A Note on the Set of After-measurement States in Generalized Quantum Measurement
aa r X i v : . [ qu a n t - ph ] A ug A Note on the Set of After-meaurement Statesin Generalized Quantum MeasurementI.D. IvanovicPhysics Department, Carleton UniversityOttawa ON. [email protected]
Abstract
The sets of after-measurement states for standard and gener-alized quantum measurements are compared. It is shown that for a SIC-POVM generalized measurement, the ratio of the volume of the set of after-measurement states and the volume of the simplex generated by individualoutcoms quckly tends to zero with increase of the number of dimensions. Thevolumes used are based on the Hilber-Schmidt norm. Some consequences onactual realizations, having finite collections of systems are discussed..
States and Standard Measurements . Few well known facts aboutstandard measurements may be useful. Let the system be described in ad-dimensional complex Hilbert space H . A pure state is a projector onto anormalized vector | Ψ i , h Ψ | Ψ i = 1, and its state is a ray projector P = | Ψ ih Ψ | .Generally, a state is an operator on H satisfying W ≥ , trW = 1. A statecan be expressed as a weighted sum of its eigen-projectors W = P k r k P k where P k r k = 1 , and P k P r = δ kr P k . The set of all states over H will bedenoted by V W = { W | W ≥ , tr ( W ) = 1 } . An orthogonal ray -resolution ofthe identity (ORRI) over H is a set of projectors { P k } satisfying X k P k = I H , P k P r = δ kr P k tr ( P k ) = 1The convex set of all convex combinations of an ORRI { P k } , conv ( { P k } ) is acommutative simplex, identical to the classical, discrete probability simplex,having pure states P k as extremal points and normalized identity W o = d I H as its baricenter. The set of all states, V W , can be obtained by applyingall unitary transformations to an initial, commutative, simplex. The pointcommon to all simplices is W o .The most natural way to look at V W is as a convex set in the space ofHermitian operators over H using Hilbert-Schmidt distance . A standardmeasurement , defined by complete, nondegenerate observable A , having1RRI { P k } is represented by a change of state W am = X k P k W pm P k = X k tr ( W P k ) P k (1)where p ( a k ) = tr ( W pm P k ) and h A i = P k p ( a k ) a k = tr ( AW pm ) = tr ( AW am ).Here W pm is a pre-measurement state and W am is the after-measurementstate of the system,or to be more precise of an infinite ensemble of systems.Formally , W am is an orthogonal projection of W pm onto the simplexdefined by the { P k } . Again, the easiest way to visualize this is to deduct the W o from all states, working in the hyperplane tr ( A ) = 1, then the simplex ofcommuting states defined by { P k } is in. e.g. two dimensions is a segment oflength √
2. The midpoint is W o . In three dimensions a commutative simplexis equilateral triangle, edge √
2, the baricenter is, as always, W o .Once an ORRI { P k } is given one may identify the three set of states:i) set of tr = 1 linear combinations of P k ’s , V ( { P k ) } ) = { W | W = X a k P k , X a k = 1 , W ≥ } , ii) set of all convex combinations of P k ’s, conv ( { P k } ) = { W | W = X k a k P k , a k ≥ , X k a k = 1 } iii) the set of all possible aftermeasurement states V am ( { P k } ) = { W | W = X k P k W pm P k , W pm ∈ V W } . All three sets are identical V ( { P k } ) = conv ( { P k } ) = V am ( { P k } ) . Furthermore, this type of measurement, corresponding to an ORRI , can beselective e.g. when the systems are ’tagged’ and states with outcome a k areselected. The other type is non-selective when W am is all we know about thestate. ORRI measurements are repeatable i.e. immediately after e.g. a k isobserved on a system, another measurement of observable A should give thesame result and a consequence it that it is also repeatable on the ensemblei.e. P k P k W am P k = W am . Obviously, almost no measurements satisfy theseconditions but as a paradigm it mirrors our ideas of distinguishability intoorthogonality. 