#
Quartic quantum theory:

an extension of the standard quantum mechanics

###### Abstract

We propose an extended quantum theory, in which the number of parameters necessary to characterize a quantum state behaves as fourth power of the number of distinguishable states. As the simplex of classical –point probability distributions can be embedded inside a higher dimensional convex body of mixed quantum states, one can further increase the dimensionality constructing the set of extended quantum states. The embedding proposed corresponds to an assumption that the physical system described in dimensional Hilbert space is coupled with an auxiliary subsystem of the same dimensionality. The extended theory works for simple quantum systems and is shown to be a non-trivial generalisation of the standard quantum theory for which . Imposing certain restrictions on initial conditions and dynamics allowed in the quartic theory one obtains quadratic theory as a special case. By imposing even stronger constraints one arrives at the classical theory, for which .

e-mail: karol@tatry.if.uj.edu.pl

## 1 Introduction

For a long time quantum mechanics (QM) belongs to the most important cornerstones of modern physics. Although predictions of quantum theory were not found to be in contradiction with results of physical experiments, there exist many reasons to look for possible generalisations of quantum mechanics– see e.g. [1, 2, 4, 3, 5].

One possible way to tackle the problem is to follow an axiomatic approach to quantum mechanics and to study consequences of relaxing some of the axioms. An axiomatic approach to quantum theory was initiated by Mackey [6], Ludwig [7, 8] and Piron [9] several decades ago and further developed in several more recent contributions [10, 11, 12, 13]. In an influential work of Hardy it is shown that quantum mechanics is a kind of probability theory for which the set of pure states is continuous [10]. This contrasts with the classical probability theory, for which pure states form a discrete set of corners of the probability simplex.

Restricting attention to the problem of finite number of distinguishable states and analyzing composite systems it is possible to conclude that the number of degrees of freedom (i.e. the number of parameters required to specify a given state) satisfies the relation with an integer exponent. The linear case, , gives the classical probability theory. The quadratic case leads to the standard quantum theory, for which it is necessary to use real parameters to characterize completely an unnormalized quantum state. However, higher order theories may exist, which include QM as a special case [14]. To single out the standard quantum theory Hardy uses a ’simplicity axiom’ and requires that the exponent takes the minimal value consistent with other axioms, which implies .

Vaguely speaking the exponent counts the number of indices decorating a mathematical object called state, used to determine probabilities associated with the outcomes of a measurement. In the classical theory one deals with probability vectors with a single index, while quantum states are described by density matrices . Do we need to work with some more complicated objects, like tensors or multi-index density matrices or ?

The main aim of this work is to propose a higher order theory, which includes QM as its special case. Instead of working with higher order tensors, the theory of which is well developed [15, 16], we remain within the known formalism of standard complex quantum mechanics, and construct an extended quartic (biquadratic) theory, for which , assuming a coupling with an auxiliary subsystem. The extended quantum mechanics (XM), reduces to the standard quantum theory in a special case of uncoupled auxiliary systems. On the other hand, XM is shown to be a non–trivial generalisation of QM. Throughout the entire work only non-relativistic version of quantum theory will be considered. For simplicity we analyze the case of finite dimensional Hilbert space. Furthermore, we restrict our attention to single–particle systems only.

Well known effects of decoherence reduce the magnitude of quantum effects and cause a quantum system to behave classically. In a similar way one can introduce analogous effects of ’hyper-decoherence’ which cause a system described in the framework of the extended theory to lose its subtle properties and behave according to predictions of standard quantum theory.

It is worth to emphasize that our approach differs from the theory of Adler based on higher order correlation tensors [17], the ’two–state vector formalism’ of Aharonov and Vaidman [18], the ’time–symmetric quantum mechanics’ of Wharton [19], and the quaternionic version of quantum theory [20, 21, 22], which is not consistent with the power like scaling, , in analogy to the quantum theory of real density matrices.

Higher order theory proposed here is also different from the algebraic approach of Uhlmann who discusses spaces of states constructed from Jordan algebra [23], and from the generalized quantum mechanics developed by Sorkin [24]. Furthermore, the extended theory by construction belongs to the class of probabilistic theories (see e.g. [25]), so it is different from the theory of hidden variables, which would allow one to predict an outcome of an individual experiment.

