## Abstract

In this paper, we present a formulation of quantum theory in terms of *bold operator tensors*. A circuit is built up of operations where an operation corresponds to a use of an apparatus. We associate collections of operator tensors (which together comprise a bold operator) with these apparatus uses. We give rules for combining bold operator tensors such that, for a circuit, they give a probability distribution over the possible outcomes. If we impose certain physicality constraints on the bold operator tensors, then we get exactly the quantum formalism. We provide both symbolic and diagrammatic ways to represent these calculations. This approach is manifestly covariant in that it does not require us to foliate the circuit into time steps and then evolve a state. Thus, the approach forms a natural starting point for an operational approach to quantum field theory.

## 1. Introduction

In this paper, we will present the bold operator tensor formulation of quantum theory. This is based on the operator tensor formulation presented in [1,2]. This formulation allows a manifestly covariant treatment of quantum theory for circuits with possible applications beyond circuits (to quantum field theory for example).

An operator tensor is a Hermitian operator with subscripts and superscripts (e.g. ) acting on a Hilbert space determined by the subscripts and superscripts (this would be in the example). A bold operator tensor is a list of operator tensors all acting on the same Hilbert space, for example, 1.1

The word ‘bold’ refers to the fact that we use bold type. These bold operator tensors will allow a more compact treatment of the physics of circuits than the operator tensor formulation.

An operation corresponds to a use of an apparatus in the laboratory. To perform an experiment, we need to wire such operations together into a circuit. Each operation has some outcome (or set of outcomes) associated with it. This might correspond to a particular light flashing. A bold operation corresponds to a set of operations associated with the same apparatus use.

We will define the notion of an outcome-complete physical bold operator as a natural mathematical object to associate with outcome-complete bold operations. With appropriate definitions in place we are able to provide the following axiom for quantum theory.

**Axiom 1**. There is a one-to-one correspondence between outcome-complete bold operations and outcome-complete physical bold operators.

This single axiom is sufficient to fully specify quantum theory and is equivalent to the usual operational formulation of quantum theory for circuits in terms of density operators, completely positive maps and positive operator-valued measures (POVMs).

## 2. Operational description

### (a) Operations and bold operations

In the laboratory, we build experiments out of *apparatuses*. We will consider experiments of the type where we have quantum systems (electrons, photons, …) passing between apparatuses. Each apparatus may have outcomes we can read off it (e.g. lights flashing, or a pointer, or clicking noises). Furthermore, each apparatus may have apertures that we can allow quantum systems to pass through.

An *operation* is a useful abstraction. It corresponds to a single use of an apparatus (i) with a specified set of outcomes and (ii) in which some apertures are nominated as inputs for given types of system (we allow this type of systems to pass into the apparatus through this aperture) and some apertures are nominated as outputs for a given type of system (we allow systems to pass out of the apparatus through this aperture). When the outcome is in the outcome set associated with the operation we will say that the operation ‘happened’. We represent an operation as
2.1where we use subscripts for input systems and superscripts for output systems. Systems are represented by a,b,…. We need to include integer subscripts in the symbolic notation for reasons we will explain shortly. We will denote the outcome set associated with a particular operation as o(A). Sometimes it is useful (especially in the bold formulation) to add a further label, *l*, to this notation. Then we put
2.2Then the outcome set is o(A[*l*]) which we will sometimes abbreviate as *o*[*l*].

We define a *bold operation* as a list of operations each having the same input and output types (so they have the same subscripts and superscripts). We denote a bold operation by
2.3Written out in full, this is
2.4The most natural bold operation is where all the operations correspond to a given apparatus but with different outcome sets such that
2.5Then we will say that is an *outcome-complete bold operation*.

### (b) Fragments and bold fragments

We can use operators to build fragments and circuits. A fragment is a bunch of operations wired together so that (i) outputs are connected to inputs for the same type of system and (ii) there are no closed loops (it is a directed acyclic graph). Here is an example of a fragment: 2.6We see here why we need integer subscripts on the type labels in the symbolic notation—they say where the wires are (in the diagrammatic notation we do not need them as we can see from the picture where the wires are). The output set associated with a fragment is the Cartesian product of the output sets associated with each of the operations that comprise it. In this example, the output set is 2.7Hence, we only say that the fragment ‘happens’ if the outcome at each operation is in the associated outcome set.

A bold fragment is where we wire a bunch of bold operations together according to the same rules. For example, 2.8Written out in full, the bold fragment is a list of operators whose outcome sets are generated by the Cartesian products of the outcome sets associated with the operations comprising the fragment. For example, 2.9We can also form bold fragments in which the outcome sets associated with the entries in the list are not Cartesian products of outcome sets from the operations. An outcome-complete bold fragment is where the outcome sets are disjoint and their union is equal to the Cartesian product of the full set of outcomes associated with each apparatus (associated with each operation).

