Postulate 4: The state space of a composite physical system is the [italics ours] tensor product of the state spaces of the component physical systems. Moreover, if we have systems numbered through , and system number is prepared in the state , then the joint state of the total system is .
Nielsen and Chuang, Quantum Computation and Information, 2010, p.94
There is no one, “true” (i.e. uniquely preferred) tensor product operator mapping a given pair of Hilbert spaces to a given target Hilbert space. This is of course well known, but the details matter, and are not disposed of by appealing to isomorphism. To do so is to ignore benefits that accrue from multiple distinct tensor product operators in the same multipartite system. None of the operators are distinguished relative to the others any more than any orthonormal basis of a Hilbert space is distinguished relative to the others. Separability of states and locality of unitary operators and observables is tensor product operator-dependent. This statement stands in tension with the statement that entanglement is physical. In this introduction to the use of multiple distinct tensor product operators in multipartite quantum systems the examples that are given focus on (1) modularity and (2) factorizing of states and operators.
A change of tensor product operator in a quantum program is a quantum program transformation. Work on quantum program verification, validation and optimization needs an underlying machine independent mathematically rigorous semantics, , , that should take account of a variety of program transformations. Due to modularity, the change of tensor product operator is among these transformations. The term modularity refers to the common-place practice that we build more complex systems out of more basic systems that have been previously constructed and supplied. In the case of quantum systems it is easy to imagine a quantum circuit built from previously independently constructed circuits, each using potentially distinctly different tensor product operators. The details of combining these previously constructed circuits into a more complex circuit is described below. In particular, change of tensor product operator for quantum circuits in “mid-circuit” can be effectively produced using swaps and, if necessary, similarity transformations. An example is given below.
1.1 Outline of the paper
Section (2) consists of two subsections. In subsection (2.1) a basis-independent construction of tensor product operators as maximally general bilinear operators is given following the procedures described in . In subsection (2.2) the dependency of separability of states on the tensor product operator is shown. Section (3) states and proves a lemma that precisely shows the unitary relationship between two distinct tensor product operators of the same type. In that section a corollary of the lemma is stated and proved that gives an equivalence between change of tensor product operators and unitary similarity for general linear operators on the tensor product operator target space. Section (4) concentrates on coordinating unitary operators and measurements with tensor product operators. In subsection (4.1) an example is given of a simple circuit to show the effect of a change of the tensor product operator based on the example of subsection (2.2). The two versions of the circuit are combined to show a how a change of tensor product operator may be uniformly implemented in “mid-circuit,” as well as to give an example of a localization of a
-qubit gate. Subsection (4.2) presents a standard example of a teleportation circuit from  but in which distinct tensor products among the three pairs of qubits are used and combined. The purpose is not to show there is a gain but that the modularity presents no difficulty and does not the change the physics. We conclude with a brief section on future work to be undertaken.
2 Multiple tensor product operators
2.1 A basis-independent approach to tensor product operators
We give here the definition of a tensor product operator as a maximally general bilinear operator. The focus is on bilinear rather than multilinear tensor product operators from which multilinear operators are derivable. The definition is mildly category-theoretic without the diagrams, is basis independent, and follows . We begin by mentioning some convenient terminology regarding relaxed but still common and rigorous use of the term type in its elementary sense to be used here and in the rest of the paper.
We are throughout this paper concerned with exploiting the multiple tensor product operators available for each type. The type of a function mapping a set to a set , denoted by
is the ordered pair of setsand is denoted . is the domain of and is the codomain of . They are components of ; if either the domain or codomain is altered, the function is altered. If the domain and codomain of a function are spaces, such as Hilbert spaces as they are here, the codomain is also called the target space.
A bilinear function is a tensor product operator if all bilinear operators defined on uniquely factorize into a linear operator and . Specifically, for every bilinear operator , there is a unique linear function such that ; i.e. for all , . This makes a tensor product operator defined on a maximally general bilinear operator on . For Hilbert spaces, a tensor product operation is also required to preserve norms: if , then .