2 eneralized Quantum Measurement In a generalized quantum mea-surement (GM) the resolution of the identity is as a rule a nonorthogonalone (NRI) and it is formally given by W am = X k A k W pm A † k where P k A † k A k = I d [1] but it is possible that P k A k A † k = I d . The possi-bility for a GM to displace W o indicates that there is a part of it which isa preparation, not simply a measurement . To make things simple we willconsider only non-orthogonal ray resolutions (NRRI), which will be ’stripped’ of their unitary part. Namely, using the polar decomposition A k = U k Q k only ray-projector factor Q k will be kept . The subset of GM we will consideris then W am = X k c k Q k W pm Q k , X k c k Q k = I, tr ( Q k ) = 1 . where Q k s are a linearly independent set. In this way, a GM is always acontraction having W o as one of the fixed points.Realizations of GM , come as a rule, come from a Naimark-like construc-tions, either by expanding the space, making H = H S a subspace of a largerspace, H = H S ⊕ H A , or by making H a factor space of H = H S ⊗ H A .In the first case[2], the original space H S is enlarged so that the NRRI { Q k } is a projection of an ORRI from the enlarged space i.e. c k Q k = P S P k P S where { P k } is an ORRI from the enlarged space H and P S is the projectoronto original space H. One must notice that the measurement should be madewith { P r } s on a state from V W and then projected or rotated back into V W .A more frequent situation is when an ancila is attached to the system. Inthis case, an ORI performed on the ancila, after a unitary transformation isperformed on H S ⊗ H A , results in an NRI measurement on the system. Themost straightforwrd construction is given in [3 ].The system in state W pm is attached to the ancila in a specifiedstate e.g. P Ao . A unitary transformation is then applied to the W pm ⊗ P Ao resulting in U S ⊗ A W pm ⊗ P Ao U † S ⊗ A such that W am = tr A ( X k ( I ⊗ P Ak )( U S ⊗ A W pm ⊗ P Ao U † S ⊗ A )( I ⊗ P Ak ) = X k c k tr ( W pm Q k ) Q k (2). 3ne should notice that the measurement of I S ⊗ { P Ak } on the ancilaserves only to tag the systems in H while the state of the system is alreadythe one given by eq.(2). So one needs a classical communication betweenthe ancila and the system to identify individual systems and their states.Strictly speaking, no actual measurement is performed on the system, whathappened is an unitary transformation and a ’distant’ selection’ [4]. Thestate of the system, after the unitary transformation is already tr A ( U S ⊗ A W pm ⊗ P Ao U † S ⊗ A ) = tr A ( X k ( I S ⊗ P Ak )( U S ⊗ A W pm ⊗ P Ao U † S ⊗ A )( I S ⊗ P Ak ) What is a measurement result in a GM ?
In the case of a GMbased on an NRRI { Q k } ),satisfying P k c k Q k = I, trQ k = 1, NRRI definesthree sets of states:i) set of all tr = 1 linear combinations of { Q k } V ( { Q k } ) = { W | W = X a k Q k ≥ , X a k = 1 , a k − real } ii) noncommutative simplex conv ( { Q k } ) = { W | W = X a k Q k ≥ , a k ≥ , X a k = 1 } andiii) the set of all possible after-measurement states V am ( { Q k } ) = { W | W = X k c k Q k W pm Q k , W pm ∈ V W } It is easy to see that V ( { Q k } ) ⊃ conv ( { Q k } ) ⊃ V am ( { Q k } )The first inclusion is obvious, the second follows if one performs a measure-ment on one of the extremal points from conv ( { Q k } ), e.g. W pm = Q k o . Thestate after the measurement is W am = X k c k tr ( Q k Q k o ) Q k = c k o Q k o + X k = k o c k tr ( Q k o Q k ) Q k In order for Q k o to remain an extremal point of conv ( { Q k } ) , c k o must be 1and tr ( Q k Q k o ) = 0. 