Explicit construction of the density matrix in Gleason's theorem
aa r X i v : . [ qu a n t - ph ] A p r Explicit construction of the densitymatrix in Gleason’s theorem
Del Rajan ID and Matt Visser ID School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand
E-mail: [email protected], [email protected] , [email protected] Abstract:
Gleason’s theorem is a fundamental 60 year old result in the foundations of quantummechanix, setting up and laying out the surprisingly minimal assumptions required todeduce the existence of quantum density matrices and the Born rule. Now Gleason’stheorem and its proof have been continuously analyzed, simplified, and revised over thelast 60 years, and we will have very little to say about the theorem and proof themselves.Instead, we find it useful, (and hopefully interesting), to make some clarifying commentsconcerning the explicit construction of the quantum density matrix that Gleason’stheorem proves exists, but that Gleason’s theorem otherwise says relatively little about.
Pacs:
Keywords:
Gleason’s theorem, probability functions, Hilbert subspaces, density matrix, Born rule,quantum probability, quantum ontology, quantum realism.
Dated: A TEX-ed April 2, 2019 ontents
Gleason’s theorem has a long 60-year-old history [1–11], and is regarded as one of thefundamental theorems in the foundations of quantum mechanix. The theorem addressesthe minimal, (in fact, quite surprisingly minimal), assumptions required to deduce theexistence of a quantum density matrix, (a unit trace Hermitian matrix encoding thenotion of quantum probability), and Gleason’s theorem pragmatically underlies thetheoretical justification for adopting the Born rule in standard quantum mechanix.With such a long history [1–11], and in view of the publication of a number of recentrelated books [12–16], it is perhaps surprising that there is anything left to say onthis subject. Early proofs of Gleason’s theorem were implicit and non-constructive,and for some time there was controversy as to whether a constructive proof was evenpossible [3, 4, 6, 7]. With hindsight, disagreement on what methods are legitimately tobe deemed “constructive” is the key point of the constructivist debate in the 1990s [3,4, 6, 7]. Even with modern constructive (in principle) proofs, the construction is notparticularly explicit, and often very little is said as to what the quantum density matrixactually looks like. Traditionally the analysis stops, and the theorem is complete, oncethe existence of the quantum density matrix ρ is established.– 1 –erein we will have very little to say about the theorem and proof themselves, focussingmore on the implications: We shall say a little more about the density matrix itself —and shall provide two constructions (one implicit, one explicit) for the quantum densitymatrix ρ in terms of the probabilities assigned to rays in the Hilbert space. An explicit statement of Gleason’s theorem runs thus [1]:
Theorem:
Suppose H is a separable Hilbert space, (either real or complex).A measure on H is defined to be a function v ( · ) that assigns a nonnegative real numberto each closed subspace of H in such a way that: If { A i } is any countable collection ofmutually orthogonal subspaces of H , and the closed linear span of this collection is B ,then v ( B ) = P i v ( A i ) . Furthermore we normalize to v ( H ) = 1 .Then if the Hilbert space H has dimension at least three, (either real or complex),every measure v ( · ) can be written in the form v ( A ) = tr( ρ P A ) , where ρ is a positivesemidefinite trace class operator with tr( ρ ) = 1 , and P A is the orthogonal projectiononto A . (cid:3) (Physicists would almost immediately focus on complex Hilbert spaces; but some ofthe mathematical literature also works with real Hilbert spaces.)The original theorem gives one very little idea of what the density matrix might looklike, and it is this topic we shall address. Indeed, the original theorem spends manypages proving that the valuation v ( P ) uniformly continuous; while this is certainly anextremely useful result, most physicists, (and applied mathematicians for that matter),would simply assume continuity on physical grounds. Our first observation is that since ρ is Hermitian we can diagonalize it and define ρ = X i λ i Q i . (3.1)Here the Q i are taken to be 1-dimensional subspaces, and the λ i