rightarrow L(H_{A})$ .

It is crucial to observe that $L(H_{B}) \otimes C(X)$ remains a C*-algebra. An element 'a' within a C*-algebra $\mathcal{A}$ is designated as positive if it can be expressed as $a = x^{*}x$ for some element $x$ . The concept of positivity for a map is defined analogously. However, it is important to acknowledge that this particular characterization is not universally embraced; the quantum instrument is sometimes put forth as a more generalized mathematical construct capable of accommodating both quantum and classical information transfer. In certain axiomatic frameworks of quantum mechanics, classical information is conveyed through a Frobenius algebra or a Frobenius category.

Examples

Time Evolution

For a purely quantum system, the evolution of its state over time, at a specified moment $t$ , is governed by the transformation:

$\rho \rightarrow U\rho U^{*}$

where $U = e^{-iHt/\hbar}$ represents the unitary evolution operator, with $H$ being the Hamiltonian of the system and $\hbar$ denoting the reduced Planck constant. This process yields a CPTP map in the Schrödinger picture, and thus qualifies as a quantum channel.[4] The corresponding dual map in the Heisenberg picture is given by:

$A \rightarrow U^{*}AU$ .

Restriction

Consider a composite quantum system whose state space is described by $H_{A} \otimes H_{B}$ . For a given state $\rho \in H_{A} \otimes H_{B}$ , the reduced state of $\rho$ pertaining to system A, denoted as $\rho^{A}$ , is obtained by applying the partial trace operation over the B system:

$\rho^{A} = \mathrm{Tr}_{B} \;\rho$ .

This partial trace operation itself constitutes a CPTP map, and therefore functions as a quantum channel in the Schrödinger picture.[5] In the Heisenberg picture, the dual map associated with this channel transforms an observable A of system A into:

$A \rightarrow A \otimes I_{B}$ ,

where $I_{B}$ is the identity operator acting on system B.

Observable

An observable fundamentally associates a numerical value, say $f_{i} \in \mathbb{C}$ , with a specific quantum mechanical effect, $F_{i}$ . The collection $\{F_{i}\}$ is assumed to comprise positive operators acting on the relevant state space, with the condition that $\sum_{i} F_{i} = I$ (such a collection is formally known as a POVM).[6][7] In the Heisenberg picture, the observable map, $\Psi$ , transforms a classical observable $f = \begin{bmatrix} f_{1} \\ \vdots \\ f_{n} \end{bmatrix} \in C(X)$ into the corresponding quantum mechanical observable:

$\Psi(f) = \sum_{i} f_{i}F_{i}$ .

In essence, this operation involves integrating the classical function $f$ with respect to the POVM elements to yield the quantum mechanical observable. It can be readily verified that $\Psi$ is both CP and unital.

The corresponding Schrödinger map, $\Psi^{*}$ , takes density matrices as input and produces classical states as output:[8]

$\Psi(\rho) = \begin{bmatrix} \langle F_{1},\rho \rangle \\ \vdots \\ \langle F_{n},\rho \rangle \end{bmatrix}$ ,

where the inner product is the Hilbert–Schmidt inner product. Furthermore, if we conceptualize states as normalized functionals and invoke the Riesz representation theorem, we can express this as:

$\Psi(\rho) = \begin{bmatrix} \rho(F_{1}) \\ \vdots \\ \rho(F_{n}) \end{bmatrix}$ .

Instrument

The observable map, when viewed in the Schrödinger picture, possesses a purely classical output algebra. Consequently, it solely delineates the statistical outcomes of measurements. To account for the concurrent change in the quantum state, we introduce the concept of a quantum instrument. Let $\{F_{1}, \dots, F_{n}\}$ represent the effects (elements of a POVM) associated with an observable. In the Schrödinger picture, an instrument is a map $\Phi$ that accepts a pure quantum state $\rho \in L(H)$ as input and produces an output in the space $C(X) \otimes L(H)$ :

$\Phi(\rho) = \begin{bmatrix} \rho(F_{1})\cdot F_{1} \\ \vdots \\ \rho(F_{n})\cdot F_{n} \end{bmatrix}$ .

Now, let $f = \begin{bmatrix} f_{1} \\ \vdots \\ f_{n} \end{bmatrix} \in C(X)$ . The dual map in the Heisenberg picture is then defined as:

$\Psi(f \otimes A) = \begin{bmatrix} f_{1}\Psi _{1}(A) \\ \vdots \\ f_{n}\Psi _{n}(A) \end{bmatrix}$ ,

where $\Psi_{i}$ is defined as follows: First, factor each effect $F_{i}$ as $F_{i} = M_{i}^{2}$ (this factorization is always possible since elements of a POVM are positive operators). Then, define $\Psi_{i}(A) = M_{i}AM_{i}$ .

