[ beforeskip=-0.3em plus 1pt,pagenumberformat=]toclinesection \stackMath
Quantum dynamics as a pseudo-density matrix
Abstract
While in relativity theory space evolves over time into a single entity known as spacetime, quantum theory lacks a standard notion of how to encapsulate the dynamical evolution of a quantum state into a single "state over time". Recently it was emphasized in the work of Fitzsimons, Jones and Vedral that if such a state over time is to encode not only spatial but also temporal correlations which exist within a quantum dynamical process, then it should be represented not by a density matrix, but rather, by a pseudo-density matrix. A pseudo-density matrix is a hermitian matrix of unit trace whose marginals are density matrices, and in this work, we make use a factorization system for quantum channels to associate a pseudo-density matrix with a quantum system which is to evolve according to a finite sequence of quantum channels. We then view such a pseudo-density matrix as the quantum analog of a local patch of spacetime, and we make an in-depth mathematical analysis of such quantum dynamical pseudo-density matrices and the properties they satisfy. We also show how to explicitly extract quantum dynamics from a given pseudo-density matrix, thus solving an open problem posed in the literature.
1 Introduction
In 1907 Hermann Minkowski showed that the work of Maxwell, Lorentz and Einstein could be viewed geometrically as a 4-dimensional theory of spacetime [MKWSKI], after which he boldly predicted that space by itself, and time by itself, were then "doomed to fade away in mere shadows". Minkowski’s prediction was indeed correct, and ever since we have only furthered our understanding of the inextricable connection between space and time. By 1927 quantum theory had been established through the revolutionary work Bohr, Heisenberg, Schrödinger, Born and others, where the fundamental view was that at the sub-atomic realm, reality was appropriately described by "quantum states" evolving in time. While debates have raged through the ages regarding the ontological status of a quantum state, the one thing that most can agree upon is that at a fundamental level a quantum state consists of information, and there is an emerging viewpoint — famously coined by Wheeler as "it from bit" — that such quantum information is the basis of our physical reality. But while our macroscopic view of the world has blossomed from Minkowski’s unification of space and time, for the most part there has not been an analogous unification of information and time in quantum theory, where we seemed to have left the insights of Minkowski behind. In particular, while in relativity theory space evolves in time to form a single mathematical object called spacetime, quantum theory lacks a standard notion of how to encapsulate the global evolution of a quantum state over successive instances of time into a single entity, or rather, as a "state over time". In this letter, we then provide an answer to the "?" in the following analogy:
Moreover, as spacetime is fundamental to our understanding of gravity, it seems only natural that a mathematically precise formulation of the above analogy will yield insights into the nature of quantum gravity.
To summarize our approach to such an analogy, consider a local patch of spacetime, which we may view as a fibration of spatial slices over an interval of time . In such a case, we may view as a cobordism, i.e., as representing the process of the spatial slice evolving over time into the spatial slice . As such, in general relativity, the objects encoding evolution over time are of the same class of entity as the objects which are evolving over time, as , and are all manifolds. However, a crucial distinction between the process manifold and the spatial slices and , is that while and are of euclidean signature, the signature of the process manifold picks up a negative sign as it extends over time.
For our quantum analogy of a local patch of spacetime, suppose there is a quantum system in state at time , which is to evolve according to a quantum channel , so that is then viewed as the state of the system at time . We also assume that is on the order of the Planck time, so that the interval may be viewed as a single discrete time-step. Then in accordance with our spacetime analogy, we wish to associate a "state over time"
encoding the dynamical evolution of according to the channel . Moreover, in analogy with the spatial slices and being the source and target of the process manifold , the states and should be the reduced density matrices of the state over time with respect to tracing out and respectively. In the case that and admit large tensor factorizations
(where denotes the matrix algebra of matrices with complex entries), each tensor factor and may be identified with a region of space at times and respectively, and in such a case, we view the state over time as a quantum mechanical unit of spacetime.
While various formulations of dynamical quantum states have appeared in the literature, including the pseudo-density operator (PDO) formalism for systems of qubits first appearing in the work of Fitzsimons, Jones and Vedral [FJV15, LQDV, ZPTGVF18, ZhangV20, Marletto2021], the causal states of Leifer and Spekkens [Le06, Le07, LeSp13], the right bloom construction of Parzygnat and Russo [PaRuBayes, PaRu19], the super-density operators of Cotler et alia [Cotler2018], the compound states of Ohya [Ohya1983], the generalized conditional expectations of Tsang [Ts22], the Wigner function approach of Wooters [Woot87] and the process states of Huang and Guo [GuoZ1], in this work we focus mainly on the state over time given by
| (1.1) |
where denotes the anti-commutator, and
is the Jamiołkowski matrix111The Jamiołkowski matrix is not to be confused with the Choi matrix . associated with the channel . The state over time given by (1.1) was shown in [FuPa22, FuPa22a] to satisfy a list of desiderata for states over time put forth in [HHPBS17], and at present it is the only known state over time construction to satisfy such properties. We note however that while the state over time given by (1.1) is hermitian and of unit trace, it is not positive in general, and as such, is often referred to as a pseudo-density matrix. In our spacetime analogy, the negative eigenvalues of a state over time are analogous to the negative sign appearing in the time-component of the spacetime metric, and further justification for such a state over time to have negative eigenvalues appears in [FJV15], where it is argued that negative eigenvalues serve as a witness to temporal correlations encoded in .