Our approach explores the geometric structures in quantum mechanics and in particular the convexity of the set of quantum states. Such a description of quantum mechanics goes back to classical papers of Ludwig [7] and Mielnik [26, 27] and was reviewed and updated in [28].

The extended quantum theory constructed here is related to the generalized quantum mechanics of Mielnik [29], in which higher order forms on the Hilbert space are considered and methods of constructing non–linear variants of quantum mechanics are discussed. On the other hand the theory analyzed here is linear, and the sets of extended states and extended measurements are precisely defined.

The paper is organized as follows. Section 2 contains a geometric review of the standard set–up of quantum mechanics in which we describe the sets of quantum states and quantum maps. Discussion of the extended, quartic theory is based on a definition of the dimensional convex set of extended quantum states, introduced in sec. 3. In section 4 we describe the set of generalized measurement operations admissible in the extended theory while section 5 concerns the corresponding discrete dynamics. Section 6 contains the evidence that the extended theory forms a non-trivial generalisation of the standard, quadratic quantum theory. The possibility of generalizing the quantum theory even further and working with higher order theories is discussed in section 7. The work is concluded with a discussion in section 8, while some information on duality between convex sets is presented in Appendix A.

## 2 Standard quantum theory: quadratic

In this section we review the standard quantum theory and present requisites necessary for its generalisation. Quantum mechanics is a probabilistic theory. Probabilities associated with outcomes of a measurement are characterized by a quantum state described by a density operator which acts on –dimensional Hilbert space . In this work we shall assume that is finite. The density operator is Hermitian and positive.

The set of normalized quantum states of size for which Tr will be denoted by . In the simplest case of the set of mixed states of a single qubit forms the Bloch ball, . Degree of mixing of a state can be characterized by the von Neumann entropy, . This quantity varies from zero for pure states, to for the maximally mixed state, , located in the center of the set .

To introduce a partial order into the set of mixed states one uses the majorization relation [31]. A density matrix of size is majorized by a state , written , if their decreasingly ordered spectra and satisfy: , for . The majorization relation implies an inequality between entropies: if then . Any mixed state satisfies relations , where denotes an arbitrary pure state – see e.g. [28].

For our purposes it is also convenient to work with subnormalized states, such that Tr. The dimensional set of subnormalized states forms a convex hull of the set of normalized states and the zero state, .

A one–step linear dynamics in may be represented in its Kraus form

(1) |

in which the number of Kraus operators can be arbitrary. Such a form ensures that the map is completely positive (CP), which means that an extended map, , sends the set of positive operators into itself for all possible dimensions of the ancilla [30]. The Kraus operators can be interpreted as measurement operators, and the above form provides a way to describe quantum measurement performed on the state : The –th outcome occurs with the probability and the measurement process transforms the initial state according to

(2) |

To assure that the trace of does not grow under the action of , the Kraus operators need to satisfy the following inequality [30],

(3) |

Usage of subnormalized states and trace non–increasing maps corresponds to a realistic physical assumption that the experimental apparatus fails to work with a certain probability and no measurement results are recorded.

The measurement process can be characterized by the elements of POVM (positive operator valued measures), . By construction these operators are hermitian and positive. Due to (3) they fulfill the relation hence each individual element satisfies . Thus the set of the elements of a POVM in the standard quantum theory can be defined as

(4) |

Since the elements of POVM are positive operators, the probability is non–negative,

(5) |

The above relation shows that the cone containing the elements of POVM is dual to the set of subnormalized states, . In the case of the standard quantum theory we work with the set of positive operators which is selfdual, so both cones are equal, – see Fig 4a. For more information on dual cones consult appendix A.

In the special case of equality in (3) the completeness relation imposes that Tr. A completely positive trace preserving map is called quantum operation or stochastic map. If the dual relation is satisfied, , the map is called unital, since it preserves the maximally mixed state, . A completely positive trace preserving unital map is called bistochastic. We are going to use an important property of these maps reviewed in [28]: Any initial state majorizes its image with respect to any bistochastic map, .

Treating as an element of the Hilbert-Schmidt space of operators, we may think of as a super–operator, (a square matrix of size ), acting in this space,

(6) |

where summation over repeated indices has to be performed. The operators acting on the vectors of Hilbert-Schmidt space are often called super–operators, in order to distinguish them from the operators of HS space itself. Let us emphasize that this common notion [32] used since the sixties [33] is not related to supersymmetric theories.