are to be repeatedwith the appropriate multiplicity. Per Gleason’s theorem, v ( Q j ) = tr( ρQ j ) = tr "X i λ i Q i Q j ! = X i λ i tr ( Q i Q j ) = λ j . (3.2)– 2 –o actually ρ = X i v ( Q i ) Q i , (3.3)which does not (yet) help unless you can somehow extract the Q i in terms of theunderlying valuation function v ( · ).Furthermore note that for each 1-dimensional subspace Q i we can identify Q i ∼ | ψ i i h ψ i | (3.4)where | ψ i i is any arbitrary vector in the 1-dimensional subspace Q i . Then v ( Q i ) = h ψ i | ρ | ψ i i . (3.5)Now let P i be any arbitrary collection of orthogonal 1-dimensional projection operators v X i P i ! = X i v ( P i ) = 1 . (3.6)Using Gleason’s theorem, we can calculate v ( P j ) = tr( ρP j ) = tr "X i v ( Q i ) Q i P j ! = X i v ( Q i ) tr ( Q i P j ) = X i v ( Q i ) S ij , (3.7)with S ij = tr ( Q i P j ) a bi-stochastic matrix. That is, Gleason’s theorem implies v ( P j ) = X i v ( Q i ) S ij ; with S ij = |h q i | p j i| = | U ij | . (3.8)So we see that the matrix S ij is actually unitary-stochastic; both unitary and unitary-stochastic matrices drop out automatically.Now pick some random basis P i and construct ρ P = X i v ( P i ) P i . (3.9)This is not ρ itself, but it is what you get from ρ by hitting it with $ P , the decoherencesuper-scattering operator with respect to the basis P i [17]. To see this note $ P ρ = X i P i tr( P i ρ ) = X i P i v ( P i ) = ρ P . (3.10)– 3 –inally consider what happens if you average over the P i : (cid:10) $ P (cid:11) ρ = *X i P i tr( P i ρ ) + = *X i P i v ( P i ) + = h ρ P i . (3.11)In d dimensions for a uniform average over the ( P i ) ab we have *X i ( P i ) ab ( P i ) cd + = δ ac δ bd + δ ab δ cd d + 1 . (3.12)This arises from symmetry plus the normalization condition h I d × d i = I d × d . But thenwe can reconstruct ρ = ( d + 1) h ρ P i − I d × d . (3.13)(Note this does have the correct trace, tr( ρ ) = 1.) So if you know all possible ways inwhich the density matrix decoheres ρ → ρ P , and uniformly average over all choices ofdecoherence basis, then one can reconstruct the full density matrix. While certainly anelegant result, this is by no means explicit. Let us now set up a reasonably explicit construction of the density matrix ρ directlyfrom the valuation function v ( P ).To construct ρ proceed as follows: First for any 1-dimensional subspace note Q ∼ | n i h n | where n can be taken to be a unit vector in S d − . This defines a valuation v ( n ) on S d − . Then find a n such that v ( Q n ) = max n ∈ S d − { v ( P n ) } = max n ∈ S d − h n | ρ | n i .Now consider the S d − perpendicular to n : Proceed as follows — find a n such that v ( Q n ) = max n ∈ S d − { v ( P n ) } . By construction n ⊥ n and P n P n = 0.Iterate this construction: Consider the S d − i perpendicular to n , n , . . . , n i − : Find a n i such that v ( Q n i ) = max n ∈ S d − i { v ( P n ) } . By construction the n j for j ∈ { , , · · · , i } are mutually perpendicular, and P n j P n k = 0 for j = k and j, k ∈ { , , · · · , i } .Ultimately we have n d = max n ∈ S { v ( P n ) } = min n ∈ S d − { v ( P n ) } .The construction terminates after d steps with an orthonormal basis n , n , . . . , n d ,and the corresponding valuations v ( Q n i ). Now construct ρ = d X i =1 v ( Q n i ) Q n i . (4.1)This is the density matrix you want. (cid:3) – 4 – roof: It is clearly a density matrix; it only remains to check that it is the density matrix.But this is obvious from the construction — the n i are the simply eigenvectors of ρ ,with the corresponding projection operators Q n i , and the v ( Q n i ) are the eigenvalues.(Basically the construction above is just an application of the Rayleigh–Ritz min-maxvariational theorem for finding eigenvectors/eigenvalues of Hermitian matrices.) Thedensity matrix is constructed in terms of the values, v ( Q n i ), and locations, n i , of themaximum, minimum, and extremal points of the valuation function v ( · ). (cid:3) Note the construction is still rather implicit. Once Gleason’s theorem guarantees theexistence of the density matrix, this construction implicitly allows one to determine thedensity matrix. The more purist of constructivist mathematicians might not call thisconstructive, but most others would. On the other hand, as we shall now show, muchbetter can be done in terms of a fully explicit construction.