One desideratum for states over time is a property often referred to as "associativity", as it allows one to unambiguously associate a "2-step state over time"
associated with the evolution of a state according to a 2-chain of quantum channels. The potential ambiguity stems from the fact that there are two parenthezations and of the matrix algebra , which lead to 2 fundamentally different constructions of a 2-step state over time . The associativity condition then ensures that these two distinct constructions of 2-step states over time are in fact equal, leading to a well-defined notion of a 2-step state over time.
In this work, we consider the case of a quantum state evolving according to an -chain of quantum channels
| (1.2) |
with which we wish to associate an "-step" state over time
Similar to how the ambiguity for 2-step states over time arises from the two different parenthezations of the matrix algebra , there is an ambiguity for -step states over time corresponding to the th Catalan number of ways to parenthesize the matrix algebra , leading to fundamentally distinct constructions of an -step state over time. We then prove that the different constructions of such an -step state over time are all in fact equal (see item ii of Theorem 7.4), leading to a well-defined notion of an -step state over time. In accordance with our spacetime analogy, we then view an -step state over time as a quantum analog of a local patch of spacetime fibered over an interval of time
where is on the order Planck time, so that is viewed as a discrete time-step corresponding to the evolution of the quantum system according to the channel .
As the -step extension of the state over time given by (1.1) is the only known construction which always yields a hermitian state over time, we denote the associated state over time by , and refer to the mapping
as the \Yinyang-function. In Example 8.20 we show that in the case of dynamically evolving systems of qubits, the output of the \Yinyang-function coincides with the coarse-grained pseudo-density operator (PDO) associated with the system, which was introduced in [FJV15] to treat temporal and spatial correlations in quantum theory on equal footing. As such, the \Yinyang-function may be viewed as a generalization to arbitrary finite-dimensional quantum systems of the pseudo-density operator formalism for systems of qubits. We also prove that given a unit-trace hermitian element satisfying some technical conditions, one may explicitly extract a dynamical process such that (see Theorem 9.5). This solves an open question recently posed in [jia2023], where it is stated "Another interesting and closely relevant open question is, for a given PDO (pseudo-density operator), how to find a quantum process to realize it.".
In what follows, we provide the necessary definitions and notation which will be used throughout in Section 2. In Section 3, we motivate the definition of quantum state over time by first analyzing the classical case of a random variable stochastically evolving into a random variable . In particular, we show that classical joint distribution associated with such stochastic evolution arises from a unique factorization of the associated classical channel, and moreover, that may be re-written in a way that is valid for any 1-step quantum process . In Section 4 we then define a factorization system for quantum channels in terms of a quantum bloom map, and then we use such a factorization system to define quantum states over time associated with 1-step processes . In Section 5 we recall the associativity property for quantum bloom maps which allow states over time for 1-step processes to extend to well-defined states over time for 2-step processes , and prove some general results regarding 2-step states over time. In Section 6 we then show how the associativity property for quantum bloom maps yields uniquely-defined extensions to quantum bloom maps for -chains of quantum channels. In Section 7 we then use the quantum bloom maps for -chains to define states over time associated with -step processes for arbitrary . In Section 8 we then focus on the -step states over time associated with the \Yinyang-function, and we prove some general properties of the \Yinyang-function. We also two distinct formulas for in terms of and the states which are Jamiołkowski-isomorphic to the channels . In Section 9 we then show that the \Yinyang-function yields a bijection when restricted to a suitably nice set of -step processes, and we prove an explicit formula for its inverse in such a case.
Acknowledgements. We thank Arthur J. Parzygnat and Franceso Buscemi for many useful discussions. This work is supported in part by the Blaumann Foundation.
2 Preliminaries
In this section we provide the basic definitions, notation and terminology which will be used throughout.
Definition 2.1.
Let be a finite set. A function will be referred to as a quasi-probability distribution if and only if . In such a case, will be denoted by for all . If for all , then will be referred to as a probability distribution.
Definition 2.2.
Let and be finite sets. A stochastic map consists of the assignment of a probability distribution for every . In such a case, will be denoted by for all and , which is interpreted as the probability of given . A stochastic map together with a prior distribution on its set of inputs will be denoted by . The set of stochastic maps from to will be denoted by .
Remark 2.3.
A stochastic map is also commonly referred to as a Markov kernel, or a discrete memoryless channel.
Notation and Terminology 2.4.
Given a natural number , the set of matrices with complex entries will be denoted by , and will be referred to as a matrix algebra. As the matrix algebra is simply the complex numbers, it will be denoted by . The matrix units in will be denoted by (or simply if is clear from the context), and for every , denotes the conjugate-transpose of . Given a finite set , a direct sum will be referred to as a multi-matrix algebra, whose multiplication and addition are defined component-wise. If and are multi-matrix algebras, then the vector space of all linear maps from to will be denoted by . The trace of an element is the complex number given by (where is the usual trace on matrices), and the dagger of is the element given by . Given with and multi-matrix algebras, we let denote the Hilbert–Schmidt dual (or adjoint) of , which is uniquely determined by the condition
for all and . The identity map between algebras will be denoted by , while the unit element in an algebra will be denoted by (subscripts such as and will be used if deemed necessary).