The superoperator can be represented by means of tensor products of the Kraus operators,

(7) |

The matrix needs not to be Hermitian. However, reshuffling its elements one defines a Hermitian dynamical matrix, [34]. Its elements read

(8) |

The symbol denotes the transformation of reshuffling of elements of a four-index matrix, which exchanges two indices, and in the formula above [28].

A theorem of Choi [35] states that the map is completely positive, if the dynamical matrix is positive, . Therefore , also called Choi matrix, might be interpreted as a Hermitian operator acting on a composed Hilbert space of size . The trace non-increasing condition (3) is equivalent to the following constraint on the dynamical matrix, Tr. Rescaling the Choi matrices , we recognize that the set of trace non–increasing maps forms a convex subset of the dimensional set of subnormalized states on [36]. This duality between linear maps and states on the enlarged system is called Jamiołkowski isomorphism, which refers to his early contribution [37].

To appreciate this duality let us look at an extended operation acting on the maximally entangled state

(9) |

from the enlarged Hilbert space, . The dynamical matrix corresponding to the map reads then

(10) |

The state–map isomorphism, written above for states with maximally mixed partial trace, Tr, can be generalized also for other states [38].

After reviewing some basic properties of discrete quantum dynamics, let us see how classical dynamics emerges as a special case of the quantum theory. Consider the set of normalized diagonal density matrices, , which forms the -dimensional simplex of classical probability distributions. If we restrict our attention to maps described by diagonal dynamical matrices, , then the diagonal structure of is preserved, so we recover the classical probability theory. Moreover, for any quantum map we obtain corresponding classical dynamics in the simplex of probability distributions, , by constructing the transition matrix out of diagonal elements of the dynamical matrix.

Lemma 1. Let be a linear quantum map acting on . Let denote a square matrix of size obtained by reshaping the diagonal elements of the corresponding dynamical matrix, , (without summation over repeating indices). If is a quantum stochastic (bistochastic) map, then is a stochastic (bistochastic) matrix.

Proof. If quantum map is completely positive, the corresponding dynamical matrix is positive definite, so all elements of its diagonal used to assemble are not negative. If is trace preserving, equality in (3) holds, and implies the relation for all . If is unital then the dual relation for partial trace of implies that for all . Thus quantum stochasticity (bistochasticity) of the map implies the classical property of the transition matrix .

Any trace non increasing map, for which the Kraus operators satisfy relation (3), can be also called sub–stochastic, since in the case of diagonal the classical transition matrix is sub–stochastic [31].

It is instructive to analyze the geometry of the sets of classical and quantum states [28]. For simplicity we have compared in Fig. 1 the sets of normalized states for . The interval of classical one–bit states can be embedded inside the 3-d Bloch ball consisting of pure and mixed states of one qubit – see Fig. 1b. On the other hand, the Bloch ball can be inscribed inside the cube, which describes the set of uncorrelated states of three classical bits – see Fig. 2a. This very cube can be embedded inside the simplex formed by corners representing all possible pure states of three classical bits. The cube would describe allowed results of fiducial measurements of three components of the spin , if they were independent classical variables [40]. Truncation of the corners of the –cube, implied by rules of quantum mechanics, reduces the cube to the ball. After this truncation procedure only two quantum states remain distinguishable. Furthermore, such a qualitative change of the symmetry of the body makes the set of extremal states continuous and allows for an arbitrary rotation of the Bloch ball [14, 39]. Rotating the initial state with respect to any axis perpendicular to the interval of classical states one generates a coherent superposition of and . Existence of such a pure state, which does not have a classical analogue, explains interference effects, typical of quantum theory.

## 3 An extended quantum theory: quartic

In order to work out a generalized, quartic theory
we need first to define a set of extended states.
In this paper we are going to consider mono–partite systems^{1}^{1}1Mono-partite systems consist of a single particle only,
while bi-partite systems consist of tow well defined subsystems.
of an arbitrary size , but to gain some intuition
we shall begin with the simplest case of .
Analyzing the normalized states of a single
qubit (quantum bit),
we will copy the embedding procedure which
blows up the interval of classical states
into the Bloch ball of quantum states.
Thus we shall put the –d Bloch ball of all states of a qubit
inside the larger –d body of an exbit
(extended bit) as sketched in Fig 1c.
(In a recent paper of Barrett [40]
a similar name of gbit, standing for generalized bit
was introduced).
Designing the shape of the set of extended bits
we have to keep in mind
that it may contain only two distinguishable states,
say and .