If are both tensor product operators, then both are bilinear and hence there are unique linear operators and such that and . Also there is are unique linear operators and such that and . It follows that and are both the identity function, and and . Thus and inverse to each other. In the case of Hilbert spaces, since and are norm-preserving, and are forced to be norm-preserving, and therefore unitary.
At this point, all bilinear tensor product operators of each type have been defined and shown to be related in a basis-independent manner by composition with a unitary operator on the target space of the type.
2.2 Tensor product operator-dependency
In this subsection an example of two tensor product operators is given for which the Bell states corresponding to each of the tensor product operators are shown to be separable with respect to the other operator. We begin by reviewing the ordered basis-dependent approach to constructing a tensor product operator of a given type and then give examples of tensor product operator-dependency of entanglement of pure states, unitary operators and observables.
A particular tensor product operator of type where , and are Hilbert spaces can be constructed by choosing ordered orthonormal bases , and for these spaces, and choosing any bijective order-preserving function . For this purpose, the order on is lexicographically induced by the orders on and . That is a bijection constrains the dimension of in relation to the dimensions of and . It is then easy to prove by using the chosen bases that the uniquely determined bilinear extension of to the whole of satisfies the basis-independent requirement for being a tensor product operator. For fixed bases of and , distinct ordered orthonormal bases in are in one-to-one correspondence with distinct tensor product operators of type .
As mentioned above, separability of states is tensor product operator-dependent. None of the multiple distinct tensor product operators of the same type
are distinguished relative to the others any more than any of the orthonormal bases for a vector space are distinguished relative to the others.
: This is an example of two tensor product operators and with unequal corresponding Bell states, each of which is separable with respect to the other’s associated tensor product operator.
Take with the usual ordered orthonormal basis, which we denote by . , and take as the tensor product space with basis to be the usual ordered orthonormal basis.
Let be the re-ordering of obtained by applying the non-local controlled-not operator , to the basis vectors in .
The column vectors and matrices in this example carry as subscripts the bases with respect to which they are written. is then given by,
is a permutation of :
A controlled-not operator is clearly dependent on the computational basis in hand. In particular, relative to is given by the familiar self-adjoint Hermitian matrix,
Since is self-adjoint, the matrix for has the same form relative to both bases in this case. (Being self-adjoint is not essential to this and subsequent examples, except where specifically noted.) Let , and therefore . In the following equation, (4), is non-local, so the result is not surprising. However, neither of these two tensor product operators is distinguished relative to the other, despite one of the ordered bases, , being standard, the other not. (Just change the coordinate system.) Then,
The equations in (4) show that the indicated Bell state corresponding to each of the tensor product operators, , , is separable relative to the other.
Note that the expected basic algebraic rules for Kronecker product operators on matrices holds, provided the matrices are written with respect to the appropriate ordered bases.
Restricting to a standard or preferred basis and preferred corresponding tensor product operator ignores the issue of what benefits may accrue from multiple tensor product operations in the same multipartite system. A better approach is to coordinate observables and unitary operators to tensor product operators, as we make clear in the examples in section (4) below.
In this subsection we have given an explicit example of two tensor product operators for which Bell states corresponding to each of the tensor product operators were shown to be separable with respect to the other operator.
3 The unitary relationship between tensor product operators
In this section we examine the relationship between two tensor product operators. Within a type , the relationship between two tensor product operators and is given by a uniquely determined unitary operator on the target space of the type. is given in terms of the action of the two tensor product operators on the bases of and .
The effect on vectors and linear transformations due to a change of tensor product operator is given by the following lemma and its proof. Additionally, we prove a corollary that exhibits an equivalence between (1) change of tensor product operators and (2) unitary similarity of linear operators on the target space of the tensor product operators.
