Theoretical framework for quantum networks

We present a framework to treat quantum networks and all possible transformations thereof, including as special cases all possible manipulations of quantum states, measurements, and channels, such as, e.g., cloning, discrimination, estimation, and tomography. Our framework is based on the concepts of quantum comb—which describes all transformations achievable by a given quantum network—and link product—the operation of connecting two quantum networks. Quantum networks are treated both from a constructive point of view—based on connections of elementary circuits—and from an axiomatic one—based on a hierarchy of admissible quantum maps. In the axiomatic context a fundamental property is shown, which we call universality of quantum memory channels: any admissible transformation of quantum networks can be realized by a suitable sequence of memory channels. The open problem whether this property fails for some nonquantum theory, e.g., for no-signaling boxes, is posed.

Figure 1

(a) Graphical representation of internal connections in a quantum network: vertices represent quantum operations, incoming and outgoing arrows represent input and output systems. The resulting diagram is a direct acyclic graph. (b) Graphical representation of a quantum network: free incoming (outgoing) arrows have been added to the diagram in (a) in order to represent input (output) systems entering (exiting) the network. (c) Totally ordered quantum network. The vertices in diagram (b) have been ordered from left to right according to a sequential ordering compatible with the causal ordering fixed by input-output relations.

Figure 2

(a) Equivalence between an arbitrary sequence of memory channels and a totally ordered quantum network: a sequence of quantum memory channels from Alice (left side) to Bob (right side) is equivalent modulo stretching and reshuffling of the quantum wires to an array of channels connected by internal ancillae, i.e., a totally ordered quantum network. (b) A sequence consisting of identical memory channels, with the memory initialized in state |0⟩ before the first use, and traced out after the last use.

Figure 3

Quantum network resulting from a concatenation of $N$ (generally different) isometric channels ${\mathcal{V}}_j\left(\rho \right)≔V_j\rho V_j^{†}$, with the last channel followed by partial trace over the ancillary degrees of freedom. Any positive operator satisfying Eq. (25) is the Choi-Jamiołkowski operator of a network of this form.

Figure 4

Quantum network resulting from a concatenation of $N$ isometric channels, with the last channel followed by a von Neumann measurement $\left\{M_i\right\}$ over the ancillary degrees of freedom. Any collection of positive operators $\left\{R_i^{\left(N\right)}\right\}$ summing up the Choi-Jamiołkowski operator of a deterministic quantum network describes a probabilistic quantum network of this form.

Figure 5

The scheme represents the connection of two networks, in which junction of two arrows means identification of the corresponding quantum systems. The final network is still a direct acyclic graph, with the set of vertices coinciding with the union of sets of vertices of the component sub-networks, and with some free input and output arrows.

Figure 6

The commutative diagram shows the relation between a transformation $\tilde{\mathcal{S}}$ from linear maps $\mathcal{T}$ to linear maps ${\mathcal{T}}^{\prime }$ and its conjugate $\mathcal{S}≔\mathfrak{C}\circ \tilde{\mathcal{S}}\circ {\mathfrak{C}}^{-1}$ through the Choi-Jamiołkowski isomorphism, transforming Choi-Jamiołkowski operators $T$ to Choi-Jamiołkowski operators $T^{\prime }$.

Figure 7

In the first row we illustrate the diagrammatic representation of combs. A quantum system is represented by a line, a quantum operation (1-comb) by a box, a 2-comb by a diagram with two teeth, and a 3-comb by a diagram with three teeth. In the second row we represent the map corresponding to a 4-comb, transforming the input 3-comb in an output 1-comb.

Figure 8

Identification of a quantum $N$-comb with the Choi-Jamiołkowski operator of an $N$-partite memory channel. The teeth of the comb correspond to the isometries $\{{\mathcal{V}}_0,\dots ,{\mathcal{V}}_{N-1}\}$ in the memory channel.

Figure 9

Realization of admissible $N$-maps by connection of memory channels. The input of the map is an $(N-1)$-comb, corresponding to a sequence of $N-1$ isometric channels $\{{\mathcal{W}}_0,\dots ,{\mathcal{W}}_{N-2}\}$. The Choi-Jamiołkowski operator of the map is an $N$-comb, corresponding to a sequence of $N$ isometric channels $\{{\mathcal{V}}_0,\dots ,{\mathcal{V}}_{N-1}\}$. The output of the map is obtained by connecting the free wires of the two memory channels.

Figure 10

Diagrammatic representation of all different quantum combs arising from the tensor product of a 2-comb $S^{\left(2\right)}$ with a 2-comb $T^{\left(2\right)}$.

Figure 11

The five possible realization schemes for admissible (2,2) maps, that transform a 2-comb in a 2-comb.

Figure 12

Realization of an $N$-instrument as a sequence of $N$ isometric channels $\{{\mathcal{V}}_0,\dots ,{\mathcal{V}}_{N-1}\}$ followed by a von Neumann measurement on the ancillary degrees of freedom. The input of the instrument is an $(N-1)$-comb, corresponding to a sequence of isometric channels $\{{\mathcal{W}}_0,\dots ,{\mathcal{W}}_{N-2}\}$. Conditionally to outcome $i$, the output of the instrument is a quantum operation, which represents the input-output transformation of the whole composite network.

Figure 13

Realization of a quantum $N$-tester as a probabilistic quantum network consisting of preparation of a pure input state $|\Psi ⟫$, isometric channels $\{{\mathcal{V}}_1,\dots ,{\mathcal{V}}_N\}$, and a final measurement with POVM $\left\{{\tilde{P}}_i\right\}$. The memory channel corresponding to the sequence of isometric channels $\{{\mathcal{W}}_0,\dots ,{\mathcal{W}}_N\}$ is tested by connecting its wires with the wires of the tester and by running the resulting quantum circuit.