The dimension of the set of normalized extended states should be equal to , since the remaining dimension is obtained by imposing a weaker condition of subnormalization. Thus it is natural to look at it as a suitable subset of the set which contains the mixed states of two quNits (systems described in -dimensional Hilbert space). As the ball of one–qubit states arises by truncating the corners of the three–bit cube, to define the set of the states of an exbit we propose to reduce the number of distinguishable states in by truncating the corners of the simplex of eigenvalues of standard quantum states for . In this way one obtains a convex -d set of these states, the spectra of which belong to the octahedron formed by centers of edges of the tetrahedron - see Fig 2b.

More formally, let us propose a general definition of the set of extended states for an arbitrary ,

(11) |

where represents an arbitrary pure state of the standard theory. The symbol denotes the majorization relation (defined in previous section), with the help of which the truncation is performed. The ordered set of eigenvalues of the extended state forms the vector of length with non-vanishing components only, eig. From a combinatorial point of view the set of all spectra majorized by this vector forms a permutation polytope called permutohedron or multipermutohedron [41, 42]. It is defined as a convex hull of all permutations of a given vector, , where the convex hull contains all permutations in the –element set of components of . In the case considered here the vector has components, but only of them are non–zero. Thus the number of corners of is given by the binomial symbol, . In the simplest case of we get and the set forms a regular octahedron shown in Fig. 2b. Thus an operator belongs to the set of extended states if its spectrum belongs to the permutohedron,

(12) |

To show that the described choice of the set is acceptable we need to discuss the number of distinguishable states it supports.

Lemma 2. The set contains exactly mutually distinguishable states.

Proof. Let represent an arbitrary orthonormal basis in . Then extended states have non—overlapping supports and can be deterministically discriminated. To show that does not contain more distinguishable states it is sufficient to apply a lemma, that the sum of the ranks of all distinguishable states is not larger than the total dimension of the Hilbert space [43]. In the case analyzed and the set does not contain any states with rank smaller than , so the maximal number of distinguishable states is equal to .

By means of a suitable mixture of unitary transformations one can send the state into any state such that . A convex mixture of unitaries is bistochastic, so any initial state majorizes its image obtained with this map. Thus both conditions are equivalent and we are in a position to formulate an alternative definition of the set of extended states,

(13) |

where is an arbitrary state in , while denotes a unitary matrix from .

Let us note that the set of extended states defined in equivalent forms (11, 12) and (13) is determined as the minimal set in dimensions which is invariant under unitary transformations and supports exactly distinguishable states.

In the simplest case of the set of extended states has an appealing property. By construction it forms a convex subset of the set of two–qubit states, which contains two sets of separable and entangled states. Consider a three dimensional subset of the set of two–qubit states defined as convex combination of four Bell states, and . Then the octahedron contained in consists of separable states only, while all other states of the tetrahedron are entangled.

For any pure state of the extended theory is represented by an mixed state of the standard theory with spectrum . The orbit of pure states of the extended theory contains unitarily equivalent states, which form an -dimensional flag manifold, in contrast to the dimensional Bloch sphere, . In general, the set of pure states of the extended theory, , has dimensions. This is exactly times more than its quantum counterpart, the complex projective space of real dimensions, . By construction the entropy of an extended state belongs to the interval , so it is convenient to define a gauged quantity which vanishes for extended pure states.

To find out how the Bloch ball is embedded inside consider a family of one qubit states with spectrum . The extension of this family has the form . The spectra of these extended states read and form the vertical diagonal of the octahedron which crosses its center and joins the edges with (see Fig. 2b). These points represent the logical states of the extended theory, and , equal to and respectively. All other points of the Bloch ball are obtained from points of the interval by local unitary rotation, . Note that the points of the octahedron beside the vertical interval do not have one–to–one analogues in the quantum theory.

In the case of an arbitrary an extension of is obtained by adding an ancilla in the maximally mixed state,

(14) |

By construction these states belong to and act on an extended Hilbert space . Moreover, pure states of the standard quantum theory with vanishing entropy are mapped into extremal states of the extended theory with entropy equal to .