This second construction is completely explicit but considerably more subtle. We assertthat within the framework of Gleason’s theorem, for any arbitrary basis on complexHilbert space we can write: ρ = X j | n j i v ( n j ) h n j | (5.1)+ 12 X j = k | n j i (cid:26) v (cid:18) n j + n k √ (cid:19) − v (cid:18) n j − n k √ (cid:19) − i v (cid:18) n j + in k √ (cid:19) + iv (cid:18) n j − in k √ (cid:19)(cid:27) h n k | . That is, to reconstruct the full density matrix we need only determine the valuations v ( · ), which is a collection of real numbers, on the specific set of unit vectors n j ; (cid:18) n j ± n k √ (cid:19) ; (cid:18) n j ± in k √ (cid:19) . (5.2)There are a total of d + d ( d −
1) + d ( d −
1) = 2 d − d such unit vectors to deal with.This formula for the density matrix can also be rearranged as follows ρ = X j v ( n j ) | n j i h n j | + 12 X j 1) = d unit vectors to deal with.This formula for the (real) density matrix can also be rearranged as follows ρ = X j v ( n j ) | n j i h n j | + 12 X j 2, and finally \ n j ± in k =( n j ± in k ) / √ 2, we have: ρ = X j | n j i v ( n j ) h n j | (5.11)+ 12 X j = k | n j i (cid:26) v (cid:18) n j + n k √ (cid:19) − v (cid:18) n j − n k √ (cid:19) − i v (cid:18) n j + in k √ (cid:19) + iv (cid:18) n j − in k √ (cid:19)(cid:27) h n k | . That is, in terms of the decohered density matrix ρ P we have: ρ = ρ P (5.12)+ 12 X j = k | n j i (cid:26) v (cid:18) n j + n k √ (cid:19) − v (cid:18) n j − n k √ (cid:19) − i v (cid:18) n j + in k √ (cid:19) + iv (cid:18) n j − in k √ (cid:19)(cid:27) h n k | . For a real Hilbert space this reduces to ρ = ρ P + 12 X j = k | n j i (cid:26) v (cid:18) n j + n k √ (cid:19) − v (cid:18) n j − n k √ (cid:19)(cid:27) h n k | . (5.13)One aspect of the “miracle” of Gleason’s theorem is that this construction is actuallyindependent of the specific basis chosen.To see why this construction works, note that from Gleason’s theorem, for unit vectorsˆ x ∼ | ˆ x i = | x i|| x || ∼ x || x || , (5.14)we have v (ˆ x ) = h ˆ x | ρ | ˆ x i = h x | ρ | x i|| x || , (5.15)or more prosaically h x | ρ | x i = || x || v (ˆ x ) . (5.16)But then h x + y | ρ | x + y i = || x + y || v ( [ x + y ) = h x | ρ | x i + h y | ρ | y i + ( h x | ρ | y i + h y | ρ | x i ) , (5.17)– 7 –nd h x − y | ρ | x − y i = || x − y || v ( [ x − y ) = h x | ρ | x i + h y | ρ | y i − ( h x | ρ | y i + h y | ρ | x i ) , (5.18)whence h x | ρ | y i + h y | ρ | x i = 12 n || x + y || v ( [ x + y ) − || x − y || v ( [ x − y ) o . (5.19)(In a real Hilbert space we could stop here since then h x | ρ | y i = h y | ρ | x i .)Similarly, in a complex Hilbert space, h x + iy | ρ | x + iy i = || x + iy || v ( \ x + iy ) = h x | ρ | x i + h y | ρ | y i + i ( h x | ρ | y i − h y | ρ | x i ) , (5.20)and h x − iy | ρ | x − iy i = || x − iy || v ( \ x − iy ) = h x | ρ | x i + h y | ρ | y i − i ( h x | ρ | y i − h y | ρ | x i ) , (5.21)whence h x | ρ | y i − h y | ρ | x i = − i n || x + iy || v ( \ x + iy ) − || x − iy || v ( \ x − iy ) o . (5.22)Combining these results h x | ρ | y i = + 14 n || x + y || v ( [ x + y ) − || x − y || v ( [ x − y ) o − i n || x + iy || v ( \ x + iy ) − || x − iy || v ( \ x − iy ) o . (5.23)This finally justifies our