We observe that $\Psi$ is both CP and unital. It's also worth noting that $\Psi(f \otimes I)$ precisely yields the observable map we discussed earlier. The map:

$\tilde{\Psi}(A) = \sum _{i}\Psi _{i}(A) = \sum _{i}M_{i}AM_{i}$

captures the overall transformation of the quantum state.

Measure-and-Prepare Channel

Imagine a scenario where two parties, A and B, wish to communicate. Party A performs a measurement on an observable and then communicates the outcome to B via a classical channel. Based on this received message, B prepares their quantum system in a specific state. In the Schrödinger picture, the first stage of this process, $\Phi_{1}$ , simply represents A's measurement, which is precisely the observable map:

$\Phi_{1}(\rho) = \begin{bmatrix} \rho(F_{1}) \\ \vdots \\ \rho(F_{n}) \end{bmatrix}$ .

If, upon obtaining the $i$ -th measurement outcome, B prepares their system in state $R_{i}$ , then the second stage of the channel, $\Phi_{2}$ , transforms the classical outcome into the density matrix:

$\Phi_{2}\left(\begin{bmatrix} \rho(F_{1}) \\ \vdots \\ \rho(F_{n}) \end{bmatrix}\right) = \sum _{i}\rho (F_{i})R_{i}$ .

The complete operation is the composition of these two stages:

$\Phi(\rho) = \Phi_{2} \circ \Phi_{1}(\rho) = \sum _{i}\rho (F_{i})R_{i}$ .

Channels exhibiting this structure are aptly termed measure-and-prepare channels, or alternatively, entanglement-breaking channels.[9][10][11][12]

In the Heisenberg picture, the dual map, $\Phi^{*} = \Phi_{1}^{*} \circ \Phi_{2}^{*}$ , is defined by:

$\Phi^{*}(A) = \sum _{i}R_{i}(A)F_{i}$ .

A crucial characteristic of a measure-and-prepare channel is that it cannot be the identity map. This fact is directly linked to the no-teleportation theorem, which asserts the impossibility of classical teleportation (distinct from entanglement-assisted teleportation). In simpler terms, it means that a quantum state cannot be reliably measured and reconstructed.

Within the framework of channel-state duality, a channel is classified as measure-and-prepare if and only if its corresponding state is separable. Indeed, all states that emerge from the partial action of a measure-and-prepare channel are separable, which is precisely why these channels are also referred to as entanglement-breaking channels.

Pure Channel

Let us now consider a purely quantum channel, $\Psi$ , within the Heisenberg picture. Assuming finite dimensionality, $\Psi$ is a unital CP map operating between spaces of matrices:

$\Psi : \mathbb{C}^{n\times n} \rightarrow \mathbb{C}^{m\times m}$ .

According to Choi's theorem on completely positive maps, such a map $\Psi$ must take the form:

$\Psi(A) = \sum_{i=1}^{N}K_{i}AK_{i}^{*}$

where $N$ is an integer such that $N \le nm$ . The matrices $K_{i}$ are known as the Kraus operators of $\Psi$ , named after the German physicist Karl Kraus who first introduced them.[13][14][15] The minimum number of Kraus operators required to represent a channel is termed its Kraus rank. A channel possessing a Kraus rank of 1 is designated as a pure channel. The time evolution of a quantum system, as discussed earlier, represents one such example of a pure channel. This terminology, too, stems from the channel-state duality: a channel is pure precisely when its associated dual state is a pure state.

Teleportation

In the context of quantum teleportation, the objective is for a sender to transmit an arbitrary quantum state of a particle to a receiver, who may be at a considerable distance. Consequently, the teleportation process itself constitutes a quantum channel. The apparatus required for this process necessitates a quantum channel for the transmission of one particle of an entangled pair to the receiver. Teleportation is accomplished through a joint measurement performed on the sent particle and the remaining particle of the entangled pair. This measurement yields classical information that must then be conveyed to the receiver to finalize the teleportation. A crucial point is that this classical information can be transmitted even after the quantum channel involved in the entanglement distribution has ceased to be active.