In general a state of the extended theory need not have the product form (14), since the state may be entangled with the ancilliary system. A bipartite state will be called an extension of the quantum state if the marginal satisfies

(15) |

Reduction by partial trace is not reversible, and a given mixed state may have several different extensions , such that . However, any pure state has a unique extension only. It has a tensor product form (14) and reads .

Extended states are defined on an bipartite system and can be interpreted in view of the Jamiołkowski isomorphism (10): A state of the extended theory may be considered as a completely positive quantum map acting on and determined by . In general, a map need not be trace preserving, since this is only true if Tr.

In particular this is the case for all product extensions (14) for which . The corresponding map acts as a complete one–step contraction, and sends any initial state into ,

(16) |

To show this let us start with the dynamical matrix of this map, . Writing down the elements of the image in the standard basis we obtain the desired result, .

Consider now an arbitrary state , prepared as an extension of . This relation can be rewritten with help of the superoperator (7), dynamical matrix (8) and Jamiołkowski isomorphism (10),

(17) |

In this way we have arrived at a dynamical interpretation of objects of the extended theory: A state can be extended to , which represents a linear map , such that its effect is equal to . It means that sends the maximally mixed state into . In particular the trivial (product) extension (14) represents the complete contraction, which sends every initial state into .

In analogy to the dimensional set of quantum subnormalized states we define subnormalized extended states, satisfying Tr. The set of all such states, has dimensions, as required. Thus the number of parameters necessary to characterize a given state of the theory behaves as the forth power of . This property justifies the name used in the title of the paper: the extended theory proposed in this work can be called quartic.

Let us compare our construction with the generalized quantum mechanics of Mielnik [29]. In his approach the set of extended states contains ’density tensors’ constructed of convex combinations of separable pure states. On the other hand the set of extended states contains also states which are not separable with respect to the fictitious splitting into the ’physical system’ and the ’hypothetical ancilliary system’. Such a possible entanglement between the ’system’ and the ’ancilla’ plays a crucial role in the dynamics: as shown in the subsequent sections it contributes to the fact that the predictions of the standard quadratic theory and the generalized quartic theory can be different.

To summarize this section, a quartic extension of the quadratic quantum theory is constructed by extending the set of admissible states. Any extended state can be interpreted as if the corresponding state of the standard theory were entangled with an auxiliary subsystem of the same size in such a way that the state of the composite system obeys (11) and its marginal is equal to . Note that such a concept of an ancillary subsystem (the presence of which might be difficult to detect) is introduced for a pedagogical purpose only: In the extended theory the system is described by a single density tensor with four indices, so in practice it cannot be divided into a ’physical particle’ and an auxiliary ’ghost-like’ subsystem.

A state of the extended theory represents a completely positive quantum map which moves the center of the body of quantum states into . As the set of classical states forms the set of diagonal density matrices embedded in , the set of quantum states forms a proper subset of containing product states, .

## 4 Extended measurements and POVMs

In analogy to the standard quantum theory of a measurement process we will assume that the generalized measurement acting on is described by the elements of an extended POVM. Conservation of probability implies a relation in analogy to the quantum case.

Furthermore, the probabilities of a single outcome have to be non–negative, so we require that

(18) |

This equation defines the set of elements of an XPOVM (an extended POVM). The key difference with respect to the quantum condition (5) is that the extended states are not only positive, but they belong to the set which arise by truncation of the set of positive operators acting on an extended system. Hence elements of an XPOVM (an extended POVM) may not be positive, provided the condition (18) holds for all admissible states. This relation shows that the set of elements of XPOVM belongs to the cone dual to the set of extended states,

(19) |

Here denotes the set dual to – see Appendix A for definition and properties of dual cones and dual sets.

The geometry of these sets is sketched in Fig 3. While in the case of standard quantum theory both sets of quantum states states and elements of POVM do coincide (panel a), in the extended theory the set of extended states does not contain all positive operators, so the dual set contains also some operators which are not positive. The boundary of the cone of the elements of POVM has to be perpendicular to the opposite boundary of the set of extended states, since the relation (18) bounds the scalar product in the Hilbert–Schmidt space of linear operators. Thus this relation can be interpreted as a condition that the angle between two corresponding vectors is not larger than .