4herefore. due to nonorthogonality between the ray-projectors from { Q k } , and in this case the lack of repeatability, the map of at least someof the extremal points of conv ( { Q k } ) can not remain extremal points, other-wise this NRRI would be an ORRI.As commented in [5], if the result of a measurement on certain numberof identically prepared systems is still outside of V am ( { Q k } ) , one shouldcontinue with measurements till the after -measurement state touches theboundary of V am ( { Q k } ) or goes into V am ( { Q k } ). Should one continue withmeasrement or stop at the boundary ? This, of course, has no bearing on aninfinite ensemble, but it may affect any actual realization.An interesting situation may occur in the following situation. Assumethat we know nothing about W pm , while the resulting W am , after certainfinite number of observations, is still outside of V am ( { Q k } ): one may beforced to change the expected values of a subset of states for the second partof the ensemble, knowing that the final result should belong to V am ( { Q k } ),or to be prepared to say that quantum mechanical description is incomplete.Furthermore an observer on S may communicate the results to ancila A ,making the future results for an ORI on the ancila also more predictable.Finally, what is actually measured? In principle, one can calculate theexpected values of all observables which are a linear combinations of { Q k } ;also, depending on the span of projectors, a position of a pre-measrementstate is reduced to a better defined subset of V W . SIC-POVM
If an NRRI is symmetric-informationally complete SIC-POVM [6 ] i.e. if d projectors { Q k } satisfy1 d X k Q k = I H ; tr ( Q k Q r ) = dδ kr + 1( d + 1)one can make some more specific conclusions.First, { Q k } spans the operator space and V ( { Q k } ) ⊃ V W . This meansthat any pre-measurement state may be written as W pm = P k a k Q k . Theafter-measurement state is then W am = X k,r a k c r tr ( Q k Q r ) Q r = 1( d + 1) I + 1( d + 1) W pm == d ( d + 1) W o + 1( d + 1) W pm tr ( A ) = 1 hyper-plane) by a factor of d +1) (cf. [7]) . First thing that one may observe is thatall after-measurement states must be nonsingular. One can say that unlessall events from { Q k } occur the state is definitely not allowed as a result.Furthermore, the set of states ”shrinks”, but the original shape of V ( d ) W ispreserved . A possible problem is that we do not have a simple characteriza-tion or parameterization of the set of states, so even if an after-measurementstate is inside the sphere of radius d +1) q ( d − d , it may not be an image ofa state, rather, one would have to ”stretch” the state to its original size toestablish was it actually a state or not.Finally, the set of admissible after-measurement states shrinks reallyquckly with incresed d. Due to the fact that all three sets V W ⊃ conv ( { Q k } ) ⊃ V am ( { Q k } )have the same dimensions , one can compare their volumes.The volume of the conv ( { Q k } ) in the hyperplane tr ( A ) = 1, which is a d − dd + 1 ! / is V ( conv { Q k } ) = d ( d + 1) ! ( d − d ( d − d − ! . The volume of states is , cf. [8], V ( V W ) = √ d (cid:16) π (cid:17) d ( d − . . . Γ( d )Γ( d )and the volume of the after-measurement states ( results ) for a SIC-POVM { Q k } is then V ( V am ( { Q k } ) = V ( V W )( d + 1) ( d − As a result, almost imediately, even for small d’s V ( conv ( { Q k } )) V ( V W ) → V ( V am ) V ( conv { Q k } ) → S z gives distribution { − a, a } . If the next measuremesnt ise.g. of S x than as long as the result is outside { / b, / − b } where − q a (1 − a ) ≤ b ≤ q a (1 − a ) , the result of the state determination isactually not a state.To conclude with, generalized measurements are indeed generalization ofstandard ORI measurements, but when they are not ORIs or combinationsof ORIs they are mostly either clever state determinations or distant statepreparations. It is indeed very difficult to change a well established name, asgeneralized measurement is, but more specifications may be necessary. NB A part of this note was presented in poster session during CAPCongress June 2010, Toronto,Canada.
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