In the Experimental Setting

From an experimental perspective, a straightforward method for implementing a quantum channel involves the transmission of single photons through fiber optic cables, or alternatively, through free space. Single photons can propagate for distances up to 100 km in standard optical fibers before signal loss becomes prohibitive.[ citation needed ] The encoding of quantum information for applications such as quantum cryptography is typically achieved by utilizing the photon's time of arrival ( time-bin entanglement ) or its polarization. Critically, such a channel is capable of transmitting not only the basis states (e.g., $|0\rangle$ , $|1\rangle$ ) but also their superpositions (e.g., $|0\rangle + |1\rangle$ ). The delicate coherence of the quantum state is preserved throughout its journey via the channel. This stands in stark contrast to the transmission of electrical pulses through wires, which constitute a classical channel and can only convey classical information (i.e., sequences of 0s and 1s).

Channel Capacity

The cb-Norm of a Channel

Before formally defining channel capacity, it is essential to introduce the preliminary concept of the norm of complete boundedness, or the cb-norm, of a channel. When evaluating the capacity of a channel $\Phi$ , we often need to compare its performance against an "ideal channel," let's denote it as $\Lambda$ . For instance, if the input and output algebras are identical, a natural choice for $\Lambda$ is the identity map. Such a comparison necessitates a well-defined metric for quantifying the distance between channels.

Given that a channel can be mathematically represented as a linear operator, it might seem intuitive to employ the standard operator norm. This would imply that the closeness of $\Phi$ to the ideal channel $\Lambda$ is defined by:

$\|\Phi - \Lambda\| = \sup \{\|(\Phi - \Lambda)(A)\| \;|\; \|A\| \leq 1\}$ .

However, this approach encounters a significant issue: the operator norm can exhibit an undesirable behavior. Specifically, when we tensor $\Phi$ with the identity map on some auxiliary system (an ancilla), the resulting operator norm, $\|\Phi \otimes I_{n}\|$ , can potentially increase without bound as the dimension $n$ of the ancilla approaches infinity.

This tendency makes the operator norm an inadequate measure for comparing channels in a way that is relevant to quantum information transmission. The resolution to this problem lies in the introduction of the cb-norm. For any linear map $\Phi$ between C*-algebras, the cb-norm is defined as:

$\|\Phi\|_{cb} = \sup _{n}\|\Phi \otimes I_{n}\|$ .

This definition ensures that the norm remains bounded, irrespective of the dimension of the auxiliary system used.

Definition of Channel Capacity

The mathematical model we are employing for a channel is analogous to the one used in classical channel capacity theory.

Let $\Psi : \mathcal{B}_{1} \rightarrow \mathcal{A}_{1}$ be a channel described in the Heisenberg picture, and let $\Psi_{id} : \mathcal{B}_{2} \rightarrow \mathcal{A}_{2}$ be a chosen ideal channel. To enable a meaningful comparison between $\Psi$ and $\Psi_{id}$ , we must consider the possibility of encoding and decoding operations. This is achieved by examining the composition:

$\hat{\Psi} = D \circ \Phi \circ E : \mathcal{B}_{2} \rightarrow \mathcal{A}_{2}$

where $E$ represents an encoder and $D$ represents a decoder. Within this context, both $E$ and $D$ are required to be unital CP maps with appropriately defined domains and codomains. The quantity of primary interest is the "best-case scenario," which is determined by minimizing the distance between the composed channel $\hat{\Psi}$ and the ideal channel $\Psi_{id}$ :

$\Delta (\hat{\Psi}, \Psi_{id}) = \inf _{E,D}\|\hat{\Psi} - \Psi_{id}\|_{cb}}$

where the infimum is taken over all permissible encoders and decoders.

To transmit sequences of $n$ symbols (words of length $n$ ), the ideal channel is conceptually applied $n$ times. Thus, we consider the tensor power of the ideal channel:

$\Psi_{id}^{\otimes n} = \Psi_{id} \otimes \cdots \otimes \Psi_{id}$ .

The tensor product operation, $\otimes$ , signifies that $n$ independent inputs are subjected to the operation $\Psi_{id}$ . This is the quantum mechanical analogue of concatenation. Similarly, $m$ applications of the channel $\Phi$ correspond to:

$\hat{\Psi}^{\otimes m}$ .

The quantity $\Delta (\hat{\Psi}^{\otimes m}, \Psi_{id}^{\otimes n})$ therefore serves as a measure of the channel's ability to faithfully transmit words of length $n$ when invoked $m$ times.

This leads us to the following formal definition:

A non-negative real number $r$ is considered an achievable rate of $\Psi$ with respect to $\Psi_{id}$ if, for all sequences $\{n_{\alpha}\}$ and $\{m_{\alpha}\}$ of natural numbers such that $m_{\alpha} \rightarrow \infty$ and $\limsup_{\alpha} (n_{\alpha} / m_{\alpha}) < r$ , the following limit holds:

$\lim_{\alpha} \Delta (\hat{\Psi}^{\otimes m_{\alpha}}, \Psi_{id}^{\otimes n_{\alpha}}) = 0$ .