Since the set of extended states is invariant with respect to unitary transformation, its structure is determined by the permutohedron containing all admissible spectra. Therefore the set of all elements of XPOVM is unitarily invariant and can be specified by defining its spectra. Lemma 4 proved in appendix A implies that

(20) |

Hence to find the set of operators belonging to it is sufficient to find a polytope dual to the permutohedron . Each corner of the permutohedron generates a face of , so the latter polytope has faces. The structure of both sets is particularly simple in case of , for which one arrives at a pair of dual regular polytopes in : an octahedron and a cube. The spectra of the states from belong to the regular octahedron , so the dual set of elements of XPOVM contains operators with spectra belonging to the cube equal to . The cube can be written as convex hull of a tetrahedron and its mirror copy, – see Fig. 4. Observe that contains also non–positive matrices. e.g. a diagonal matrix . However, due to duality relation , inequality (18) is by construction fulfilled for any extended state .

## 5 Extended dynamics and super-maps

To complete the construction of the quartic theory we have to allow for some action in the set of extended states. As in the case of standard quantum theory we shall discuss only discrete linear maps.

An extended state may be considered as a map on , so we are going to analyze a transformation which sends a quantum map into a quantum map. Since in the physics literature a map sending operators into operators is called a ’super–operator’, we shall call a super-map. It is represented by a matrix of size which acts on an extended state in analogy to (6),

(21) |

A supermap corresponds to the concept of motion, which transforms the set of extended pure states in the generalized quantum mechanics of Mielnik [29]. Some properties of supermaps were independently investigated in a very recent work by Chiribella, D’Ariano and Perinotti [44].

Investigating linear maps in the set of extended states we aim to accomplish two complementary tasks: i) For any quantum operation construct a corresponding supermap which preserves the set of quantum states embedded inside the set of extended states. ii) For any admissible trace-preserving supermap which acts on the set of extended states find a reduced quantum map , which acts the set and forms a quantum operation, (is completely positive and trace preserving).

Let us first consider the product extension of quantum operations,

(22) |

If the initial state has the product form, , then . The maps of the form (22) preserve thus the structure of the set of quantum states, and in this way any quantum operation can be realized in the extended set–up. In this way we have arrived at

Proposition 1. The extended, quartic theory is a generalisation of the standard quantum theory. In the special case of the tensor product structure of initial states and supermaps, XM reduces to QM.

The special case of supermaps of the product form (22) has a simple interpretation in view of the Jamiołkowski isomorphism. Associating by means of (10) initial and final states, and , with the maps and we realize that the map acts in the space of extended states (identified with the set of maps) as a composition, . Going back into the space of states with Jamiołkowski isomorphism into the space of quantum states one can define in this way a composition of states [45], .

Let us now relax the assumption (22) on the product form and look for a more general class of linear maps. In general we need to work with maps that preserve the set of extended states; if then . This property parallels the positivity of quantum maps which preserve the set . However, analyzing dynamics of a quantum system one takes into account the possible presence of an ancilla and defines completely positive maps. Hence we advance the following notion of completely preserving maps, related to the concept of well defined transformations used by Barrett in [40].

Definition. Consider a given sequence of convex sets labeled by an integer and a map defined on . The map is called preserving if . We say that the map is completely preserving if its extension acting on is preserving for an arbitrary .

Taking for the set of quantum states we get back the standard definition of CP maps. However, if we put for the set , we get the characterization of the maps which completely preserve the set of extended states, so are admissible in the quartic theory.

It is not difficult to show that the set of supermaps completely preserving the structure of is not empty. It contains for instance all maps of the product form (22), and also all maps acting on which are bistochastic. Any extension of a bistochastic map is bistochastic, and this property guarantees that any initial state majorizes its image, so the structure (11) of the set of is preserved.

On the other hand, the problem of deciding, whether a given map acting in the space of extended states is preserving (completely preserving) is in general not simple, and it is not determined by the (complete) positivity of the map. For instance, a completely positive map which sends all states of into a pure state (where ), is not preserving, since the pure state does not belong to , so it can not be completely preserving. However, the reflection with respect to the maximally mixed state, , is not positive in , but it preserves the smaller set of extended states. It is also known that allowing for a non–product extension followed by a global unitary dynamics and partial trace over the auxiliary subsystem may lead to non completely positive dynamics [46]. We close the discussion here admitting that the problem of finding an efficient criterion to distinguish the preserving and completely preserving maps remains open.