Remark 2.5.
Every finite-dimensional -algebra is isomorphic to a multi-matrix algebra [Fa01].
Notation 2.6.
Given a finite set , the multi-matrix algebra is canonically isomorphic to algebra of complex-valued functions on . As such, we will write an element as
where is the Dirac-delta of , which is the function taking the value 1 at and 0 otherwise. In such a case, will be referred to as the -component of for all .
Definition 2.7.
Let be a finite set and let be a multi-matrix algebra. An element is said to be
-
•
self-adjoint if and only if for all .
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•
positive if and only if is self-adjoint and has non-negative eigenvalues for all .
-
•
a state if and only if is positive and of unit trace. If contains only one element, then a state will often be referred to as a density matrix. The set of all states on will be denoted by .
Remark 2.8.
If is a finite set and , then a state is of the form with and . As such, any state on may be identified with a probability distribution on .
Definition 2.9.
Let and . The tensor product of and is the multi-matrix algebra given by
where is the usual tensor product of matrix algebras. Given elements and , the element is the multi-matrix given by
where is the usual tensor product of matrices. Given maps and , then is the map corresponding to the linear extension of the assignment .
Definition 2.10.
Given multi-matrix algebras and , an element consists of elements such that
In such a case, the will be referred to as the component functions of .
Definition 2.11.
Let and be multi-matrix algebras. A map is said to be
-
•
-preserving if and only if for all .
-
•
trace-preserving if and only if for all .
-
•
positive if and only if is positive whenever is positive.
-
•
completely positive if and only if is positive for every multi-matrix algebra .
Notation 2.12.
Let be pair of multi-matrix algebras. The subset of consisting of completely positive, trace-preserving maps will be denoted by , while the subset of consisting of trace-preserving maps will be denoted by . An element of with and matrix algebras will often be referred to as a quantum channel.
Definition 2.13.
Let and be finite sets. An element will be referred to as a classical channel. In such a case, we let denote the elements such that
In such a case, the elements will be referred to as the conditional probabilities associated with .
Definition 2.14.
Given a pair of multi-matrix algebras, an element will be referred to as a process, and the set of processes will be denoted by . The subset of consisting of processes with invertible will be denoted by . When and for finite sets and , then will be referred to as a classical process.
Definition 2.15.
Let be a pair of multi-matrix algebras, and let . The channel state of is the element given by
where is the Hilbert-Schmidt dual of the multiplication map .
Remark 2.16.
Given a pair of multi-matrix algebras, the map given by is a linear isomorphism, which we refer to as the Jamiołkowski isomorphism [Jam72]. In this work we will also make heavy use of the the inverse of the Jamiołowski isomorphism, which is given by
Definition 2.17.
If , and are matrix algebras, there exists an associator isomorphism which then allows one to define tensor products of of a finite number of matrix algebras iteratively. We can then extend such a constriction to multi-matrix algebras as follows. Let be a collection of multi-matrix algebras indexed by a finite set , where . The tensor product of the algebras is the multi-matrix algebra indexed by the set given by
If the index set is of the form , then we will write as .
Definition 2.18.
Let be a tensor product of multi-matrix algebras. The th partial trace for is the map given by the linear extension of the assignment
where denotes the empty matrix for all .
Notation 2.19.
If , then the partial trace maps and will be denoted by and respectively. If and , then the partial trace maps and will be denoted by and . If , then we will frequently make use of the th partial trace , and as such, we will often denote simply by tr.
Definition 2.20.
Let be a finite set and let be a multi-matrix algebra. A self-adjoint element is said to be a pseudo-density operator with respect to the factorization if and only if for all we have
If is a matrix algebra (i.e., when consists of a single element), then a pseudo-density operator with respect to the factorization will be referred to as a pseudo-density matrix. The set of all pseudo-density operators with respect to the factorization will be denoted by .
Remark 2.21.
A pseudo-density operator with respect to the trivial factorization is simply a state, so that .
Remark 2.22.
Since all of the marginals of a pseudo-density operator are of unit trace, it follows that a pseudo-density operator is necessarily of unit trace as well.
3 Classical probability as CPTP dynamics
In this section we recall how classical probability may be recast in terms of CPTP maps between multi-matrix algebras. In particular, we will show how the joint distribution associated with a classical channel arises from a unique factorization of the channel of the form , where is a map referred to as the bloom of . If is a state, then the element is the joint distribution associated with the classical process , which in a more dynamical language we refer to as the state over time associated with . We will then re-write the classical state over time in a way that is valid for any quantum process with and arbitrary multi-matrix algebras, thus paving the way for a quantum generalization of the classical state over time .
The following proposition shows how classical stochastic maps may be recast in terms of CPTP dynamics.
Proposition 3.1.
Let be a pair of finite sets, and let be the map given by
Then is a continuous bijection.
Proof.