: For , let be two tensor product operators, and for , let be any orthonormal basis of . Then
Proof: The existence and uniqueness of a unitary operator satisfying (5) has been proved in the argument of subsection (2.1). Equation (6), explicitly useful for matrix calculations, is a trivial consequence of (5). For (7) we have,
For any bipartite quantum circuit C using , consider the circuit C that results from C obtained by replacing by and every gate and observable by the result of the similarity transform . The corollary below shows that is equivalent to C in the sense that the evolution of any state through C to a state corresponds to the evolution of through C to . This is shown by (9) below.
: Suppose are tensor product operators with , is a linear operator on , and and are linear operators on and , respectively. Then
and for any vector
Proof: For (8) we have,
The unitary relationship between tensor product operators and the effect on vectors and linear transformations due to a change of tensor product operator has been made explicit by the above lemma and its corollary. These results reveal the equivalence between change of tensor product operators and unitary similarity of linear operators and observables.
4 Coordination of gates, measurements and tensor product operators
In this section we present examples showing how quantum gates and measurements are affected by several different tensor product operators applied to different pairs of qubits in the same circuit. In particular, basis-dependent gates such as and observables need to be coordinated, in a manner to be made precise, with tensor product operators, since separability of states and operators is tensor product operator-dependent.
With respect to a single tensor product operator, any general linear operator may fail to be local, yet may be unitarily similar to a local operator. The term localizable operator is used in that case.
There exists operators, e.g. and , that are nonlocalizable relative to any ordered orthonormal basis are said to be non-localazable. Since and are self-adjoint they also serve as nonlocalizable observables.
4.1 An example of coordination
In this subsection we present a standard example of a teleportation circuit from  but in which distinct tensor products among the three pairs of qubits are used and combined. The purpose is not to show there is a gain but that the modularity presents no difficulty and does not the change the physics (as expected). A nontrivial example of a localizable non-local gate is given in this subsection.
The formation of a Bell state, such as , is often given by applying a controlled-not operator to the result of applying a Hadamard operator to one of a pair of qubits initially in state :
In the example below we will use the two tensor product operators and from (4). If is the tensor product operator for the pair of qubits in the circuit below, then of course the Bell state is formed via the standard gate sequence (see, e.g., p27, )
If for the pair of qubits , the operator is replaced by , then the Bell state is formed. By (4), the state is also formed because it is identical to . Similarly, if in addition to replacing by , the gate is eliminated, then the gate is effectively built into the tensor product operator for the pair and the state formed, , is as given by the circuit below:
The point of this example is not to show there is a gain by eliminating the gate. Rather, it is to show that both operators (gates in the case of circuits) and observables for multipartite systems must be chosen and coordinated with the tensor product operators that combine the states of the separate parts of such systems.
It might be argued that when we construct circuits we do not explicitly build in the tensor product operators among the qubits. But in fact we do implicitly build in the tensor product operators by the choice of gates and observables in the circuit that are tensor product operator-dependent. To change the tensor product operator in a circuit is to change the ordered computational basis for the combined circuit, by re-ordering the computational basis, or changing the computational basis set altogether, or both.
Suppose now that the state space of the circuit of both qubits is with each of the qubits having state space . In modifying (12) to obtain (13), the change from tensor product operator to re-orders the computational basis so that for example is and results in the overall state in circuit (13) immediately after the Hadamard gate being the same as it is in (12) after the gate, as shown in the circuit diagram. Suppose is measured into its computational basis immediately following the gate in (12) and the Hadamard gate in (13). In (12) we obtain or
with equal probability. In (13) we obtain with certainty even though we are measuring the same state of the overall state of the circuit in each case. We get different results because the (implicit) observables are different in the two cases. Specifically, an observable for the projective measurement into the computational basis of , considered in isolation, is . An observable for the measurement of the combined system of both qubits in (12) is given by . Let and be the orthogonal projection operators in the spectral decomposition of . In (13), by lemma (3.1) the observable is given by with corresponding orthogonal projection operators and . Then for the circuit (12) we have the two possible post-measurement states
In both cases the denominators, being show the post-measurement outcomes to be equally likely. In (13) the possible post-measurement states are
The second of these two post-measurement states is undefined, and the first has unit probably. The distinct observables for these two circuits show that if the projective measurement into the computational basis of as part (13) is carried out with respect to the observable corresponding to while holds between and , the improper coordination of the observable with the tensor product operator results in an unintended distribution on the possible results of the measurement.