In order to compare predictions of quartic and quadratic theory we need to find a way to associate with a given supermap, , a quantum map . For a moment let us restrict our attention to completely positive maps which act in the set . It can be represented in the standard Kraus form,

(23) |

The Kraus operators act now in the extended Hilbert space . In contrast to the form (1), which describes all completely positive maps admissible by standard quantum theory, the Kraus form (23) of a supermap provides only a class of measurement processes admissible within the extended quantum theory.

In analogy to (8) we may represent such a supermap by its dynamical matrix

(24) |

The operators can be chosen to be orthogonal, so their number will not be larger than . The Choi matrix of size is Hermitian and it acts on . Here label denotes the principal system, its extension which generates the state in the quartic theory, while and represent their counterparts used to apply the Jamiołkowski isomorphism (10).

To simplify the dynamics in the space of extended states it is enough to consider the Choi matrix obtained by normalizing the partial trace of the Choi matrix representing a supermap, . The resulting quantum map, , inherits its properties from the corresponding supermap.

Lemma 3. Let be a linear supermap acting on , so it can be represented by a Hermitian dynamical matrix . Construct a quantum map acting on by performing the partial trace of the dynamical matrix, where . If is a completely positive (stochastic, bistochastic) supermap, so is the quantum map .

Proof. If a supermap is completely positive, then due to Choi theorem the corresponding dynamical matrix is positive, . So is its partial trace, , which implies complete positivity of . If is trace preserving then , hence , which implies trace preserving condition for . Analogously, if is unital then so which implies unitality of . Thus stochasticity (bistochasticity) of the supermap implies the same property of the associated quantum map .

If the supermap has a product form, , and all its Kraus operators have the tensor product structure, the corresponding quantum dynamics is given by reduced Kraus operators, . However, in the case of an arbitrary stochastic this relation does not hold.

In general one may also consider a wider class of supermaps which preserve the set of extended states, but for which the extended Choi matrix is not positive. However we need to require that the induced quantum dynamics is completely positive. This implies a condition for the partial trace

(25) |

which is obviously fulfilled for any positive . On the other hand relation (25) is satisfied for a large class of operators which are not positive. This shows that the class of admissible dynamics in the extended theory is wider than in the standard quantum theory.

##
6 Classical, quantum and extended theories:

a comparison

The standard quantum theory reduces to a classical theory if one takes into account only the diagonal parts of a state and restricts the space of operations. Technically, one may define an operation of coarse–graining with respect to a given Hermitian operator , which is assumed to be non-degenerate. This operation can be represented as a sum of projectors onto eigenstates of ,

(26) |

In other words, this map deletes all off-diagonal elements from a density matrix, if represented in the eigenbasis of and produces a probability vector . It consist of non-negative components, the sum of which is not larger than unity, so lives in the simplex . Since off-diagonal elements are called quantum coherences, the process induced by coarse graining is called decoherence. The effects of decoherence play a key role in quantum theory and their presence explains why effects of quantum coherence are not easy to register.

In a similar way, for each quantum map one may obtain reduced, classical dynamics, by taking diagonal elements of dynamical matrix . The classical transition matrix, , inherits properties of , as stated in Lemma 1. In particular, if is a stochastic map, then forms a stochastic matrix, while if is a trace non–increasing map, then is substochastic, what means that the sum of all elements in each its column is not larger than unity.

Consider an arbitrary quantum state , transform it by a stochastic map into and perform coarse graining to obtain a classical state . Alternatively, get the classical vector by coarse–graining, , and transform it by reduced dynamics to arrive at . In general, both vectors are not equal,

(27) |

which is a consequence of the known fact that classical and quantum dynamics do differ. Such a direct comparison between discrete classical and quantum dynamics may be succinctly summarized in a non–commutative diagram:

(28) |

Horizontal arrows represent quantum (classical) discrete dynamics, while vertical arrows can be interpreted as the action of the coarse–graining operation defined in Eq. (26), which reduces quantum theory to classical.

In an analogous way one can compare dynamics with respect to the extended and standard quantum theories. The transition from a state of the quartic theory to a standard quantum mechanical state occurs by taking the partial trace, . This process can be called a hyper–decoherence, since it corresponds to the decoherence which induces the quantum–classical transition. As standard decoherence effects make the observation of the quantum effects difficult, the hyper–decoherence reduces the magnitude of effects unique to the extended theory.