This follows form the fact that stochastic maps and CPTP maps are completely determined by their associated conditional probabilities and . ∎
The next proposition shows that factorizations of classical channels are essentially unique, which we will use to characterize joint distributions associated with stochastic dynamics.
Proposition 3.2.
Let be a pair of finite sets, let , and let be such that and . Then is the linear map given by
| (3.3) |
where are the conditional probabilities associated with the map .
Proof.
We show that given , we have
The result then follows as is assumed to be linear and Dirac deltas form a basis of . So let be arbitrary, and let be the conditional probabilities associated with the map , so that
From the conditions and we then have
and
where is the Kronecker delta. For all we then have
and for all we have
It then follows that for all we have , thus
as desired. ∎
Definition 3.4.
Let be a pair of finite sets and let be a classical process.
-
•
The map defined by (3.3) will be referred to as the bloom of .
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•
The element will be referred to as the state over time associated with the classical process .
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•
The map on classical processes given by will be referred to as the classical state over time function.
|
|
Remark 3.5.
Given a pair of finite sets and a classical process , if we identify with conditional probabilities , with a probability distribution on , and with a joint probability distribution on , then the equation
may be re-written as
| (3.6) |
which is the classical equation relating a joint distribution with a conditional distribution and a prior. As such, Proposition 3.2 may be interpreted as saying that the joint distribution associated with a classical process arises as an intermediary step in a canonical factorization of the associated channel. This observation will be crucial for generalizing joint distributions to the quantum domain.
Next we show that in the classical domain, joint distributions are in a bijective correspondence with classical processes.
Notation 3.7.
Given a pair of finite sets, let denote the subset given by
Proposition 3.8.
Let be a pair of finite sets, and let
be the map given by . Then is a bijection, whose inverse is determined by the condition
Proof.
Let be a pair of finite sets, and let be the map uniquely determined by the condition
We will show and . For the former case, let , and let . By the definition of we have
thus
Moreover, by definition of , for all we have
thus .
Now let , let , and let . Then for all we have
thus , as desired. ∎
Remark 3.9.
In light of Proposition 3.8, it follows that a state may either be viewed as a joint distribution associated with two space-like separated random variables and occurring in parallel, or as a state over time associated with the process . These two equivalent viewpoints are what we refer to as the static-dynamic duality for classical joint states in . Another way to think of the static-dynamic duality for classical joint states, is that classical probability does not distinguish between temporal correlations and spatial correlations, revealing a certain symmetry between space and time for classical random variables. For quantum systems however it turns out that such a symmetry between space and time does not exist, as there exists quantum processes for which temporal correlations may not be viewed as spatial correlations and vice-versa [FJV15]. As such, the static-dynamic duality for classical joint states does not generalize to the quantum domain, and we will see that this is manifested in the fact that quantum states over time are not positive in general. This fact will play a crucial role in the general theory of states over time for quantum processes.
In the next proposition, we show how the state over time associated with a classical process may be re-written in a way that is valid for all quantum processes, which we will use in the next section for a quantum generalization of states over time.
Proposition 3.10.
Let be a pair of finite sets, and let . Then
| (3.11) |
for all . In particular, if , then is the state over time associated with the classical process .
Proof.
Let denote the multiplication map. From the definition of the Hilbert-Schmidt dual we have
thus
And since for all we have
it follows that
as desired. ∎
Remark 3.12.
While up to now the left-hand side of equation (3.11) is only defined for classical processes, the right-hand side is defined for any quantum process . Moreover, since for classical processes we have , the right-hand side of equation (3.11) coincides with
| (3.13) |
for all , which is also well-defined for any quantum process . However when and the expression (3.13) is distinct from , thus in the quantum domain, (3.13) yields a parametric family of states over time associated with a general quantum process which generalizes the classical state over time . Another crucial point, is that while the multi-matrix (3.13) is of unit trace for all , in the quantum domain it is rarely positive (even for ), and is only guaranteed to be self-adjoint for . As such, when generalizing states over time to the quantum domain, we will relax the positivity condition and only require states over time to be self-adjoint elements of unit trace. In fact, as first pointed out in [FJV15], it is the negative eigenvalues of a quantum state over time which act as a witness to causal correlations which exist in the associated process, and are a feature of the theory rather than a defect.
4 In quantum bloom
The classical bloom map from Proposition 3.2 may be viewed as the output of a mapping that associates every pair of finite sets with a function
| (4.1) |
such that and for all . Moreover, the bloom map naturally extends to a map on all of , which together with the partial trace provides a factorization system on which yields the state over time associated with every classical process . In this section, we extend the classical bloom map (4.1) to the quantum domain. Though there are various such extensions, the extension referred to as the symmetric bloom yields a state over time which is always self-adjoint, and will be the focus of our work.
Definition 4.2.
A bloom map associates every pair of multi-matrix algebras with a map such that and for all .
Definition 4.3.
If is a quantum process, and is a bloom map, then will be referred to as the state over time associated with the process and the bloom map , and will be denoted by . The map on quantum processes given by
will then be referred to as the state over time function associated with .
Definition 4.4.
Given a bloom map , the factorization will be referred to as the bloom-shriek factorization of .
Remark 4.5.
An analogue of bloom-shriek factorization for free gs-monoidal categories appears in [fritz2023free] under the name bloom-circuitry factorization.