The sequence $\{n_{\alpha}\}$ can be interpreted as representing a message composed of potentially infinitely many words. The condition involving the limit supremum signifies that, in the limit, faithful transmission can be achieved by invoking the channel no more than $r$ times for each word transmitted. Alternatively, $r$ can be understood as the number of symbols per channel invocation that can be sent without error.

The channel capacity of $\Psi$ with respect to $\Psi_{id}$ , denoted by $C(\Psi, \Psi_{id})$ , is defined as the supremum of all achievable rates.

From the very definition, it is trivially true that $0$ is an achievable rate for any channel.

Important Examples

As previously mentioned, for a system characterized by the observable algebra $\mathcal{B}$ , the ideal channel $\Psi_{id}$ is, by definition, the identity map $I_{\mathcal{B}}$ . Consequently, for a purely $n$ -dimensional quantum system, the ideal channel is the identity map acting on the space of $n \times n$ matrices, $\mathbb{C}^{n\times n}$ . With a slight notational convention, this ideal quantum channel will also be denoted simply as $\mathbb{C}^{n\times n}$ . Similarly, a classical system with an output algebra $\mathbb{C}^{m}$ will have an ideal channel represented by the same symbol. We can now state some fundamental channel capacities:

The channel capacity of the classical ideal channel $\mathbb{C}^{m}$ with respect to a quantum ideal channel $\mathbb{C}^{n\times n}$ is:

$C(\mathbb{C}^{m}, \mathbb{C}^{n\times n}) = 0$ .

This result is a direct restatement of the no-teleportation theorem: it is fundamentally impossible to transmit quantum information solely through a classical channel.

Furthermore, the following equalities hold:

$C(\mathbb{C}^{m}, \mathbb{C}^{n}) = C(\mathbb{C}^{m\times m}, \mathbb{C}^{n\times n}) = C(\mathbb{C}^{m\times m}, \mathbb{C}^{n}) = \frac{\log n}{\log m}$ .

The first part of this equality signifies, for instance, that an ideal quantum channel offers no advantage over an ideal classical channel in terms of transmitting classical information. When $n = m$ , the most efficient rate achievable is one classical bit per qubit.

It is pertinent to highlight that both of the aforementioned capacity bounds can be surpassed with the strategic utilization of quantum entanglement. The entanglement-assisted teleportation scheme provides a mechanism for transmitting quantum information even when employing a classical channel. Similarly, superdense coding demonstrates the remarkable feat of achieving two classical bits of information per qubit. These profound results underscore the indispensable role that entanglement plays in the realm of quantum communication.

Classical and Quantum Channel Capacities

Using the same notation as in the preceding subsection, the classical capacity of a channel $\Psi$ is defined as:

$C(\Psi, \mathbb{C}^{2})$ ,

meaning it is the capacity of $\Psi$ when compared against the ideal channel operating on the classical one-bit system, $\mathbb{C}^{2}$ .

Analogously, the quantum capacity of $\Psi$ is defined as:

$C(\Psi, \mathbb{C}^{2\times 2})$ ,

where the reference system is now the one-qubit system, $\mathbb{C}^{2\times 2}$ .

Channel Fidelity

Another crucial metric for quantifying how well a quantum channel preserves information is known as channel fidelity. This concept is derived from the more general notion of the fidelity of quantum states. Given two pure states, $|\psi\rangle$ and $|\phi\rangle$ , their fidelity is defined as the probability that one state, when subjected to a test designed to identify the other, will yield a positive result:

$F(|\psi\rangle, |\phi\rangle) = |\langle \psi |\phi \rangle |^{2}$ .

This definition can be extended to encompass the comparison of two mixed states, which are represented by density matrices $\rho$ and $\sigma$ :[16][17]

$F(\rho, \sigma) = \left(\mathrm{tr} \sqrt{{\sqrt {\rho }}\sigma {\sqrt {\rho }}}\right)^{2}$ .

The channel fidelity for a specific quantum channel is determined by transmitting one half of a maximally entangled pair of systems through that channel. Subsequently, the fidelity is calculated between the resulting state and the original input state.[18]

Bistochastic Quantum Channel

A bistochastic quantum channel is a quantum channel $\Phi(\rho)$ that possesses the property of being unital. This means that $\Phi(I) = I$ . These types of channels encompass a range of transformations, including unitary evolutions, convex combinations of unitaries, and, in dimensions greater than 2, other possibilities as well.[19][20]