construction of the density matrix ρ as presented above. Although Gleason’s theorem does not apply in two dimensions, there are improvedversions of Gleason’s theorem based on POVMs (positive operator valued measures),see [8, 9], that do apply to 2-dimensional Hilbert space. In this case the formalismsimplifies even further: Let ˆ x and ˆ y be any orthonormal basis for the 2-dimensionalHilbert space. Then in terms of the valuation v ( · ) the density matrix is ρ = v (ˆ x ) | ˆ x i h ˆ x | + v (ˆ y ) | ˆ y i h ˆ y | + 12 (cid:26) v (cid:18) ˆ x + ˆ y √ (cid:19) − v (cid:18) ˆ x − ˆ y √ (cid:19)(cid:27) (cid:16) | ˆ x i h ˆ y | + | ˆ y i h ˆ x | (cid:17) − i (cid:26) v (cid:18) ˆ x + i ˆ y √ (cid:19) − v (cid:18) ˆ x − i ˆ y √ (cid:19)(cid:27) (cid:16) | ˆ x i h ˆ y | − | ˆ y i h ˆ x | (cid:17) . (6.1)– 8 –f desired one can further rewrite this in terms of the Pauli σ matrices ρ = v (ˆ x ) + v (ˆ y )2 I × + v (ˆ x ) − v (ˆ y )2 σ z + 12 (cid:26) v (cid:18) ˆ x + ˆ y √ (cid:19) − v (cid:18) ˆ x − ˆ y √ (cid:19)(cid:27) σ x − i (cid:26) v (cid:18) ˆ x + i ˆ y √ (cid:19) − v (cid:18) ˆ x − i ˆ y √ (cid:19)(cid:27) σ y . (6.2)For real 2-dimensional Hilbert space this further simplifies to ρ = v (ˆ x ) | ˆ x i h ˆ x | + v (ˆ y ) | ˆ y i h ˆ y | + 12 (cid:26) v (cid:18) ˆ x + ˆ y √ (cid:19) − v (cid:18) ˆ x − ˆ y √ (cid:19)(cid:27) (cid:16) | ˆ x i h ˆ y | + | ˆ y i h ˆ x | (cid:17) . (6.3)(For completeness, note that for one dimension the valuation trivializes to v ( · ) ≡ ρ ≡ I × .) We have not attempted to provided a new proof of Gleason’s theorem. Instead wehave in mind a much more modest attempt at trying to understand what the densitymatrix actually looks like directly in terms of the probability valuations v ( · ) on a limitednumber of subspaces of the Hilbert space.Gleason’s theorem is profound that it shapes the probabilistic nature of quantum theory.It places strong constraints on any attempts to modify this formalism, and it alsogives a fundamental reason for why density operators play such an important role. Avast amount of literature has been accrued on Gleason’s theorem and its applications.Many physicists and mathematicians have tried to simplify the proof and extend itto more generalized structures. For a complete treatment, refer to the monograph byHamhalter [15].Future work regarding this explicit construction of the density operator may involveapplications to quantum information theory. This may reveal interesting links betweenquantum foundations, and to the fundamental results of quantum information theorysuch as no-cloning or no-broadcasting [18]. Such a direction would allow the reach ofGleason’s theorem to extend further into the modern information-theoretic setting ofquantum physics. – 9 – cknowledgments DR is indirectly supported by the Marsden fund,administered by the Royal Society of New Zealand.MV is directly supported by the Marsden fund,administered by the Royal Society of New Zealand. Background resources For a general introduction to Gleason’s theorem, see: • https://en.wikipedia.org/wiki/Gleason’s_theorem • https://ncatlab.org/nlab/show/Gleason’s+theorem • https://plato.stanford.edu/entries/qt-quantlog/ References [1] Andrew M. Gleason, “Measures