Example 4.6.
Let be a pair of multi-matrix algebras, and let
be the map given by . When is a matrix algebra it was shown in [PaRuBayes] that , where are the matrix units in . We then have
from which it follows that
and
thus defines a bloom map in this case. When and are both multi-matrix algebras, then
where are the component functions of . We then have
thus the above arguments in the case when is a matrix algebra may be applied component-wise to deduce and , thus defines a bloom map. Moreover, it follows from Proposition 3.10 that this bloom map recovers the classical state over time associated with classical processes .
In light of the previous example, the following definition provides three examples of bloom maps.
Definition 4.7.
Let be a pair of multi-matrix algebras.
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The right bloom is the map given by , where
for all .
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The left bloom is the map given by , where
for all .
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The symmetric bloom is the map given by , where
for all .
Remark 4.8.
The symmetric bloom was introduced in [FuPa22], where it was proved that the associated state over time function satisfies a list of axioms set forth in [HHPBS17]. The list of axioms include hermiticity of , bilinearity of , a classical limit axiom, and an associativity axiom which ensures that \Yinyang extends in a well-defined way to -step processes . In a later work we will prove that the symmetric bloom is in fact the only state over time function satisfying the aforementioned list of axioms.
The following statement yields an important property of the symmetric bloom which will be useful for our purposes.
Proposition 4.9.
The be a pair of multi-matrix algebras, and let . Then the following statements are equivalent.
-
i.
is -preserving.
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ii.
is self-adjoint.
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iii.
is -preserving.
Proof.
We prove the statement in the case of matrix algebras.
Definition 4.10.
A bloom map is said to be
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•
classically reducible if and only if .
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•
hermitian if and only if is -preserving whenever is -preserving.
Remark 4.11.
By Proposition 4.9 it follows that the symmetric bloom is hermitian, and it follows directly from its definition that the symmetric bloom is classically reducible. These are two key properties one should expect from a quantum bloom map if it is to be viewed as generalization of the classical bloom. While we have reason to believe that the symmetric bloom is the only bloom map which is both hermitian and classically reducible, we have yet to prove such a result.
We now generalize Proposition 3.8 using the symmetric bloom. In particular, given a pseudo-density operator on , we derive an explicit formula for a process such that . For this, we recall that denotes the set of pseudo-density operators with respect to the tensor factorization (see Definition 2.20).
Lemma 4.12.
Let be such that is invertible. Then there exists a unique element such that
| (4.13) |
Proof.
We prove the statement in the case of matrix algebras, where we will use the fact from linear algebra that if , then the Sylvester equation has a unique solution if and only if and have disjoint spectrums. So let be such that is invertible. Since is a state which is invertible, it follows that the eigenvalues of are strictly positive, thus the spectrums of and are disjoint. It then follows that the Sylvester equation
has a unique solution , thus proving the statement. ∎
Notation 4.14.
Given a pair of multi-matrix algebras, we let
| (4.15) |
denote the subsets given by
and
where is the unique element satisfying (4.13).
Lemma 4.16.
Let be a pair of multi-matrix algebras. Then for all .
Proof.
Let and let . To show we need to show that is a pseudo-density operator such that is invertible and is CPTP. To show is a pseuso-density operator we first show is self-adjoint. Now since is CPTP it is -preserving thus is -preserving by Proposition 4.9. We then have
thus is self-adjoint. And since the marginals of are and with invertible (since ), it follows that . Moreover, since and
it follows that and , thus , as desired. ∎
Theorem 4.17.
Let be a pair of multi-matrix algebras, and let be the map given by . Then is a bijection, whose inverse is given by
| (4.18) |
where is the unique element satisfying (4.13).
Proof.
Let be the map given by . We now show and . So let and let , so that and
We then have
thus . Now let , so that
We then have
thus , as desired. ∎
5 Blooming 2-chains
Our next goal is to extend bloom maps to -chains, but first we need to consider the case . In this section we will show is that if a bloom map satisfies an associativity condition on -chains, then it naturally extends to a unique bloom map on -chains. We then show that the right bloom, left bloom and symmetric bloom are all associative, and as such, yield well-defined state over time functions for 2-step processes. In such a case we also prove an explicit formula for the symmetric bloom state over time function, and show that is satisfies a certain compositionality property. This will provide us with the mathematical foundation for using bloom maps to define explicit states over time associated with the dynamics of -step quantum processes for all .
Definition 5.1.
Given a triple of multi-matrix algebras, a -chain consists of a pair . The set of all such -chains will be denoted by .
Example 5.2 (Blooming 2-chains).
Let be a 2-chain, let be a bloom map, and consider the following diagram.
| (5.3) |
Using only maps in the above diagram, there are two maps one may construct in , namely, and . As the map has codomain and the map has codomain , there is a bijection between 2-blooms constructed from the above diagram and parenthezations of the multi-matrix algebra . It is then a natural to question whether or not the two maps and are equal. If the two maps are in fact equal, then the bloom map naturally extends to a unique bloom map on 2-chains. Such considerations then motivate the following definition.
Definition 5.4.