Let us start with an arbitrary state of the quartic theory, transform it by an admissible linear supermap map into and perform a reduction to obtain the quantum state . Alternatively, get the quantum state by reduction, , and transform it by reduced quantum map , characterized in Lemma 3, to arrive at . In general, both quantum states are different,

(29) |

In this way we have justified

Proposition 2. The extended quartic theory forms a nontrivial generalisation of the standard quantum theory. In particular, there exist experimental schemes (consisting of an initial state and the measurement operators) for which both theories give different predictions concerning probabilities recorded.

The comparison between dynamics in quartic and quadratic theories is visualized in a non–commutative diagram analogous to (28),

(30) |

The vertical arrows denote here the operation of partial trace over auxiliary subsystem and reduction of extended theory to quantum theory, while horizontal arrows represent dynamics in the space of extended (quantum) states.

Thus the classical theory describing dynamics inside the dimensional simplex of subnormalized probability vectors remains a special case of the quantum theory, in which dynamics takes place in dimensional set of subnormalized quantum states. In a very similar manner, the standard quantum theory may be considered as a special case of the extended theory, obtained by projecting down the dimensional set of extended states into . A comparison between all three approaches is summarized in Table 1. The symbol denotes here the transformation of reshuffling of a matrix defined in Eq. (8).

Theory |
Classical (linear) |
Quantum (quadratic) |
Extended (quartic) |
---|---|---|---|

pure states | corners of | ||

mixed states | |||

dimensionality | |||

dynamics |

reduction from |
quantum to classical |
extended to quantum |
---|---|---|

of the state | ||

of the map |

To reveal similarities between both decoherence processes let us formulate two analogous statements.

Proposition 3: Decoherence. A classical state obtained by a decoherence of an arbitrary quantum state corresponding to the classical state satisfies the majorization relation

(31) |

where is a unitary matrix of size and stands for a diagonal matrix with vector on the diagonal.

The proof consists in an application of the Schur lemma, which states that the diagonal of a positive Hermitian matrix is majorized by its spectrum. This statement follows also from the Horn–Littlewood–Polya lemma, which says that if there exists a bistochastic matrix such that – see e.g. [28].

Proposition 4: Hyper–decoherence. A quantum state obtained by a hyper–decoherence from an arbitrary extension of a quantum state satisfies the majorization relation

(32) |

where is a unitary matrix of size .

To prove this statement it is enough to observe that the map is bistochastic, so according to the quantum analogue of the Horn–Littlewood–Polya lemma (see e.g. [28]), the majorization relation (32) holds. Alternatively, for small system sizes one may use results of the quantum marginal problem: inequalities of Bravyi [47] for and inequalities of Klyachko [48] for concerning constraints between the spectra of a composite system and its partial traces imply relation (32).

Note that the unitary matrix is arbitrary, it may in particular represent the swap operation, which exchanges both subsystems. Thus the extended state should not be treated as a merely composition of a ’physical particle’ with an ’auxiliary ghost’: They are intirinsicly intertwined into a single entity representing an extended state , which may be reduced to the standard quantum state due to the process of hyper–decoherence.

## 7 Higher order theories

Iterating the extension procedure one can construct higher-order theories, in which the number of degrees of freedom scales with dimensionality as for any even . Let us rename the set of quantum states into , and the set of extended states into . Then we may define the set of states in a -th order generalized theory as in (11),

(33) |

where . The parameter represents the number of additional ancilliary states: It is equal to zero for the standard quantum theory, for the extended quartic theory discussed in this work, and for the next, extended quantum theory for which . In general, the number of degrees of freedom behaves as , so the exponent is equal to . The standard quantum state is obtained by the partial trace over an auxiliary system of size .

The spectra of the states of the higher order theories also form a permutohedron , defined by the vector of length containing non–zero equal components. From a geometric point of view increasing the number of ancillas corresponds to increasing the dimensionality of the Hilbert space and continuing the procedure of truncation of the set of positive operators, which represent quantum states. The larger , the more faces and corners of the permutohedron which becomes closer to the ball in dimensions.

The set of pure states of the higher order theory forms the flag manifold of dimensions. The entropy of any state of such a theory belongs to the interval , so its degree of mixing can be characterized by the gauged entropy, , which is equal to zero for extended pure states.