on the closed subspaces of a Hilbert space”.Indiana University Mathematics Journal (1957) 885–893.MR 0096113. doi: https://doi.org/10.1512/iumj.1957.6.56050 [2] Roger Cooke, Michael Keane, and William Moran,“An elementary proof of Gleason’s theorem”,Mathematical Proceedings of the Cambridge Philosophical Society (1985) 117–128;doi: https://doi.org/10.1017/S0305004100063313 [3] G. Hellman, “Gleason’s theorem is not constructively provable”,Journal of Philosophical Logic, (1993) 193–203; doi: https://doi.org/10.1007/BF01049261 [4] Helen Billinge, “A Constructive Formulation of Gleason’s Theorem”Journal of Philosophical Logic (1997) 661–670 URL: [5] Itamar Pitowsky,“Infinite and finite Gleason’s theorems and the logic of indeterminacy”,Journal of Mathematical Physics (1998) 218; doi: http://dx.doi.org/10.1063/1.532334 – 10 – 6] Fred Richman, Douglas Bridges, “A Constructive Proof of Gleason’s Theorem”,Journal of Functional Analysis (1999) 287-312. https://doi.org/10.1006/jfan.1998.3372 [7] Fred Richman, “Gleason’s Theorem Has a Constructive Proof”,Journal of Philosophical Logic , No. 4 (Aug 2000) 425–431; doi: https://doi.org/10.1023/A:1004791723301 ; [8] Paul Busch, “Quantum States and Generalized Observables: A Simple Proof ofGleason’s Theorem”, Physical Review Letters (2003) 120403. doi: https://doi.org/10.1103/PhysRevLett.91.120403 ; [arXiv:quant-ph/9909073].[9] Carlton M. Caves, Christopher A. Fuchs, Kiran K. Manne, Joseph M. Renes,“Gleason-Type Derivations of the Quantum Probability Rule for GeneralizedMeasurements”,Foundations of Physics (2004) 193–209.doi: https://doi.org/10.1023/B:FOOP.0000019581.00318.a5 [arXiv:quant-ph/0306179].[10] David Buhagiar, Emmanuel Chetcuti, and Anatolij Dvurecenskij,“On Gleason’s Theorem without Gleason”,Foundations of Physics (2009) 550–558 https://doi.org/10.1007/s10701-008-9265-6 [11] Victoria J Wright and Stefan Weigert,“A Gleason-type theorem for qubits based on mixtures of projective measurements”,J. Phys. A: Math. Theor. (2019) 055301 https://doi.org/10.1088/1751-8121/aaf93d [12] C.J. Isham, Lectures on quantum theory: Mathematical and structural foundations.(Imperial College Press, London, 1995).[13] T. Heinosaari and M. Ziman,The mathematical language of quantum theory: From uncertainty to entanglement.(Cambridge University Press, England, 2012).[14] M.D. Chiara, R. Giuntini, and R. Greechie,Reasoning in quantum theory: Sharp and unsharp quantum logics.(Springer Netherlands, Dordrecht, 2013). doi: 10.1007/978-94-017-0526-4[15] J. Hamhalter, Quantum measure theory.(Springer Netherlands, Dordrecht, 2013). doi: 10.1007/978-94-017-0119-8[16] D.W. Cohen, An introduction to Hilbert space and quantum logic.(Springer Verlag, New York, 2012). doi: 10.1007/978-1-4613-8841-8 – 11 – 17] A. Alonso-Serrano and M. Visser,“Coarse graining Shannon and von Neumann entropies”,Entropy 19 (2017) 207. doi: https://doi.org/10.3390/e19050207 [arXiv:1704.00237 [quant-ph]].[18] Howard Barnum, Jonathan Barrett, Matthew Leifer, and Alexander Wilce,“Generalized No-Broadcasting Theorem”, Phys. Rev. Lett. (2007) 240501 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.99.240501https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.99.240501