A bloom map is said to be associative if and only if for every 2-chain
we have
| (5.5) |
It turns out that the right bloom, left bloom and symmetric bloom are indeed associative, but before giving the proof, we will first prove the following lemma, which will also be useful later on.
Lemma 5.6.
Let be a triple of matrix algebras, and let be a 2-chain. Then the following statements hold.
-
i.
-
ii.
-
iii.
-
iv.
-
v.
Proof.
Proposition 5.7.
The right bloom, left bloom and symmetric bloom are all associative.
Proof.
Right bloom: Indeed, for all we have
where the third and fourth equalities follow from items iii and i of Lemma 5.6. It then follows that , as desired.
Left bloom: The proof is similar to that for the right bloom.
Associative bloom maps yield well-defined state over time functions on 2-step processes, as we now show.
Definition 5.8.
Given a triple of multi-matrix algebras, an element will be referred to as a 2-step process, and the set of 2-step processes will be denoted by . The subset of consisting of processes with invertible will be denoted by .
Definition 5.9.
Suppose is associative. If is a 2-step process, then
will be referred to as the state over time associated with the 2-step process and the bloom map , and will be denoted by . The map on 2-step quantum processes given by
will then be referred to as the 2-step state over time function associated with .
The next result shows that the state over time has the expected marginals.
Proposition 5.10.
Suppose is associative, and let be a 2-step process. Then the following statements hold.
-
i.
-
ii.
-
iii.
-
iv.
-
v.
Proof.
Associativity is used in the proofs by identifying with either or .
We now derive a formula for the symmetric bloom state over time in terms of , and .
Definition 5.11.
Let be a multi-matrix algebra. The normalized Jordan product is the map given by
Lemma 5.12.
Let be a 2-chain. Then
Proposition 5.13.
Let be a 2-step process. Then
| (5.14) |
Proof.
To conclude the section, we prove a result which will be useful later on.
Proposition 5.15 (Compositionality).
Let be a two-step process. Then
| (5.16) |
Proof.
It follows by direct computation that
thus
Similarly, we have
thus
as desired. ∎
6 Blooming -chains
In this section we show how associative bloom maps naturally extend to bloom maps for arbitrary -chains.
Definition 6.1.
Given an -tuple of multi-matrix algebras, an -chain consists of a -tuple
The set of all such -chains will be denoted by .
Definition 6.2.
An -bloom associates every -tuple of multi-matrix algebras with a map such that
| (6.3) |
for all .
Example 6.4 (Blooming 3-chains).
Let be a bloom map, and let
be a 3-chain, which after incorporating bloom-shriek factorizations yields the following diagram:
| (6.5) |
Similar to the case of 2-chains, there are then five 3-blooms in one may construct from the above diagram (6.5), each of which corresponds to a parenthezation of .
For example, the 3-bloom associated with the parenthezation is constructed as follows. We first associate a map with what is inside the outermost parantheses of , namely, , which is a parenthezation of the multi-matrix algebra . As we’ve already seen in the case of 2-chains, the 2-bloom associated with is then
and then post-composing with yields
so that taking a final bloom yields
which is the desired 3-bloom. The 3-blooms associated with the other 4 parenthezations of may then be obtained by considering the pentagon where each vertex corresponds to one of the 5 parenthezations of , and a directed edge is drawn between two vertices if and only if they are related by a single application of the associator transformation :
The associated 3-blooms for the other 4 parenthezations of can then be obtained from by applying corresponding associator transformations of maps along the pentagon:
This procedure then yields the following 3-blooms associated with each parenthezation of :
-
i.
-
ii.
-
iii.
-
iv.
-
v.
If is assumed to be associative, then this implies that these five 3-blooms are all equal:
Remark 6.6.
We now generalize the previous two examples to -chains for all . Let be a bloom map, let , and let
| (6.7) |
be an -chain. With every parenthezation of the multi-matrix algebra we associate a map
constructed from the bloom map , the maps for and the partial trace tr. For the case , we know from Example 5.2 that there are two parenthezations of , namely,
and we let
In such a case we have that may be obtained from by a transformation of the form
with and . For general we will build up recursively from this case.
So now let and let be an arbitrary parenthezation of . It then follows that is in one of the three forms:
where and , , and are the parenthezations of the multi-matrix algebras , , and for which the above equations hold. In each of the three cases we define the associated -blooms recursively as follows.
The next proposition shows that , when viewed as a map on -chains, is an actual -bloom, i.e., that satisfies the condition (6.3) in Definition 6.2.
Proposition 6.8.
Let , let be an -tuple of multi-matrix algebras, let be a parenthezation of , and let be a bloom map. Then is an -bloom, i.e.,
| (6.9) |
for all .
Proof.
We use induction on . For the statement follows from Proposition 5.10. Now suppose the result holds for , and let be a parenthezation of . We then consider the three cases as for as defined above.
Case I: In this case we have and
for all -chains . For we then have
where the last equality follows from the definition of a bloom map. As for , we have
thus is an -bloom.
Case II: In this case we have and
for all -chains , where . For we then have
As for , we have
thus is an -bloom.
Case III: In this case we have for some and
for all -chains , where . Now let . Then for we have
As for , we have
thus is an -bloom, as desired. ∎
Remark 6.10.