Change of tensor product operator from to in either of these circuits can result in the localization of some measurements. Consider a joint projective measurement of the qubits into the eigenstates of observable
which is a local with respect to since
The eigenvalues ofand are and , each with multiplicity . The corresponding measurement only nontrivially involves and measures into the eigenstates of . Thus, if the original eigenstates of are needed for further processing, they are known from the known observable and the post-measurement state that would have resulted from the original measurement. Further, they recoverable from the known eigenstates of the observable.
The relationship between and in circuit (12) suggests that changing tensor products in “mid-circuit” can be useful. Changing tensor product operators in this way is effectively achievable by keeping another related circuit but using a different tensor product operator in the other circuit. One can then transfer qubit states by swapping relative to the appropriate tensor product operators that connect the two circuits by connecting the corresponding qubits with swap gates. For example, consider the following circuit
The tensor product operator to connect with and with can be any tensor product operator one likes and could even be distinct for each of the qubit pairs to be connected with swap gates. In practice, that operator would be determined by what unitary operator one chooses to be the swap gates, but one might as well choose, say, .
In the following subsection, we show how to combine these extra tensor product operators with and to produce states of the -qubit circuit.
4.2 Mixing tensor products: example with teleportation
This subsection presents a standard example of a teleportation circuit from  but in which distinct tensor products among the three pairs of qubits are used and combined. The purpose of this exercise is not to show there is a gain but rather to demonstrate that the modularity presents no difficulty and does not change the physics. In addition, a nontrivial example of a localizable non-local operator is given in this subsection.
Consider the following well-known 3-qubit quantum circuit to teleport the state of one of Alice’s qubits to Bob’s qubit where Alice and Bob share the Bell state at the start of the process, .
Below, we explore the consequences for this teleportation circuit arising from using distinct tensor product operators for each of the three 2-qubit subsystems. The purpose is to exhibit in detail how potentially distinct tensor product operations among different pairs of qubits of a circuit affect the circuit’s action. It will be seen that if the gates and observables on the -qubit subsystems are coordinated with the tensor product operators on each of the -qubit subsystems, then the net action of the circuit is to produce the same -qubit states , and as it would if it employed the same tensor product operator on all three -qubit subsystems.
The state spaces of each of the three qubits is with the usual standard ordered orthonormal basis which we have labeled , above. For illustration purposes, we choose three distinct ordered orthonormal bases for the state spaces of the qubit pairs: , and . The tensor product symbol in the expressions for each of these state spaces for the qubit pairs does not denote an operator on pairs of vectors. Instead, the symbol denotes the tensor product space to serve as the codomain for tensor product operators on the vectors in the spaces being joined. For example, denotes the tensor product space that serves as the codomain for all tensor product operators of type such as the operators and that were defined above in example (2.1). and have the same type: , where it is assumed the tensor product space has been constructed to be .
Two of the bases are given by: and and the third is given by , where is derived from the Bell basis with respect to and is given by
Thus, combines the subsystem consisting of qubits and , combines and , and combines and . The corresponding matrix for the similarity transformation for a change of basis from to is
yields a third tensor product operator such that
Let be the state space of the circuit, and choose any fixed ordered orthonormal basis of and denote it by
An expression such as
is not rigorously defined. To be rigorously defined the operator would have to map, in particular, a pair of states, one of which is -level, and the other -level, to an -level state while also mapping a pair of -level states to a -level state. Thus, the expression is ill-typed due to both an implicit overloading of the symbol and an ambiguous grouping. A rigorous means of treating this obstacle is to define several tensor product operators as in the following. Define three tensor product operators: For each ,
Prefix notation is used in these auxiliary tensor product operator definitions since it is necessary in the case of . With these operators the successive states of (19) can be calculated. Let the initial state of be . We then have