A circuit is a fragment with no open inputs or outputs left over. A bold circuit is a bold fragment with no inputs or outputs left open. 2.10Associated with a circuit is a probability. Associated with a bold circuit is a list of probabilities (one for each circuit). Associated with an outcome-complete bold circuit is a probability distribution. Our task now is to say how to calculate these in quantum theory. We can break a circuit up into fragments in any way we wish and then analyse the parts and how they fit back together.

## 3. Bold operator tensors

Associated with each system type, a, is an input Hilbert space, , and an output Hilbert space, . These have the same dimension. We define as the space of Hermitian operators acting on and as the space of Hermitian operators acting on . We write the tensor product as 3.1Here we have to introduce integer subscripts on the type labels so we know which Hilbert space is which. For complex Hilbert spaces we have the interesting property that 3.2An operator tensor is an element of the space . We can represent it by diagrammatic notation 3.3A bold operator tensor is a list of operator tensors that are all elements of the same space (see the example in (1.1)). We can use diagrammatic and symbolic notation 3.4

## 4. Bold operator fragments

We can join two or more operator tensors by using the partial trace operation where there are repeated indices (or wires). First consider a simple example. The expression is the trace of the operators and . In standard notation it is equal to
4.1We put the (!) to denote that the right hand side is not written in the notation being developed in this paper. Clearly,
4.2is just equal to times the trace of and . Any operator can be written as a linear sum of product operators such as . Hence, we can give meaning to
4.3simply by demanding that the contraction operation on the a_{1} index respects this linearity. This is a partial trace operation being applied in the space associated with the a_{1} label. We get an operator associated with the space determined by the open input and output type labels (in this case, ). This method clearly extends so we can evaluate an arbitrary expression of this type.

We call an arbitrary expression of this type an *operator fragment*. We can represent operator fragments diagrammatically and symbolically. For example,
4.4This object is an operator belonging to (these are the types left open).

We can similarly define *bold operator fragments*. For example,
4.5We can represent *bold operator fragments* diagrammatically. For example,
4.6If there are no open inputs or outputs (so all indices are contracted over) then we have a set of real numbers.

## 5. Formulating quantum theory

Here we will simply give an axiom for quantum theory in the bold operator language using the results of [1,2] where an axiom for quantum theory was given for operator tensors. We will provide some definitions and a theorem first to put these axioms in context.

*Definition of physicality*. An operator, , is *physical* iff
5.1and
5.2for all rank one projection operators and and all types c. Here is the identity operator on .

The motivation for this definition is that we want to admit into our framework preparations and measurement outcomes associated with rank one projectors (in the case of preparations these are the pure states). We also want to allow the identity operator as this corresponds to tracing out—i.e. absorbing the system and ignoring any outcomes on this absorbing apparatus. Given that we admit these preparations and measurement outcomes, it is clearly necessary to require physicality if we want to give the probability interpretation to operator circuits. The following can be shown.

### Theorem 5.1

*An operator,* *is physical iff
*5.3*and
*5.4

The T superscript in (5.3) indicates we are taking the partial transpose in the corresponding space. We require an explicit choice of basis to take the partial transpose. However, it can be shown that the set of operators that are positive under transpose on the input is the same for any choice of basis. We can also show the following.

### Theorem 5.2

*Any operator circuit built only from physical operators will be bounded between 0 and 1. Furthermore, given any nonphysical operator, there exists an operator circuit built out of otherwise physical operators which is less than zero or greater than one.*

We give one further definition.

*Definition of outcome-complete set of physical operators*. A bold operator, , consisting of operators , is an outcome-complete physical bold operator iff
5.5and
5.6

A few more notions are necessary.

*Equivalence classes*. We regard two operations as being the same if they have the same input/output structure and have the same probabilities whenever one is substituted by the other for all circuits.

*Correspondence*. Under a *correspondence* from bold operations to bold operators applied to any circuit we obtain the correct probability distribution.

With these theorems and definitions in place we state the following axiom for quantum theory.

**Axiom 1**. There is a one-to-one correspondence between outcome-complete bold operations and outcome-complete physical bold operators.

This axiom is equivalent to standard operational quantum theory (expressed in terms of density matrices, completely positive maps and POVMs). This axiom requires a little unpacking. First, it implies that every outcome-complete bold operation has a physical outcome-complete physical bold operator associated with it. In the case of a single outcome preparation, this would correspond to a single element bold operator. This would simply be a density matrix. A transformation would be associated with a collection of operators that are positive under partial transpose over the inputs. This would correspond to a set of completely positive maps in the usual formulation. The trace non-increasing condition on completely positive maps in the standard formulation is equivalent to condition (5.4). A POVM is provided by the collection of elements of the outcome-complete bold operator associated with a measurement operation (having no output systems). It follows from the definition of an outcome-complete physical bold operator that these elements constitute a POVM. This makes it clear that we have all the usual mathematical objects we require for representing preparations, transformations and measurements as a consequence of axiom 1. The second implication of axiom 1 is that there exists an outcome-complete bold operation for every outcome-complete physical bold operator. This is the statement that there exists some way of realizing the corresponding bold operation for any mathematically sensible bold operator. A similar statement is true in the standard operational formulation of quantum theory.