If and are two parenthezations of such that is obtained from by an associator transformation of the form , then there exists a 2-chain
such that is obtained from by an associator transformation at the level of maps associated with the 2-chain , i.e.,
The previous remark motivates the following definition.
Definition 6.11.
Let and let . The associator relation on the set of maps is the subset
given by if and only if is obtained from from successive transformations of the form
In such a case, we use the notation to denote the fact that .
Lemma 6.12.
Let . Then .
Proof.
We use induction on . The case holds by definition of the associator relation, so now assume the result holds for . Then
thus the result holds for , as desired. ∎
Proposition 6.13.
Let , and let be a parenthezation of . Then
Proof.
We will consider each of the three cases given above for the definition of , and use strong induction on . The case follows by definition, so now assume the result holds for all with .
Case I: Indeed,
where in the last line we note that we have repeatedly used the elision , thus the result holds.
Case II: Indeed,
as desired.
Case III: Let . Then
where again in the last line we note that we have repeatedly used the elision , thus the result holds. ∎
The following theorem follows directly from Proposition 6.13, and will form the basis for a definition of states over time associated with -step processes, as we will see in the next section.
Theorem 6.14.
If is associative, then
for every parenthezation of .
7 States over time for -step processes
In this section we use bloom maps for -chains to define states over time for arbitrary finite-step quantum processes.
Definition 7.1.
Given an -tuple of multi-matrix algebras, an element
will be referred to as an -step process, and the set of all such -step processes will be denoted by . Given an -step process , we let and for all .
Definition 7.2.
An -step state over time function associates every -tuple of multi-matrix algebras with a map
such that
| (7.3) |
for all . In such a case, the pseudo-density operator will be referred to as the state over time associated with the -step process . For an -step state over time function will be referred to simply as a state over time function.
Theorem 7.4.
Let be a bloom map which is hermitian, let , and let be the map on -step processes given by
| (7.5) |
Then the following statements hold.
-
i.
The map is an -step state over time function.
-
ii.
If is also associative and , then for every parenthezation of .
Proof.
Item i: Let be an -tuple of multi-matrix algebras, let be an -step process. Then for all we have
thus satisfies condition (7.3) of being a state over time function.
To conclude the proof, we must show that is self-adjoint, for which we use induction on . For we have , and since is CPTP it is -preserving, thus is -preserving by the hermitian assumption on . We then have
where the last equality follows from the fact that is a state (and so self-adjoint), thus is self-adjoint. Now assume the result holds for , and let , so that
Now since and tr are both completely positive, it follows that is -preserving, thus by the hermitian assumption on it follows that is -preserving as well. We then have
where the last equality follows from the fact that is self-adjoint by the inductive hypothesis, thus is self-adjoint.
8 The \Yinyang-function
Since the symmetric bloom \Yinyang is hermitian and associative, it follows from Theorem 7.4 that \Yinyang extends to a uniquely defined -step state over time function for all , which we refer to as the \Yinyang-function. In this section we will prove various properties the \Yinyang-function satisfies, including multi-linearity, classical reducibility and a multi-marginal property that will be used in the next section to solve an open problem from the literature. We also prove a general formula for the \Yinyang-function evaluated on an -step process in terms of and the channel states . To conclude the section, we show that in the case of a closed system of dynamically evolving qubits, the \Yinyang-function recovers the pseudo-density matrix formalism first introduced by Fitzsimons, Jones and Vedral [FJV15].
Definition 8.1.
The \Yinyang-function is the function on all finite-step processes given by
| (8.2) |
Remark 8.3.
Note that the \Yinyang-function as given by (8.2) is well-defined even if is not a state and even if is not CPTP for any . As such, we may view the \Yinyang-function as being defined on the set
for any -tuple of multi-matrix algebras . However, the element will only be viewed as a state over time when .
We now prove various properties the \Yinyang-function satisfies, but first we introduce some notation.
Notation 8.4.
Let . Given , we let
denote the partial trace map.
Theorem 8.5.
The \Yinyang-function satisfies the following properties.
-
i.
(Hermiticity) is self-adjoint of unit trace for all finite step processes .
-
ii.
(Multi-Linearity) The \Yinyang-function is multi-linear, i.e.,
(8.6) and
(8.7) for all and for all .
-
iii.
(Reduction) for all .
-
iv.
(The Multi-Marginal Property) Let . Then
(8.8) -
v.
(Classical Reducibility) If is a classical process, then
where is the classical distribution associated with and are conditional probabilities associated with .
Proof.
Item ii: Equation (8.6) follows from the fact that is a linear map, while equation (8.7) follows from the fact that
for all and all .
Item iii: Let , and let . Then
where the third equality follows from the associativity of the symmetric bloom.
Item iv: The proof relies on two lemmas, which we will prove after we finish proving the theorem. The case follows from item i of Theorem 7.4, so now suppose . The partial trace map may be written as
where if , we set . Now by items i and ii Lemma 8.9 we have
thus
where the last equality follows from Lemma 8.10.
We now prove the lemmas needed for the proof of the multi-marginal property of the \Yinyang-function (item iv of Theorem 8.5).
Lemma 8.9.
Let be an -step process. Then the following statements hold.
-
i.
for all .
-
ii.
for all .
Proof.
Item i: We use induction on . For we have
where the second equality follows from Lemma 6.12 and the third equality follows from the definition of a bloom map, thus the result holds for . Now suppose the result holds for with . Then
where the last equality follows from a calculation similar to the case. It then follows that the result holds for , as desired.
Item ii: We prove the equivalent statement for all by induction on . For we have
where the last equality follows from the definition of bloom map, thus the result holds for . Now suppose the result holds for with . Then
where the last equality follows from a calculation similar to the case. It then follows that the result holds for , as desired. ∎
Lemma 8.10.
Let be an -chain, let , let , and let be the element given by
where we set if . Then
| (8.11) |
Proof.
We use induction on . The case follows from item ii of Lemma 8.9. So assume the result holds for , and let be the element given by
The result then follows once we show
For this, we first consider the 2-chain
where and . Then after letting , we have
where the fourth equality follows from the associativity of the symmetric bloom. Then after letting , we then have
where the fifth equality follows from the fact that , and the seventh equality follows from the associativity of symmetric bloom. Now let , so that
We then have
where the final equality follows from item iii of Theorem ii, thus concluding the proof. ∎
We now prove a formula for in terms of and the channel states , …, .
Notation 8.12.
Let for every natural number . For every subset , the subscripts are chosen so that . In such a case, we denote the complement by , with the same convention that .
Notation 8.13.
Given an -chain and an element , we let be the element given by
Theorem 8.14.
Let be an -step process. Then
| (8.15) |
Proof.
We use induction on . For the result holds by definition of the \Yinyang-function. So now suppose the result holds for . Then after letting , we have
where the second to last equality follows from the inductive assumption. It then follows that the result holds for , as desired. ∎
We now extend the normalized Jordan product to an -ary operation for all , which we will then use to derive a formula for the \Yinyang-function which may be viewed as the -step generalization of the 2-step formula (5.14).
Definition 8.16.
Give a multi-matrix algebra , the extended Jordan product is the map
given by
| (8.17) |
Theorem 8.18.
Let be an -step process. Then
| (8.19) |
Proof.
The statement follows from Theorem 8.14 together with the fact that commutes with for all with . ∎
To conclude this section, we show how the \Yinyang-function recovers the pseudo-density matrix formalism for qubits.
Example 8.20 (The pseudo-density matrix formalism for systems of qubits).
Let be the initial state of an -qubit system, let be the Pauli basis of , and let be an -step process with initial state . In the formalism introduced by Fitzsimons, Jones and Vedral [FJV15], the pseudo-density matrix associated with is the matrix given by
where and denotes the expectation value associated with the observable . In [LQDV], it was recently proved that
and since
it follows via induction that (the final equality follows from item iii of Lemma 5.6). As such, the \Yinyang-function recovers the pseudo-density matrix formalism for systems of qubits first introduced in [FJV15], and moreover, Theorems 8.14 and 8.18 yield new formulas for .
9 Extracting dynamics from a pseudo-density matrix
In this section, we derive an explicit expression for the inverse of the \Yinyang-function restricted to a suitably nice subset of -step processes. In particular, given a pseudo-density matrix , we show how to identify with an -step process such that . This solves an open problem from [jia2023], where it is stated "Another interesting and closely relevant open question is, for a given PDO (pseudo-density matrix), how to find a quantum process to realize it.".
Notation 9.1.
Given an element with invertible, we let be the unique element such that .
Notation 9.2.
Given an -tuple of multi-matrix algebras, we let denote the subset given by
and we let be the subset consisting of elements such that
-
•
for , where is as defined by (4.15).
-
•
where and is the partial trace map.
Remark 9.3.
We have reason to believe that the two conditions defining are in fact equivalent, but we have yet to find a proof.
We now are going to prove that the \Yinyang-function restricted to defines a bijection onto , and we will also give a precise formula for the inverse. Before doing so however, we first prove that the image of the \Yinyang-function restricted to indeed lies in .
Proposition 9.4.
Let . Then .
Proof.
Let , let , and let
for all . Then it follows from the multi-marginal property of the \Yinyang-function that , and since is invertible by the definition of , it follows that , thus for by Lemma 4.16, showing that satisfies the first condition for being an element of . Moreover, it follows from Theorem 4.17 that for all , thus
thus satisfies the second condition for being an element of . It then follows that , as desired.
∎
Theorem 9.5.
Let be an -tuple of multi-matrix algebras, let
be the restriction of the \Yinyang-function to , and let
be the map given by
| (9.6) |
where and is defined via (4.13) for all . Then is a bijection, and .
Proof.
Let , let , and let
for all . Then we have seen in the proof of Proposition 9.4 that , thus
It then follows that , thus is injective.
Now let , so that
From the definition of we have for all , from which it follows that is CPTP for all and also that is invertible, thus . We then have
where the second to last equality follows from the fact that . It then follows that , thus concluding the proof. ∎
Remark 9.7.
As the inverse of the Jamiołowski isomorphism admits the explicit formula
and since there are algorithms to compute with , the formula (9.6) for the inverse of the \Yinyang-function is easily implementable in symbolic computing software.