## 6. Connection with other work

This approach originated with the causaloid formalism [3,4]. The causaloid formalism is a general probabilistic framework for operational physical theories that does not assume definite causal structure. However, with the circuit framework we need to assume a certain amount of definite causal structure as the wiring is fixed in advance (much like formulating a physical theory on a fixed metric background). In [3], it was shown how to formulate quantum theory for interacting qubits on some fixed circuit of interactions. This framework allows one to consider arbitrary fragments of the circuit and to see how to ‘glue’ them back together using the *causaloid product*. The objects associated with fragments were certain vectors, **r**, rather than operators.

The pictorial approach adopted here was motivated by the categorical formulation of Abramsky & Coecke [5] (see also [6]).

The notation used here (both symbolic and diagrammatic) is motivated by Penrose's work (his abstract tensor notation and his diagrammatic tensor notation) [7,8].

Chiribella *et al.* [9] gave the quantum combs formulation. They associated operators with *combs* (their name for fragments) and showed how to glue them back together using the *link product*. Their operators are positive (effectively they work with the partial transpose objects) and they do not exploit the Penrose style notation for operators. However, there is a clear dictionary between their approach and the approach presented here (see [1] for more details on this).

There are various approaches to formulating quantum theory of a similar nature that deserve mention. Markopoulou [10] developed the quantum causal histories approach that used completely positive maps on a graph-like structure and a dual point of view was advanced by Blute *et al.* [11].

There is also a certain similarity between this approach and the approaches of Aharonov *et al.* [12,13], of topological quantum field theory [14,15] and related approaches [16], and also with more recent work by various researchers [17,18].

Brody & Hughston (BH) [19] develop a statistical geometric approach to quantum theory. They start by reformulating classical statistical theory in terms a real Hilbert space, , using abstract tensor style notation. Since we also use abstract tensor notation it is interesting to compare their approach with the present approach (as we will see, the two approaches are very different). In the BH approach, a state, *ξ*^{a}, is an element of this space. Normalization is given by
where *g*_{ab} defines a symmetric inner product on . The expectation value of a random variable, *X*_{ab}, is given by
In classical statistics, BH identify *ξ*^{a} with and *X*_{ab} with *xδ*(*x*−*y*) so that *X*_{ab}*ξ*^{a}*ξ*^{b} is given by
To go to quantum theory BH stick with real Hilbert space and introduce complex structure through a tensor satisfying . A symmetric operator, *X*_{ab}, is said to be Hermitian if it satisfies
The Dirac product from standard quantum theory can be expressed in this notation as
so that
The only points in common between the approach of BH and the approach of this paper are that (i) an abstract tensor notation is used and (ii) quantum theory is treated. However, the ways in which the abstract tensor notation is used are very different.

— The BH states,

*ξ*^{a}, go as the square root of probability whereas the states in the operator tensor approach (and, indeed, all operator tensors) are linear in probabilities.— The inner product of BH is defined via a metric

*g*_{ab}on whereas the inner product for operator tensors is defined through multiplying and taking the partial trace in the appropriate part of the space (e.g. in the simplest case).

Furthermore, the operator tensor approach is fundamentally about circuits, whereas circuits are not treated in BH. This is important because the correspondence is very natural. It induces a mapping between circuit descriptions and operator tensor calculations in which the abstract tensor notation really comes into its own. Note finally that the physicality constraint for operator tensors does not appear in the approach of BH as their approach is not about circuits.

## 7. Discussion

It is striking that the expression we write down to calculate the probability distribution for a circuit looks the same as the expression we write down to describe the circuit. It is this that allows us to simply replace bold operations with bold operators to do the calculation. This suggests a general approach to physics in which we do calculations by means of expressions that, structurally, look just like the object we are doing the calculation for. In [20], this idea was explored further.

This approach is, in a certain sense, as manifestly covariant as one could expect for circuits. When we do a calculation we do not need to evolve a state in time through the circuit. Rather, we simply evaluate the bold operator circuit. We can do this evaluation in any order we wish (corresponding to some order on the integer labels for the wires).

This manifest covariance suggests an approach to quantum field theory in which we go from circuits to regions of space–time. We could think of this in the following terms. We can imagine a grid of interacting qubits. If we ‘zoom out’ then this will look like a continuous system. Now associated with regions of space–time will be fragments built out of many numbers of operations. Associated with an arbitrary region of space–time will, then, be a bold operator. To make this work in the limit, we would want to replace finite dimensional Hilbert spaces with continuous dimensional ones and to write down the conditions on these operators that correspond to the physicality condition we gave earlier.

## Competing interests

I declare I have no competing interests.

## Funding

This project was made possible in part through the support of a grant from the John Templeton Foundation.

## Acknowledgements

Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

## Footnotes

One contribution of 13 to a theme issue ‘New geometric concepts in the foundations of physics’.

- Accepted March 3, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.